Nonlocal-to-local $L^p$-convergence of convolution operators with singular, anisotropic kernels

Nonlo cal-to-lo cal L p -con v ergence of con v olution op erators with singular, anisotropic k ernels Helm ut Ab els ∗ Christoph Hurm ∗ P atrik Knopf ∗ † This is a pr eprint version of the p aper. Ple ase cite as: C. Hurm [Journal] xx :xx 000-000 (2026), https://doi.org/... Abstract. W e study nonlocal con volution-t yp e operators with singular, possibly anisotropic kernels. Our main ob jective is to establish and quan tify their nonlo cal- to-lo cal con vergence to a lo cal diﬀerential op erator with natural b oundary condi- tions, as the k ernels concen trate at the origin in a suitable wa y . Suc h conv ergence results provide a useful to ol for the physical justiﬁcation of mathematical mo d- els, particularly in situations where the desired local diﬀeren tial operator cannot b e directly deriv ed from microscopic laws. The present work substantially ex- tends previous results by allo wing kernels with stronger singularities (comparable to those of fractional Laplacians), anisotropic and non-localized k ernels, and b y pro ving strong conv ergence in general L p spaces together with explicit conv ergence rates. Keyw ords: Nonlo cal-to-lo cal conv ergence, singular in tegral operators, nonlo cal op erators, anisotropic kernels, conv ergence rates, singular limits. Mathematics Sub ject Classiﬁcation: Primary: 47G10; Secondary: 35B40, 35J25, 47B38, Con tents 1 In tro duction 2 2 Assumptions and preliminaries 6 3 Nonlo cal-to-lo cal conv ergence of the nonlo cal op erator 9 3.1 W ell-deﬁnedness of the nonlo cal operator . . . . . . . . . . . . . . . . . . . . 9 3.2 Con v ergence on the full space . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Con v ergence on a curv ed half space . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 Con v ergence on a domain with compact b oundary . . . . . . . . . . . . . . . 38 ∗ F aculty of Mathematics, Universit y of Regensburg, D-93053 Regensburg, Germany ( helmut.abels@ur.de , christoph.hurm@ur.de , patrik.knopf@ur.de ). † Institute for Analysis, Department of Mathematics, Karlsruhe Institute of T ec hnology (KIT), D-76128 Karlsruhe, Germany ( patrik.knopf@kit.edu ). 1 1 In tro duction W e consider nonlo cal con v olution type op erators of the form L Ω ∗ u ( x ) = P . V . Z Ω J ( x − y )  u ( x ) − u ( y )  d y = lim r ↘ 0 Z Ω ∩{| x − y |≥ r } J ( x − y )  u ( x ) − u ( y )  d y (1.1) for all x ∈ Ω and any suitable function u : Ω → R , where Ω ⊆ R n is a suﬃcien tly smo oth domain and J : R n \ { 0 } → R , J ( x ) = ρ ( x ) | x | 2 , (1.2) is a prescrib ed interaction kernel. Throughout this pap er, the function ρ : R n → R is supp osed to b e ev e n and has to fulﬁll certain in tegrability conditions that will b e sp eciﬁed in Section 2 . In particular, the conv olution k ernel J is also even. Because of this symmetry , the nonlo cal energy asso ciated with L Ω ∗ is given by E Ω ∗ ( u ) = Z Ω L Ω ∗ u ( x ) u ( x ) d x = 1 2 Z Ω Z Ω J ( x − y )   u ( x ) − u ( y )   2 d x for any suitable function u : Ω → R . Main ob jective of the paper. The main goal of this pap er is to study the nonlo c al-to-lo c al c onver genc e of nonlo cal operators as introduced in ( 1.1 ). This means that we intend to show that suc h a nonlocal op erator con v erges to a local diﬀeren tial op erator in some suitable sense as the function ρ approaches a multiple of the δ -distribution. T o formulate this in a mathematically precise wa y , we deﬁne the functions ρ ε : R n → R , ρ ε ( x ) : = ε − n ρ  x ε  , (1.3) J ε : R n \ { 0 } → R , J ε ( x ) = ρ ( x ) | x | 2 (1.4) for an y ε > 0. This means that ( ρ ε ) ε> 0 is a Dir ac se quenc e , i.e., it approaches a m ultiple of the δ -distribution in the limit ε → 0 (cf. Lemma 2.4 ). F or an y ε > 0, we write L Ω ε u ( x ) : = P . V . Z Ω J ε ( x − y )  u ( x ) − u ( y )  d y (1.5) for all x ∈ Ω, to denote the nonlocal op erator corresp onding to ρ ε . W e w an t to show that in the limit ε → 0, the nonlo cal operator L ε con verges to a lo cal diﬀeren tial op erator L of the type L Ω u ( x ) : = − div( M ∇ u ) = − n X i,j =1 M ij ∂ x j ∂ x k u ( x ) for all x ∈ Ω , (1.6) where M ∈ R n × n is a symmetric matrix with entries M ij : = 1 2 Z R n J ( z ) z i z j d z for all i, j ∈ { 1 , . . . , n } . (1.7) 2 M is usually referred to as the momentum matrix . In case Ω ⊊ R n , which means that ∂ Ω  = ∅ , we in terpret the second-order diﬀerential op erator L Ω to b e equipp ed with the natural b oundary condition M ∇ u · n ∂ Ω = 0 on ∂ Ω . (1.8) Here, n ∂ Ω denotes a unit normal vector ﬁeld on ∂ Ω. Under suitable assumptions on the domain Ω and on ρ , J and p , we intend to show that L Ω ε u → L Ω u in L p (Ω) as ε → 0 (1.9) for all functions u ∈ W 2 ,p (Ω) with ( 1.8 ) if Ω ⊊ R n . In addition, we are planning to establish the conv ergence rate   L Ω ε u − L Ω u   L p (Ω) ≤ C Ω p √ ε ∥ u ∥ W 3 ,p (Ω) . (1.10) for all functions u ∈ W 3 ,p (Ω) with ( 1.8 ) if Ω ⊊ R n . Here, C is a positive constant that dep ends only on Ω, ρ and p . State of the art and related results. The nonlocal-to-lo cal con v ergence of L Ω ε to L Ω w as ﬁrst established in the w eak sense. In this case, it can be deriv ed from the con vergence of the corresp onding energies. The energy functional asso ciated with the nonlo cal op erator L Ω ε is given by E Ω ε ( u ) = Z Ω L Ω ε u ( x ) u ( x ) d x = 1 2 Z Ω Z Ω J ε ( x − y )   u ( x ) − u ( y )   2 d x = 1 2 Z Ω Z Ω ρ ε ( x − y )  | u ( x ) − u ( y ) | | x − y |  2 d x. and the energy functional asso ciated with the local diﬀeren tial op erator L Ω reads as E Ω ( u ) = 1 2 Z Ω M ∇ u ( x ) · ∇ u ( x ) d x. Giv en these energies, one may in tuitiv ely expect that the w eighted squared diﬀerence quotien t ρ ε ( x − y )  | u ( x ) − u ( y ) | | x − y |  2 approac hes an expression of the form M ∇ u ( x ) · ∇ u ( x ) as the k ernel ρ ε concen trates at the origin, since in this regime only points y close to x con tribute signiﬁcan tly to the in tegral. Here, the matrix M accoun ts for anisotropic eﬀects when ρ is ev en but not radially symmetric. In case ρ is radially symmetric, w e exp ect M to b e a scalar multiple of the identit y matrix and thus, L Ω is a scalar m ultiple of the negative Laplacian. A rigorous pro of of the nonlo cal-to-lo cal conv ergence of such energies in a more general setting was pro vided by Ponce [ 29 ] (see also [ 27 , 28 ]). These inv estigations of n onlo cal energies pro vide an extension of earlier w orks b y Bourgain , Brezis and Mironescu [ 6 ]. Some impro vemen ts w ere obtained later, for example, by Brezis and Nguyen [ 7 ]. In particular, it w as shown in [ 29 ] that E Ω ε ( u ) → E Ω ( u ) as ε → 0 for all u ∈ H 1 (Ω). (1.11) Moreo ver, also the Γ conv ergence of such energies w as discussed in [ 29 ]. F urther results on nonlocal-to-lo cal Γ-con vergence w ere found b y Brezis and Nguyen [ 8 , 9 ]. 3 F rom the p oint wise conv ergence ( 1.11 ), the weak conv ergence L Ω ε u  L Ω u w eakly in H 1 (Ω) ′ as ε → 0 for all u ∈ H 1 (Ω) (1.12) w as deriv ed by Melchoinna , Ranetbauer , Scarp a and Tr ussardi [ 26 ] for ra- dially symmetric, suﬃcien tly regular k ernels and Ω b eing a ﬂat torus. An analogous result for b ounded domains Ω ⊂ R n and radially symmetric k ernels J ε ∈ W 1 , 1 loc ( R n ) w as established b y Da voli , Scarp a and Tr ussardi [ 14 ]. The main goal of these w orks w as to relate the nonlo c al Cahn–Hil liar d e quation to the classical (lo c al) Cahn– Hil liar d e quation by a nonlocal-to-lo cal limit. This is an imp ortan t adv ance because, as shown by Giacomin and Lebowitz [ 20 ] (see also [ 21 , 22 ]), the lo cal Cahn–Hilliard equation, in con trast to its nonlo cal counterpart, cannot b e deriv ed directly from microscopic physical principles. In this sense, the nonlo cal-to-lo cal con vergence oﬀers a ph ysical justiﬁcation for the local Cahn–Hilliard equation. Therefore, b e- sides [ 14 , 26 ], the weak conv ergence ( 1.12 ) for radially symmetric W 1 , 1 loc -k ernels has b een applied to v ariants of the Cahn–Hilliard equation and related models in v arious settings. W e refer, for example, to [ 4 , 11 – 13 , 16 , 17 , 25 ]. Ho wev er, the aforementioned results are merely qualitative and do not yield any quan titative information on the nonlocal-to-lo cal con vergence, such as conv ergence rates. Also, at least for more regular functions u , it ma y b e exp ected that the con vergence L Ω ε u → L Ω u actually holds in a strong top ology . A ﬁrst step in this direction was achiev ed in [ 1 ] by the ﬁrst tw o authors of the presen t pap er. F or radi- ally symmetric, lo calized (i.e., compactly supp orted) W 1 , 1 -k ernels and a suﬃciently regular b ounded domain Ω, it w as shown in [ 1 ] that L Ω ε u → L Ω u strongly in L 2 (Ω) as ε → 0 (1.13) for all u ∈ H 2 (Ω) with ∇ u · n ∂ Ω = 0 on ∂ Ω, and ∥L Ω ε u − L Ω u ∥ L 2 (Ω) ≤ C ε γ ∥ u ∥ H 3 (Ω) (1.14) for all u ∈ H 3 (Ω) with ∇ u · n ∂ Ω = 0 on ∂ Ω. Here, C denotes a p ositiv e constan t that ma y dep end on Ω but is indep endent of ε and u . The exp onent γ is giv en by γ = 1 if Ω = R n or Ω = T n (a ﬂat torus), and γ = 1 2 if Ω is a suﬃciently regular b ounded domain. In [ 1 ], these con vergence results were also applied to prov e nonlo cal-to- lo cal con vergence of the Cahn–Hilliard equation and the Allen–Cahn equation. In subsequen t works, this technique was applied in [ 2 , 15 , 23 , 24 ] to analyze further mo dels related to the Cahn–Hilliard equation. F urthermore, we remark that the kernels in the nonlo cal-to-lo cal conv ergence results men tioned so far were alwa ys assumed to b e radially symmetric. How ever, in concrete applications, this migh t not alwa ys be the case. F or example, mo dels related to crystal growth usually in volv e anisotropic interactions. W e p oin t out that in the con tributions [ 20 – 22 ] b y Giacomin and Lebowitz , the interaction k ernel w as merely assumed to b e ev en, but not radially symmetric. Also in the w ork [ 29 ] b y Ponce on the con vergence of nonlo cal energies, radial symmetry of the k ernel is not required. This leads to the conjecture that it should b e p ossible to generalize the aforementioned nonlo cal-to-lo cal conv ergence results to anisotropic k ernels. A ﬁrst step in this direction was achiev ed v ery recently by the ﬁrst author of the present pap er and Terasa w a [ 5 ] by establishing the weak nonlo cal-to-lo cal con vergence for an anisotropic v ariant of the Cahn–Hilliard equation. It is also 4 w orth mentioning that the strong W 1 , 1 loc -regularit y assumption on the con volution k ernel was also relieved there. Instead, weak er assumptions similar to those by Gounoue , Kassmann and Voigt [ 18 ] were imp osed. In the context of our w ork, we also wan t to mention the recent contribution [ 10 ] by Bunger t and del Teso , where nonlo cal-to-lo cal con vergence rates in the W 2 ,s ( R n )-norm w ere established for the the Dirichlet problem to the fractional Laplacian ( − ∆) s as s → 1. F or a discussion of nonlo cal Neumann problems in a broader sense with general measurable, nonnegative conv olution k ernels, we refer to the work [ 19 ] by Frerick , Vollmann and Vu . No velties and main results of our pap er. In the present pap er, we intend to pro vide a generalization of the aforementioned results on nonlocal-to-lo cal conv er- gence. Under suitable assumptions, we will show the conv ergences L Ω ε u → L Ω u strongly in L p (Ω) as ε → 0 (1.15) for all u ∈ W 2 ,p (Ω) with M ∇ u · n ∂ Ω = 0 on ∂ Ω, and ∥L Ω ε u − L Ω u ∥ L p (Ω) ≤ C ε γ ( p ) ∥ u ∥ W 3 ,p (Ω) (1.16) for all u ∈ W 3 ,p (Ω) with M ∇ u · n ∂ Ω = 0 on ∂ Ω. Here, p ∈ [1 , ∞ ) and γ ( p ) = ( 1 if Ω = R n , 1 p if Ω ⊊ R n is a suﬃciently regular domain. Compared with the previous results in [ 1 ], our new results pro vide the follo wing impro vemen ts: (i) The domain Ω do es not need to b e b ounded, but we merely require Ω to ha ve a suﬃciently regular, compact b oundary . This means that exterior domains (e.g., R n \ B 1 (0)) can also b e considered. (ii) The strong nonlocal-to-lo cal con vergence is established in a general L p frame- w ork. This enhances the applicabilit y of the results, as in concrete applica- tions one may need con vergence in an L p space other than L 2 , or the relev ant function u may lie only in W k,p instead of H k for k ∈ { 2 , 3 } . Note that the rate obtained in ( 1.16 ) is consistent with the one from [ 1 ] (cf. ( 1.14 )) for p = 2. (iii) W e demand signiﬁcan tly w eaker assumptions on ρ and J , see (A1) and (A2) . In particular, W 1 , 1 loc -regularit y of J ε is not required, and we allow for a m uch stronger singularit y at the origin, comparable to that of a fractional Laplacian ( − ∆) s of order s ∈ (0 , 1) (cf. Remark 2.2 ). (iv) As in [ 5 ], w e also do not require ρ to b e radially symmetric. This allows for anisotropic eﬀects, since diﬀerent directions ma y b e w eighted diﬀerently , resulting in a non-diagonal momentum matrix M . Compared to [ 5 ], our adv ance is that w e establish the nonlocal-to-lo cal conv ergence in the strong sense as well as asso ciated conv ergence rates. Esp ecially (iii) and (iv) giv e rise to considerable technical diﬃculties in our pro ofs. 5 Structure of the pap er. In Section 2 , we ﬁrst in tro duce our fundamental as- sumptions on the conv olution k ernel. W e also giv e a concrete example of class of k ernels, whic h satisfy these assumptions. F urthermore, we explain in Remark 2.2 , that our assumptions allo w the kernels to exhibit singularities at the origin that are comparable to those of fractional Laplacians. Our main results are successively stated and prov ed in Section 3 . In Subsec- tion 3.1 , we ﬁrst sho w that the nonlo cal op erator ( 1.5 ) is actually well deﬁned under our assumptions on the con volution k ernel. Then in Subsection 3.2 , we establish our main result in the case Ω = R n . Afterwards, we prov e the conv ergences ( 1.15 ) and ( 1.16 ) in the sp ecial case that Ω is a suﬃcien tly regular curved half-space. Finally , in Subsection 3.4 , this allows us to establish our main result in the general case that Ω ⊂ R n is a domain with suﬃcien tly regular, compact b oundary . 2 Assumptions and preliminaries In the remainder of this pap er, let the dimension n ∈ N b e arbitrary . W e mak e the follo wing general assumptions on the conv olution kernel. (A1) W e assume that ρ : R n → [0 , ∞ ), ρ ≡ 0 is measurable and even (i.e., ρ ( x ) = ρ ( − x ) for almost all x ∈ R n ) with Z R n ρ ( x )(1 + | x | ) d x < ∞ . (2.1) The asso ciated kernel is giv en by J : R n \ { 0 } → R , J ( x ) = ρ ( x ) | x | 2 . (2.2) F or any ε > 0, we in tro duce the functions ρ ε : R n → R , ρ ε ( x ) : = ε − n ρ  x ε  , (2.3) J ε : R n \ { 0 } → R , J ε ( x ) = ρ ε ( x ) | x | 2 . (2.4) Since ρ ≥ 0 and ρ ≡ 0, the momentum matrix M : = 1 2 Z R n J ( z ) z ⊗ z d z (2.5) is p ositive deﬁnite. (A2) In addition to Assumption (A1) , we assume that ρ ∈ C 1 ( R n \ { 0 } ) and that there exist c 0 , c 1 > 0, α ∈ (0 , 2) with N > 3 − α and | ρ ( x ) | ≤ c 0 | x | 2 − α − n (1 + | x | ) − N |∇ ρ ( x ) | ≤ c 1 | x | 1 − α − n (1 + | x | ) − N (2.6) for all x ∈ B 1 (0) \ { 0 } . Moreo ver, w e assume that the asso ciated kernel J fulﬁlls the conditions Z R n − 1 J  AQ  z ′ z n  z ′ d z ′ = 0 for all z n ∈ R and all Q ∈ SO( n ), (2.7) 6 where A : = √ M . (2.8) Remark 2.1. W e make the following observ ations. (a) If ρ ∈ C 1 ( R n \ { 0 } ) with compact supp ort in R n , it suﬃces to demand | ρ ( x ) | ≤ c 0 | x | 2 − α − n |∇ ρ ( x ) | ≤ c 1 | x | 1 − α − n (2.9) instead of ( 2.6 ). In this situation, ( 2.9 ) implies that ( 2.6 ) holds for ev ery N > 0, pro vided that c 0 and c 1 are chosen suﬃciently large. (b) If ρ is radially symmetric, so is J . In this case, condition ( 2.7 ) is automatically fulﬁlled. This is b ecause A = a I for some a > 0, where I denotes the iden tity matrix, which implies that Z R n − 1 J ( AQz ) z ′ d z ′ = Z R n − 1 J ( az ) z ′ d z ′ = 0 . Here, the second equalit y can b e shown b y applying the change of v ariables z ′ 7→ − z ′ . Remark 2.2. Recall that, in case Ω = R n , the fr actional L aplac e op er ator of order s ∈ (0 , 1) can b e represented as ( − ∆) s Ω u ( x ) = 4 s Γ  n 2 + s  π n 2 | Γ( − s ) | P . V . Z Ω ρ ( x − y ) | x − y | 2  u ( x ) − u ( y )) d y , with ρ ( x ) = | x | 2 − n − 2 s (2.10) for all x ∈ R n , where Γ denotes the Gamma function. If Ω ⊊ R n , the ab ov e op erator is referred to as a r e gional fr actional L aplac e op er ator of order s . W e notice that the singularity of ρ at the origin is as demanded in ( 2.6 ) with α = 2 s . How ever, ρ needs to b e mo diﬁed for large v alues of | x | in a suitable wa y in order to satisfy ( 2.1 ) and ( 2.6 ). In case Ω is a b ounded domain, such a mo diﬁcation for large | x | does not constitute a true res triction. If Ω is b ounded, we can ﬁnd a radius R > 0 suc h that Ω ⊂ B R (0). This means that for an y x ∈ Ω, ( − ∆) s Ω u ( x ) do es not depend on the v alues of ρ on R n \ B 2 R (0). Therefore, w e can simply replace ρ by ˜ ρ ( x ) = ρ ( x ) χ ( x ) = | x | 2 − n − 2 s χ ( x ) , where χ ∈ C ∞ c ( R n ) with χ = 1 on B 2 R (0), without changing the v alues of ( − ∆) s Ω u in Ω. It is straightforw ard to chec k that ˜ ρ satisﬁes (A1) and (A2) with α = 2 s and an y N > 3 − α = 3 − 2 s . 7 Remark 2.3. In order to construct a non trivial, anisotr opic density function ρ with asso ciated k ernel J , such that (A1) and (A2) are fulﬁlled, we pro ceed as follo ws. Let ˚ ρ ∈ C 1  R n \ { 0 } ; [0 , ∞ )  b e a radially symmetric function, whic h satisﬁes ( 2.1 ) and ( 2.6 ). Without loss of generality , w e assume that ∥ ˚ ρ ∥ L 1 ( R n ) = 2 n . This implies that 1 2 Z R n ˚ ρ ( z ) | z | 2 z ⊗ z d z = I . Moreo ver, let B ∈ R n × n b e a symmetric, positive deﬁnite matrix. T o simplify the computations, we further assume, without loss of generalit y , that det B = 1. W e no w deﬁne ρ ∈ C 1  R n \ { 0 } ; [0 , ∞ )  with ρ ( x ) : = ˚ ρ ( B x ) | x | 2 | B x | 2 for all x ∈ R n \ { 0 } . In this w ay , ρ is clearly ev en but, in general, not any more radially symmetric. Its asso ciated kernel can b e expressed as J ( x ) = ˚ ρ ( B x ) | B x | 2 for all x ∈ R n \ { 0 } . It is easy to chec k that ρ and J satisfy (A1) and Condition ( 2.6 ). Moreo ver, a straigh tforward computation yields M = B − 2 and A = B − 1 . T o verify ( 2.7 ), let Q ∈ SO( n ) b e arbitrary . Recalling that ˚ ρ is radially symmetric, we deduce that Z R n − 1 J  AQ  z ′ z n  z ′ d z ′ = Z R n − 1 ˚ ρ ( B AQz ) | B AQz | 2 z ′ d z ′ = Z R n − 1 ˚ ρ ( Qz ) | Qz | 2 z ′ d z ′ = Z R n − 1 ˚ ρ ( z ) | z | 2 z ′ d z ′ = 0 for all z n ∈ R . This shows that J satisﬁes Condition ( 2.7 ) and consequen tly , (A2) is also fulﬁlled. The following lemma pro vides tw o imp ortant prop erties of the family { ρ ε } ε> 0 . Lemma 2.4. Supp ose that Assumption (A1) holds. (a) F or any ε > 0 , it holds ρ ε ∈ L 1 ( R n ) with ∥ ρ ε ∥ L 1 ( R n ) = ∥ ρ ∥ L 1 ( R n ) < ∞ . (2.11) (b) F or any δ > 0 , it holds that lim ε → 0 Z R n \ B δ (0) ρ ε ( x ) d x = 0 . (2.12) Pro of. Let ε > 0 b e arbitrary . Since ρ ∈ L 1 ( R n ) and the identit y in ( 2.11 ) follows from the change of v ariables x 7→ x ε . Thus, assertion (a) is v eriﬁed. Let now δ > 0 b e arbitrary . The change of v ariables x 7→ x ε yields Z R n \ B δ (0) ρ ε ( x ) = Z R n \ B δ/ε (0) ρ ( x ) d x. Since ρ ∈ L 1 ( R n ), the left-hand side con verges to zero as ε → 0 due to Leb esgue’s dominated conv ergence theorem. This veriﬁes (b). 8 3 Nonlo cal-to-lo cal conv ergence of the nonlo cal op erator In general, w e assume that (A1) holds and w e consider a suﬃcien tly smooth domain Ω ⊆ R n . The main goal of this pap er is to in vestigate the con vergence of the nonlo cal op erator L Ω ε , which is (at ﬁrst formally) given b y L Ω ε u ( x ) = P . V . Z Ω J ε ( x − y )  u ( x ) − u ( y )  d y (3.1) for all x ∈ Ω and any suﬃcien tly regular function u : Ω → R , to the lo cal diﬀeren tial op erator L , which is deﬁned as L Ω u ( x ) = − div  M ∇ u ( x )  = − n X i,j =1 M ij ∂ x j ∂ x k u ( x ) (3.2) for all x ∈ Ω and an y suﬃcien tly regular function u : Ω → R . Here, M ∈ R n × n is the symmetric matrix introduced in (A1) , whic h is p ositiv e deﬁnite. If the b oundary of Ω is not empt y , the diﬀerential operator L Ω is equipp ed with the natural b oundary condition M ∇ u · n ∂ Ω = 0 on ∂ Ω , (3.3) where n ∂ Ω denotes the a unit normal vector ﬁeld on ∂ Ω. F or any k ≥ 2 and p ∈ [1 , ∞ ], we deﬁne the corresp onding Sob olev spaces as W k,p M (Ω) =  u ∈ W k,p (Ω) : M ∇ u · n ∂ Ω = 0 on ∂ Ω  . (3.4) 3.1 W ell-deﬁnedness of the nonlo cal op erator In this subsection, we ﬁrst show that the expression L Ω ε u ( x ) = P . V . Z R n J ε ( x − y )  u ( x ) − u ( y )  d y is actually w ell deﬁned if Assumption (A1) is fulﬁlled. In case Ω = R n , the follo wing prop osition shows that L Ω ε actually deﬁnes a b ounded linear op erator. Prop osition 3.1. L et Ω = R n , supp ose that Assumption (A1) holds and let p ∈ [1 , ∞ ) and ε > 0 b e arbitr ary. Then, the fol lowing statements hold. (a) The op er ator L Ω ε : C 2 c ( R n ) → L p ( R n ) , L Ω ε u ( x ) = Z R n J ε ( x − y )  u ( x ) − u ( y ) − ∇ u ( y ) · ( x − y )  d y (3.5) is wel l-deﬁne d and line ar. Mor e over, for any u ∈ C 2 c ( R n ) , L Ω ε u c an b e expr esse d as L Ω ε u ( x ) = P . V . Z R n J ε ( x − y )  u ( x ) − u ( y )  d y (3.6) for almost al l x ∈ R n . 9 (b) The op er ator intr o duc e d in (a) c an b e extende d to a b ounde d line ar op er ator L Ω ε : W 2 ,p ( R n ) → L p ( R n ) with ∥L Ω ε u ∥ L p ( R n ) ≤ C ∗ ∥ u ∥ W 2 ,p ( R n ) , for al l u ∈ W 2 ,p ( R n ) , wher e the c onstant C ∗ > 0 dep ends only on ρ and p . By means of Prop osition 3.1 , we can draw conclusions on the well-deﬁnedness of the nonlo cal op erator L Ω ε if Ω is a domain of class C 2 . Corollary 3.2. L et Ω b e a (not ne c essarily b ounde d) domain with C 2 -b oundary, supp ose that (A1) holds, and let p ∈ [1 , ∞ ) and ε > 0 b e arbitr ary. Then, the op er ator L Ω ε : C 2 c (Ω) → L p loc (Ω) , L Ω ε u ( x ) = P . V . Z Ω J ε ( x − y )  u ( x ) − u ( y )  d y (3.7) is wel l-deﬁne d and line ar. Pro vided that (A1) and (A2) and certain assumptions on the domain are ful- ﬁlled, w e will see later that the op erator in tro duced in ( 3.7 ) can actually b e extended to a linear and b ounded op erator mapping from W 2 ,p (Ω) to L p (Ω). In the remainder of this subsection, w e present the pro ofs of Proposition 3.1 and Corollary 3.2 . Pro of of Prop osition 3.1 . Let p ∈ [1 , ∞ ) and ε > 0 be arbitrary . In the follo wing, the letter C will denote generic p ositiv e constants dep ending only on ρ and p . The concrete v alue of C ma y v ary throughout the pro of. Pro of of (a) . Let u ∈ C 2 c ( R n ) b e arbitrary . By means of T aylor’s theorem, we hav e u ( x ) − u ( y ) − ∇ u ( x ) · ( x − y ) = − R 2 ( x, y ) , (3.8) where the error term is given by R 2 ( x, y ) : = X | β | =2 2 β ! Z 1 0 (1 − t ) D β u  y + t ( x − y )  d t ( x − y ) β . (3.9) Therefore, we obtain Z R n   J ε ( x − y )  u ( x ) − u ( y ) − ∇ u ( x ) · ( x − y )    d y ≤ X | β | =2 2 β ! Z R n Z 1 0   J ε ( x − y ) (1 − t ) D β u  y + t ( x − y )  ( x − y ) β   d t d y ≤ C ∥ u ∥ W 2 , ∞ ( R n ) Z R n | ρ ε ( x − y ) | d y = C ∥ u ∥ W 2 , ∞ ( R n ) ∥ ρ ∥ L 1 ( R n ) . This prov es that for almost all x ∈ R n , J ε ( x − · )  u ( x ) − u ( · ) − ∇ u ( x ) · ( x − · )  ∈ L 1 ( R n ) (3.10) 10 and therefore, the in tegral in the deﬁnition of L Ω ε actually exists. Recalling ( 3.8 ) and ( 3.9 ), we use H¨ older’s inequality and F ubini’s theorem to deduce ∥L Ω ε u ∥ p L p ( R n ) = Z R n      X | β | =2 2 β ! Z R n Z 1 0 J ε ( x − y ) (1 − t ) D β u  y + t ( x − y )  ( x − y ) β d t d y      p d x ≤ C Z R n X | β | =2  Z R n Z 1 0 | ρ ε ( x − y ) || D β u  y + t ( x − y )  | d t d y  p d x ≤ C Z R n ∥ ρ ∥ p − 1 p L 1 ( R n ) X | β | =2 Z R n Z 1 0 | ρ ε ( x − y ) || D β u  y + t ( x − y )  | p d t d y d x ≤ C X | β | =2 Z 1 0 Z R n Z R n | ρ ε ( x − y ) || D β u  y + t ( x − y )  | p d y d x d t. Applying ﬁrst the change of v ariables y 7→ z = x − y and afterw ards the c hange of v ariables x 7→ w = x − z + tz , we infer ∥L Ω ε u ∥ p L p ( R n ) ≤ C X | β | =2 Z 1 0  Z R n | ρ ε ( z ) | d z   Z R n | D β u ( w ) | p d w  d t ≤ C ∥ ρ ∥ L 1 ( R n ) X | β | =2  Z R n | D β u ( w ) | p d w  ≤ C p ∗ ∥ u ∥ p W 2 ,p ( R n ) (3.11) for some constan t C ∗ > 0 dep ending only on ρ and p . This pro ves that the op erator L Ω ε is w ell-deﬁned and b ounded in the sense of ( 3.11 ). Moreov er, it is easy to chec k that the op erator L Ω ε is linear. It remains to verify the representation ( 3.6 ). T o this end, let r > 0 b e arbitrary . Recalling ( 3.10 ), w e deduce Z | x − y |≥ r J ε ( x − y )  u ( x ) − u ( y )  d y = Z | x − y |≥ r J ε ( x − y )  u ( x ) − u ( y ) − ∇ u ( x ) · ( x − y )  d y + Z | x − y |≥ r J ε ( x − y ) ∇ u ( x ) · ( x − y ) d y (3.12) for almost all x ∈ R n . W e p oint out that Z | x − y |≥ r   J ε ( x − y ) ∇ u ( x ) · ( x − y )   d y ≤ Z | x − y |≥ r | ρ ε ( x − y ) | |∇ u ( x ) | 1 | x − y | d y ≤ 1 r ∥ u ∥ W 1 , ∞ ( R n ) ∥ ρ ∥ L 1 ( R n ) . This means that the second integral on the right-hand side of ( 3.12 ) actually exists and therefore, the identit y ( 3.12 ) is justiﬁed. As ρ is an even function so is the k ernel J ε . This implies that Z | x − y |≥ r J ε ( x − y ) ∇ u ( x ) · ( x − y ) d y = 0 . (3.13) 11 Moreo ver, inv oking ( 3.10 ), we obtain Z | x − y |≥ r J ε ( x − y )  u ( x ) − u ( y ) − ∇ u ( x ) · ( x − y )  d y → L Ω ε u ( x ) (3.14) as r → 0 for almost all x ∈ R n b y means of Lebesgue’s dominated con vergence theorem. Even tually , combining ( 3.12 ), ( 3.13 ) and ( 3.14 ), we conclude that L Ω ε u ( x ) = lim r ↘ 0 Z | x − y |≥ r J ε ( x − y )  u ( x ) − u ( y )  d y . By the deﬁnition of the principal v alue, this is exactly ( 3.6 ). Pro of of (b) . W e already know from (a) that the op erator L Ω ε : C 2 c ( R n ) → L p ( R n ) is w ell-deﬁned, linear and b ounded in the sense of ( 3.11 ). As C 2 c ( R n ) is dense in W 2 ,p ( R n ), w e conclude via estimate ( 3.11 ) that L Ω ε can be extended to a b ounded linear op erator L Ω ε : W 2 ,p ( R n ) → L p ( R n ) with ∥L Ω ε u ∥ L p ( R n ) ≤ C ∗ ∥ u ∥ W 2 ,p ( R n ) . Hence, the pro of is complete. Pro of of Corollary 3.2 . Without loss of generalit y , w e assume that Ω ⊊ R n as the case Ω = R n w as already handled in Prop osition 3.1 . Let u ∈ C 2 c (Ω) be arbitrary . Since Ω is of class C 2 , we can ﬁnd an extension ˜ u ∈ C 2 c ( R n ) with ˜ u | Ω = u . W e now ﬁx an arbitrary x ∈ Ω. Since Ω ⊂ R n is open, w e can ﬁnd a radius r ( x ) > 0 suc h that B r ( x ) ( x ) ⊂ Ω. Then, for any r ∈ (0 , r ( x )], we hav e B r ( x ) ⊂ Ω and thus, Z Ω \ B r ( x ) J ε ( x − y )  u ( x ) − u ( y )  d y = Z R n \ B r ( x ) J ε ( x − y )  ˜ u ( x ) − ˜ u ( y )  d y − Z R n \ Ω J ε ( x − y )  ˜ u ( x ) − ˜ u ( y )  d y (3.15) W e already know from Prop osition 3.1(a) that Z R n \ B r ( x ) J ε ( x − y )  ˜ u ( x ) − ˜ u ( y )  d y → L R n ε ˜ u ( x ) as r → 0. Therefore, it remains to show that the second integral on the righ t-hand side of ( 3.15 ) actually exists. Since B r ( x ) ( x ) ⊂ Ω, we obviously ha ve | x − y | ≥ r ( x ) for all y ∈ R n \ Ω . Recalling the deﬁnition of J ε , this implies that      Z R n \ Ω J ε ( x − y )  ˜ u ( x ) − ˜ u ( y )  d y      ≤ Z R n \ Ω J ε ( x − y )   ˜ u ( x ) − ˜ u ( y )   d y ≤ 2 r ( x ) 2 ∥ ˜ u ∥ L ∞ ( R n ) ∥ ρ ε ∥ L 1 ( R n ) = 2 r ( x ) 2 ∥ ˜ u ∥ L ∞ ( R n ) ∥ ρ ∥ L 1 ( R n ) < ∞ . 12 Hence, sending r → 0 in ( 3.15 ), we conclude that the expression L Ω ε u ( x ) = P . V . Z Ω J ε ( x − y )  u ( x ) − u ( y )  d y = lim r ↘ 0 Z Ω ∩{| x − y |≥ r } J ε ( x − y )  u ( x ) − u ( y )  d y = L R n ε ˜ u ( x ) − Z R n \ Ω J ε ( x − y )  ˜ u ( x ) − ˜ u ( y )  d y (3.16) is well-deﬁned for ev ery x ∈ Ω provided that u ∈ C 2 c (Ω). Let no w K b e an arbitrary compact subset of Ω. Since Ω is op en, we know that r ( K ) : = dist ( R n \ Ω , K ) > 0. As the p oint wise limit of a sequence of measurable functions is measurable, we further know that the mapping Ω ∋ x 7→ L Ω ε u ( x ) ∈ R is measurable. Using ( 3.16 ) along with Prop osition 3.1 , and proceeding similarly as ab o v e, we infer that ∥L Ω ε u ∥ p L p ( K ) ≤ C ∥L R n ε ˜ u ∥ p L p ( R n ) + Z K      Z R n \ Ω J ε ( x − y )  ˜ u ( x ) − ˜ u ( y )  d y      p d x ≤ C ∥L R n ε ˜ u ∥ p L p ( R n ) + 2 p | K | r ( K ) 2 p ∥ ˜ u ∥ p L ∞ ( R n ) ∥ ρ ∥ p L 1 ( R n ) < ∞ . As the compact subset K w as arbitrary , this pro ves that L Ω ε u ∈ L p loc (Ω). This means that the op erator L Ω ε : C 2 c (Ω) → L p loc (Ω) , u 7→ L Ω ε u is well-deﬁned. Moreo ver, the op erator is ob viously linear and thus, the pro of is complete. 3.2 Con vergence on the full space In this subsection, w e consider the case where Ω is the full space, i.e., Ω = R n . Based on the results established in Proposition 3.1 , w e obtain the following nonlo cal-to- lo cal conv ergence prop erties. Theorem 3.3. We c onsider Ω = R n . Supp ose that Assumption (A1) holds and let p ∈ [1 , ∞ ) b e arbitr ary. Then, the fol lowing statements hold. (a) F or al l u ∈ W 2 ,p ( R n ) , L Ω ε u → L Ω u in L p ( R n ) as ε → 0 . (b) Ther e exists a c onstant C > 0 such that for al l ε > 0 and al l u ∈ W 3 ,p ( R n ) , it holds ∥L Ω ε u − L Ω u ∥ L p ( R n ) ≤ C ε ∥ u ∥ W 3 ,p ( R n ) . 13 Pro of of Theorem 3.3 . Let p ∈ [1 , ∞ ) and ε > 0 b e arbitrary . In the follo wing, the letter C will denote generic p ositiv e constants dep ending only on ρ and p . The concrete v alue of C ma y v ary throughout the pro of. Pro of of (b) . Since C ∞ c ( R n ) is dense in W 3 ,p ( R n ), w e ﬁrst verify the assertion for functions u ∈ C ∞ c ( R n ). Therefore, let u ∈ C ∞ c ( R n ) b e arbitrary . By the deﬁnitions of L Ω ε and L (see ( 3.5 ) and ( 3.2 )), we obtain ∥L Ω ε u − L Ω u ∥ p L p ( R n ) = Z R n      Z R n J ε ( x − y )  u ( x ) − u ( y ) + ∇ u ( x ) · ( x − y )  d y + M : D 2 u ( x )      p d x ≤ C  I 1 ,ε + I 2 ,ε  , (3.17) where I 1 ,ε : = Z R n      Z B 1 ( x ) J ε ( x − y )  u ( x ) − u ( y ) + ∇ u ( x ) · ( x − y )  d y + M : D 2 u ( x )      p d x I 2 ,ε : = Z R n      Z R n \ B 1 ( x ) J ε ( x − y )  u ( x ) − u ( y ) + ∇ u ( x ) · ( x − y )  d y      p d x. W e now estimate the terms I 1 ,ε and I 2 ,ε separately . A d I 1 ,ε : Applying T a ylor’s theorem, w e obtain the expansion u ( x ) − u ( y ) + ∇ u ( x ) · ( x − y ) = − 1 2 ( x − y ) T D 2 u ( x )( x − y ) − R 3 ( x, y ) , (3.18) where the error term is given by R 3 ( x, y ) = X | β | =3 3 β ! Z 1 0 (1 − t ) D β u  y + t ( x − y )  d t ( x − y ) β . (3.19) Therefore, I 1 ,ε can b e estimated as I 1 ,ε ≤ C Z R n      − Z B 1 ( x ) 1 2 J ε ( x − y )( x − y ) T D 2 u ( x )( x − y ) d y + M : D 2 u ( x )      p d x + C Z R n      Z B 1 ( x ) J ε ( x − y ) R 3 ( x, y ) d y      p d x. (3.20) Emplo ying the change of v ariables x 7→ z = x − y and recalling the deﬁnition of M (see ( 2.5 )), we observ e that − Z B 1 ( x ) 1 2 J ε ( x − y )( x − y ) T D 2 u ( x )( x − y ) d y + M : D 2 u ( x ) d x = − 1 2 Z B 1 (0) J ε ( z ) z T D 2 u ( x ) z d z − 1 2 Z R n J ε ( z ) z T D 2 u ( x ) z d z = Z R n \ B 1 (0) J ε ( z ) z T D 2 u ( x ) z d z . (3.21) 14 for all x ∈ R n . Plugging this identit y into ( 3.20 ), we obtain I 1 ,ε ≤ C Z R n      Z R n \ B 1 (0) J ε ( z ) z T D 2 u ( x ) z d z      p d x + C Z R n      Z B 1 ( x ) J ε ( x − y ) R 3 ( x, y ) d y      p d x. (3.22) Since | z | ≥ 1 for all z ∈ R n \ B 1 (0), the ﬁrst summand on the right-hand side of ( 3.22 ) can b e estimated as Z R n      Z R n \ B 1 (0) J ε ( z ) z T D 2 u ( x ) z d z      p d x ≤ Z R n Z R n \ B 1 (0) | ρ ε ( z ) | | z |   D 2 u ( x )   d z ! p d x ≤ C Z R n  Z R n | ρ ε ( z ) | | z | d z  p   D 2 u ( x )   p d x ≤ C ε p ∥ u ∥ p W 2 ,p ( R n ) . Recalling the deﬁnition of the error term, w e obtain the follo wing estimate for the the second summand on the right-hand side of ( 3.22 ): Z R n      Z B 1 ( x ) J ε ( x − y ) R 3 ( x, y ) d y      p d x = Z R n       X | β | =3 3 β ! Z B 1 ( x ) Z 1 0 J ε ( x − y )(1 − t ) D β u  y + t ( x − y )  ( x − y ) β d t d y       p d x ≤ C X | β | =3 Z R n Z B 1 ( x ) Z 1 0 | ρ ε ( x − y ) | | x − y | | D β u  y + t ( x − y )  | d t d y ! p d x Next, we use the splitting | ρ ε ( x − y ) | | x − y | =  | ρ ε ( x − y ) | | x − y |  1 q  | ρ ε ( x − y ) | | x − y |  1 p , (3.23) where q denotes the conjugate exp onen t to p , i.e., 1 q + 1 p = 1. Then, using H¨ older’s inequalit y as well as the changes of v ariables y 7→ z = x − y and x 7→ w = x − z + tz , w e infer Z R n      Z B 1 ( x ) J ε ( x − y ) R 3 ( x, y ) d y      p d x ≤ C X | β | =3 Z R n  Z R n | ρ ε ( z ) | | z | d z  p − 1 ·  Z R n Z 1 0 | ρ ε ( z ) | | z |    D β u ( x − z + tz )    p d t d z  d x ≤ C ε p − 1 X | β | =3 Z 1 0 Z R n Z R n | ρ ε ( z ) | | z |    D β u ( w )    p d z d w d t ≤ C ε p ∥ u ∥ p W 3 ,p ( R n ) . 15 Altogether, this shows that I 1 ,ε ≤ C ε p ∥ u ∥ p W 3 ,p ( R n ) . (3.24) A d I 2 ,ε : Applying T a ylor’s theorem, w e obtain the expansion u ( x ) − u ( y ) − ∇ u ( x ) · ( x − y ) = − R 2 ( x, y ) , (3.25) where the error term is given by R 2 ( x, y ) : = X | β | =2 2 β ! Z 1 0 (1 − t ) D β u  y + t ( x − y )  d t ( x − y ) β . (3.26) Hence, I 2 ,ε can b e estimated as I 2 ,ε = C Z R n      Z R n \ B 1 ( x ) J ε ( x − y ) R 2 ( x, y ) d y      p d x ≤ C X | β | =2 Z R n Z R n \ B 1 ( x ) Z 1 0 | ρ ε ( x − y ) |    D β u  y + t ( x − y )     d y d t ! p d x Using the splitting | ρ ε ( x − y ) | = | ρ ε ( x − y ) | 1 q | ρ ε ( x − y ) | 1 p , (3.27) where 1 q + 1 p = 1, H¨ older’s inequalit y , and the c hanges of v ariables y 7→ z = x − y and x 7→ w = x − z + tz , w e infer I 2 ,ε ≤ C X | β | =2 Z R n  Z R n | ρ ε ( z ) | d z  p − 1 · Z R n \ B 1 (0) Z 1 0 | ρ ε ( z ) |    D β u ( x − z + tz )    p d t d z ! d x ≤ C ε p − 1 X | β | =2 Z 1 0 Z R n Z R n \ B 1 (0) | ρ ε ( z ) | | z |    D β u ( w )    p d z d w d t ≤ C ε p ∥ u ∥ p W 2 ,p ( R n ) , (3.28) Com bining ( 3.17 ) with ( 3.24 ) and ( 3.28 ), we ev en tually conclude that ∥L Ω ε u − L Ω u ∥ L p ( R n ) ≤ C ε ∥ u ∥ W 3 ,p ( R n ) (3.29) for all u ∈ C ∞ c ( R n ). Let no w u ∈ W 3 ,p ( R n ) b e arbitrary . Then, b ecause of density , there exists a sequence ( u k ) k ∈ N ⊆ C ∞ c ( R n ) suc h that u k → u in W 3 ,p ( R n ) as k → ∞ . In particular, this entails that ( u k ) k ∈ N is b ounded in W 3 ,p ( R n ). 16 Th us, it follows from ( 3.29 ) that  L Ω ε u k − L Ω u k  k ∈ N ⊆ L p ( R n ) is b ounded. Hence, according to the Banach–Alaoglu theorem, there exists a func- tion w ∈ L p ( R n ) such that, up to subsequence extraction, L Ω ε u k − L Ω u k  w in L p ( R n ) as k → ∞ . Because of Prop osition 3.1 , we kno w that L Ω ε u k → L Ω ε u in L p ( R n ) as k → ∞ . Moreo ver, recalling the deﬁnition of L , it is further clear that L Ω u k → L Ω u in L p ( R n ) as k → ∞ . Consequen tly , due to the uniqueness of the weak limit, we ha v e w = L Ω ε u − L Ω u . Th us, by means of weak low er semi-contin uity , we conclude ∥L Ω ε u − L Ω u ∥ L p ( R n ) ≤ lim inf k →∞ ∥L Ω ε u k − L Ω u k ∥ L p ( R n ) ≤ lim sup k →∞ C ε ∥ u k ∥ W 3 ,p ( R n ) = C ε ∥ u ∥ W 3 ,p ( R n ) . Since u ∈ W 3 ,p ( R n ) was arbitrary , this prov es (b). Pro of of (a) . W e already kno w from Proposition 3.1(b) that the operator norm of the nonlo cal op erator L Ω ε : W 2 ,p ( R n ) → L p ( R n ) is b ounded b y the constan t C ∗ , whic h is indep endent of ε . T ogether with the deﬁnition of L Ω u (see ( 3.2 )), this implies that ∥L Ω ε u − L Ω u ∥ L p ( R n ) ≤ ∥L Ω ε u ∥ L p ( R n ) + ∥L Ω u ∥ L p ( R n ) ≤ C ∥ u ∥ W 2 ,p ( R n ) for all u ∈ W 2 ,p ( R n ). W e further know from part (b) that for any u ∈ W 3 ,p ( R n ), it holds that L Ω ε u − L Ω u → 0 in L p ( R n ) as ε → 0. As the inclusion W 3 ,p ( R n ) ⊂ W 2 ,p ( R n ) is dense, the assertion of (a) directly follows from the Banach–Steinhaus theorem. Hence, the pro of of Theorem 3.3 is complete. 3.3 Con vergence on a curv ed half space In this section, we next consider the case where our domain is a curv ed half space. More precisely , we consider R n γ =  x ∈ R n : x n > γ ( x 1 , . . . , x n − 1 )  (3.30) for a prescrib ed function γ ∈ C 3 b ( R n − 1 ). 17 Theorem 3.4. L et γ ∈ C 3 b ( R n − 1 ) , p ∈ [1 , ∞ ) , ε ∈ (0 , 1] , and supp ose that (A1) and (A2) hold with M = I . Then, if ∥ γ ∥ C 1 b ( R n − 1 ) is suﬃc ently smal l, the op er ator intr o duc e d in ( 3.7 )  r estricte d to C 2 c ( R n γ ) ∩ W 2 ,p I ( R n γ )  c an b e extende d to a b ounde d line ar op er ator L R n γ ε : W 2 ,p I ( R n γ ) → L p ( R n γ ) (3.31) with ∥L R n γ ε u ∥ L p ( R n γ ) ≤ c γ ∥ u ∥ W 2 ,p ( R n γ ) (3.32) for al l u ∈ W 2 ,p I ( R n γ ) , wher e c γ is a p ositive c onstant dep ending only on γ , ρ and p . Mor e over, ther e exists a p ositive c onstant C γ dep ending only on γ , ρ and p such that for al l ε > 0 and al l u ∈ W 3 ,p I ( R n γ ) , it holds   L R n γ ε u + ∆ u   L p ( R n γ ) ≤ C γ p √ ε ∥ u ∥ W 3 ,p ( R n γ ) . (3.33) F urthermor e, for al l u ∈ W 2 ,p I ( R n γ ) , it holds L R n γ ε u → − ∆ u in L p ( R n γ ) as ε → 0 . (3.34) Corollary 3.5. L et γ ∈ C 3 b ( R n − 1 ) , Q ∈ SO( n ) , ε ∈ (0 , 1] , and supp ose that (A1) and (A2) hold with M = I . Then, the r esults of The or em 3.4 hold true for Q R n γ inste ad of R n γ and W k,p Q ( Q R n γ ) inste ad of W k,p I ( R n γ ) for k = 2 , 3 . Pro of of Theorem 3.4 . T o pro vide a cleaner presentation, we will usually refrain from indicating the principal v alue by the symbol P . V . whenever the meaning is clear. In the following, the letter C will denote generic p ositive constan ts dep ending only on γ , ρ and p . The concrete v alue of C may v ary throughout the pro of. The pro of is split into three steps. Step 1: Pro of of Estimate ( 3.32 ) for u ∈ C 2 c ( R n γ ) ∩ W 2 ,p I ( R n γ ) . Let u ∈ C 2 c ( R n γ ) ∩ W 2 ,p I ( R n γ ) b e arbitrary . Hence, according to Corollary 3.2 , the expression L R n γ ε u ( x ) = P . V . Z R n γ J ε ( x − y )  u ( x ) − u ( y )  d y is well-deﬁned. Our ﬁrst goal is to show that L R n γ ε u ∈ L p ( R n γ ) and that ∥L R n γ ε u ∥ L p ( R n γ ) ≤ C ∥ u ∥ W 2 ,p ( R n γ ) . (3.35) By construction, it is clear that L R n γ ε u is measurable. Since R n γ is of class C 3 , we can ﬁnd an extension ˜ u ∈ C 2 c ( R n ) with u | R n γ = u . In particular, ˜ u can b e chosen in suc h a wa y that ∥ ˜ u ∥ W 2 ,p ( R n ) ≤ C ∥ u ∥ W 2 ,p ( R n γ ) . (3.36) 18 Using Prop osition 3.1 , we obtain the estimate ∥L R n γ ε u ∥ p L p ( R n γ ) ≤ C ∥L R n ε ˜ u ∥ p L p ( R n ) + C ∥R ε ˜ u ∥ p L p ( R n γ ) ≤ C ∥ u ∥ p W 2 ,p ( R n γ ) + C ∥R ε ˜ u ∥ p L p ( R n γ ) , (3.37) where the error term is given by R ε ˜ u ( x ) : = Z ( R n γ ) c J ε ( x − y )  u ( x ) − ˜ u ( y )  d y for a.e. x ∈ R n γ . Therefore, it remains to sho w that ∥R ε ˜ u ∥ p L p ( R n γ ) ≤ C ∥ u ∥ p W 2 ,p ( R n γ ) (3.38) since ( 3.35 ) then follows b y combining ( 3.37 ) and ( 3.38 ). Since γ ∈ C 3 b ( R n − 1 ), we infer from [ 30 , Lemma 2.1] that there exists a C 2 , 1 - diﬀeomorphism F γ : R n → R n with F γ ( R n + ) = R n γ , which satisﬁes F γ ( x ′ , 0) =  x ′ γ ( x ′ )  and ∂ x n F γ ( x ) | x n =0 = − n ∂ R n γ ( x ′ , γ ( x ′ )) (3.39) for all x ′ ∈ R n − 1 . Since F γ ∈ C 2 , 1 ( R n ; R n ), we further ha ve D F γ ∈ W 2 , ∞ ( R n ; R n × n ) with ∥ D F γ ∥ W 2 , ∞ ( R n ; R n × n ) ≤ C ∥ γ ∥ C 3 b ( R n − 1 ) ≤ C. (3.40) In the following, w e write d F γ : = | det D F γ | as an abbreviation. In particular, due to ( 3.40 ), we hav e ∥ d F γ ∥ L ∞ ( R n ) ≤ C. (3.41) Moreo ver, assuming ∥ γ ∥ C 1 b ( R n − 1 ) to b e suﬃciently small, w e can ensure that sup x ∈ R n | D F γ ( x ) − I | ≤ 1 6 . (3.42) No w, by the c hange of v ariables b y x 7→ F γ ( x ), we infer that ∥R ε ˜ u ∥ p L p ( R n γ ) = Z R n γ      Z ( R n γ ) c J ε ( x − y )  u ( x ) − ˜ u ( y )  d y      p d x (3.43) = Z R n +      Z R n − J ε  F γ ( x ) − F γ ( y )   u ( F γ ( x )) − ˜ u ( F γ ( y ))  d F γ ( y ) d y      p d F γ ( x ) d x, T o simplify the notation, we introduce the functions w : R n → R n , w ( x ) = ˜ u  F γ ( x )  , (3.44) G γ : R n × R n → R n × n , G γ ( x, y ) : = Z 1 0 D F γ ( y + t ( x − y )) d t. (3.45) 19 In particular, this means that G γ ( x, x ) = D F γ ( x ) for all x ∈ R n . Using the chain rule along with ( 3.36 ), ( 3.40 ) and ( 3.41 ), we deduce ∥ w ∥ W 2 ,p ( R n ) ≤ C ∥ ˜ u ∥ W 2 ,p ( R n ) ≤ C ∥ u ∥ W 2 ,p ( R n γ ) . (3.46) Moreo ver, by means of the fundamental theorem of calculus, we obtain F γ ( x ) − F γ ( y ) = G γ ( x, y ) ( x − y ) . Let now x, ˜ x, y , ˜ y ∈ R n b e arbitrary . Recalling that G γ ( ˜ x, ˜ x ) = D F γ ( ˜ x ), we infer from ( 3.42 ) that    G γ ( ˜ x, ˜ x ) − I  ( x − y )   ≤ 1 6 | x − y | . (3.47) This implies that   G γ ( ˜ x, ˜ x )( x − y )   ≥    | x − y | −    G γ ( ˜ x, ˜ x ) − I  ( x − y )      ≥ 5 6 | x − y | . (3.48) Moreo ver, recalling the deﬁnition of G γ in ( 3.45 ) and in v oking once more ( 3.42 ), we deduce that    G γ ( ˜ x, ˜ y ) − G γ ( ˜ x, ˜ x )  ( x − y )   ≤ h   G γ ( ˜ x, ˜ y ) − I   +   G γ ( ˜ x, ˜ x ) − I   i | x − y | ≤ 1 3 | x − y | . (3.49) Com bining ( 3.48 ) and ( 3.49 ), we conclude that   G γ ( ˜ x, ˜ y )( x − y )   ≥      G γ ( ˜ x, ˜ x )( x − y )   −    G γ ( ˜ x, ˜ y ) − G γ ( ˜ x, ˜ x )  ( x − y )      ≥ 1 2 | x − y | (3.50) for all x, ˜ x, y , ˜ y ∈ R n . Next, using ( 3.45 ), w e can reformulate ( 3.43 ) as ∥R ε ˜ u ∥ p L p ( R n γ ) = Z R n +      Z R n − J ε  G γ ( x, y )( x − y )  w ( x ) − w ( y )  d F γ ( y ) d y      p d F γ ( x ) d x. (3.51) No w, we introduce the sets A : = R n − 1 × (0 , 2) ⊂ R n + and B : = R n − 1 × ( − 2 , 0) ⊂ R n − . (3.52) Hence, from ( 3.51 ) we infer that ∥R ε ˜ u ∥ p L p ( R n γ ) ≤ C  I 1 ε + I 2 ε + I 3 ε  , (3.53) where I 1 ε : = Z R n + \A     Z B J ε  G γ ( x, y )( x − y )  ( w ( x ) − w ( y )) d F γ ( y ) d y     p d F γ ( x ) d x , I 2 ε : = Z R n +      Z R n − \B J ε  G γ ( x, y )( x − y )  ( w ( x ) − w ( y )) d F γ ( y ) d y      p d F γ ( x ) d x , 20 I 3 ε : = Z A     Z B J ε  G γ ( x, y )( x − y )  ( w ( x ) − w ( y )) d F γ ( y ) d y     p d F γ ( x ) d x . These integral terms will now b e handled separately . A d I 1 ε : W e ﬁrst observe that I 1 ε ≤ C ( I 1 , 1 ε + I 1 , 2 ε ) , where I 1 , 1 ε : = Z R n + \A  Z B J ε  G γ ( x, y )( x − y )  | w ( x ) | d F γ ( y ) d y  p d x, I 1 , 2 ε : = Z R n + \A  Z B J ε  G γ ( x, y )( x − y )  | w ( y ) | d F γ ( y ) d y  p d x. W e further observe that for all x ∈ R n + \ A and all y ∈ B , we hav e | x − y | ≥ | x n − y n | = x n − y n ≥ 2 . for all x ∈ R n + \ A and all y ∈ B . Hence, due to ( 3.50 ), we ha v e   G γ ( x, y )( x − y )   ≥ 1 2 | x − y | ≥ 1 . (3.54) Hence, using Assumption (A2) and applying the change of v ariables y 7→ z = x − y ε , w e deduce that Z B ρ ε  G γ ( x, y )( x − y )  d y ≤ C ε − n Z B    x − y ε    2 − α − n  1 +    x − y ε     − N d y = C Z R n | z | 2 − α − n (1 + | z | ) − N d z ≤ C. (3.55) By ( 3.41 ), ( 3.55 ) and H¨ older’s inequality , we ﬁnd that I 1 , 1 ε ≤ C Z R n + \A | w ( x ) | p  Z B ρ ε  G γ ( x, y )( x − y )  d y  p d x ≤ C ∥ w ∥ p L p ( R n + ) . Pro ceding similarly , we use ( 3.41 ), ( 3.55 ) and H¨ older’s inequalit y to obtain I 1 , 2 ε ≤ C Z R n + \A  Z B ρ ε  G γ ( x, y )( x − y )  d y  p − 1 ·  Z B ρ ε  G γ ( x, y )( x − y )  | w ( y ) | p d y  d x ≤ C Z R n + \A Z B ρ ε  G γ ( x, y )( x − y )  | w ( y ) | p d y d x. Finally , recalling Assumption (A2) , using F ubini’s theorem, and preo ceeding simi- larly to ( 3.55 ), we conclude I 1 , 2 ε ≤ C ε − n Z R n + \A Z B    x − y ε    2 − α − n  1 +    x − y ε     − N | w ( y ) | p d y d x 21 ≤ C ε − n Z B | w ( y ) | p Z R n + \A    x − y ε    2 − α − n  1 +    x − y ε     − N d x ! d y ≤ C ∥ w ∥ p L p ( R n ) . In summary , this sho ws I 1 ε ≤ C ∥ w ∥ p L p ( R n ) . A d I 2 ε : Here, we notice that for all x ∈ R n + and all y ∈ R n − \ B , it holds | x − y | ≥ | x n − y n | = x n − y n ≥ 2 . Hence, I 2 ε can b e estimated in a similar manner as I 1 ε . In this w a y , we conclude that I 2 ε ≤ C ∥ w ∥ p L p ( R n ) . A d I 3 ε : A straightforw ard computation yields I 3 ε ≤ C  I 3 , 2 ε + I 3 , 1 ε + I 3 , 3 ε  (3.56) where I 3 , 1 ε : = Z A      Z B J ε ( G γ ( x, x )( x − y ))( w ( x ) − w ( y )) d F γ ( x ) d y      p d x I 3 , 2 ε : = Z A      Z B J ε  G γ ( x, y )( x − y )  ( w ( x ) − w ( y ))  d F γ ( y ) − d F γ ( x )  d y      p d x I 3 , 3 ε : = Z A      Z B  J ε  G γ ( x, y )( x − y )  − J ε ( G γ ( x, x )( x − y ))  · ( w ( x ) − w ( y )) d F γ ( x ) d y      p d x These terms will b e handled separately . A d I 3 , 1 ε : Due to the construction of F γ , there exist con tinuously diﬀeren tiable func- tions U : R n → SO( n ), H : R n → GL n ( R ) and H ′ : R n → GL n − 1 ( R ) with H ( x ) =      0 H ′ ( x ) . . . 0 0 . . . 0 1      for all x ∈ R n (3.57) suc h that G γ ( x, x ) = D F γ ( x ) = U ( x ) H ( x ) for all x ∈ R n . (3.58) F or more details on this decomp osition, we refer to [ 3 , Pro of of Corollary 2]. In 22 particular, we hav e H ( x ) − 1 =      0 H ′ ( x ) − 1 . . . 0 0 . . . 0 1      for all x ∈ R n (3.59) and since det U ( x ) = 1 for all x ∈ R n , it further holds that det H ( x ) = d F γ ( x ) and det  H ( x ) − 1  =  d F γ ( x )  − 1 for all x ∈ R n . Recalling G γ ( x, x ) = D F γ ( x ), we no w apply T a ylor’s theorem to derive the estimate I 3 , 1 ε ≤ C  I 3 , 3 , 1 ε + I 3 , 3 , 2 ε  , (3.60) where I 3 , 1 , 1 ε : = Z A     Z B J ε  D F γ ( x )( x − y )  ∇ w ( x ) · ( x − y ) d F γ ( x ) d y     p d x, I 3 , 1 , 2 ε : = Z A     Z B J ε  D F γ ( x )( x − y )  R 2 ( x, y ) d F γ ( x ) d y     p d x, and the error term is given by R 2 ( x, y ) = X | β | =2 2 β !  Z 1 0 (1 − t ) D β w  y + t ( x − y )  d t  ( x − y ) β . (3.61) A d I 3 , 1 , 1 ε : Using the c hange of v ariables y 7→ z = x − y , then z 7→ ˜ z = H ( x ) z , and ﬁnally ˜ z 7→ y = x − ˜ z , the term I 3 , 1 , 1 ε can b e reformulated as I 3 , 1 , 1 ε = Z A      Z R n − 1 × ( x n ,x n +2) J ε  U ( x ) H ( x ) z  ∇ w ( x ) · z d z d F γ ( x )      p d x = Z A      Z R n − 1 × ( x n ,x n +2) J ε  U ( x ) ˜ z  H ( x ) − T ∇ w ( x ) · ˜ z d ˜ z      p d x = Z A     Z B J ε  U ( x )( x − y )  H ( x ) − T ∇ w ( x ) · ( x − y ) d y     p d x = Z A     H ( x ) − T ∇ w ( x ) · Z B J ε  U ( x )( x − y )  ( x − y ) d y     p d x. Th us, exploiting the structure of H ( x ) − T , we use Condition ( 2.7 ) to infer that I 3 , 1 , 1 ε = Z A     ∂ x n w ( x ) Z B J ε  U ( x )( x − y )  ( x n − y n ) d y     p d x. Since u ∈ C 2 c ( R n γ ) ∩ W 2 ,p I ( R n γ ), we deduce by means of the chain rule and ( 3.39 ) that ∂ x n w ( x ′ , 0) = ∂ x n  u ◦ F γ  ( x ′ , 0) = −∇ u ( x ′ , γ ( x ′ )) · n  x ′ , γ ( x ′ )  = 0 (3.62) 23 for all x ′ ∈ R n − 1 . Moreov er, we hav e | x n | ≤ | x n − y n | for all x ∈ A and y ∈ B . Th us, inv oking the fundamental theorem of calculus, we hav e ∂ x n w ( x ) = ∂ x n w ( x ) − ∂ x n w ( x ′ , 0) = Z 1 0 ∂ 2 x n w ( x ′ , tx n ) x n d t for all x = ( x ′ , x n ) ∈ R n . Hence, w e obtain I 3 , 1 , 1 ε = Z A      ∂ x n w ( x ) − ∂ x n w ( x ′ , 0)  Z B J ε  U ( x )( x − y )  ( x n − y n ) d y     p d x = Z A     Z 1 0 Z B J ε  U ( x )( x − y )  ∂ 2 x n w ( x ′ , tx n ) x n ( x n − y n ) d y d t     p d x ≤ Z A  Z 1 0 Z B ρ ε  U ( x )( x − y )    ∂ 2 x n w ( x ′ , tx n )   d y d t  p d x = Z A  Z 1 0   ∂ 2 x n w ( x ′ , tx n )   d t   Z B ρ ε  U ( x )( x − y )  d y  p d x (3.63) Applying the change of v ariables y 7→ z = U ( x )( x − y ) along with Lemma 2.4(a) , w e deduce that Z B ρ ε  U ( x )( x − y )  d y ≤ Z R n ρ ε ( z ) d z = ∥ ρ ∥ L 1 ( R n ) . (3.64) Hence, recalling the deﬁnition of A and using the c hange of v ariables t 7→ s = tx n , w e get I 3 , 1 , 1 ε ≤ C Z R n − 1 Z ∞ 0  1 x n Z x n 0 | ∂ 2 x n w ( x ′ , s ) | d s  p d x n d x ′ (3.65) Finally , applying Hardy’s inequalit y , we conclude that I 3 , 1 , 1 ε ≤ C  p p − 1  p Z R n − 1 Z ∞ 0 | ∂ 2 x n w ( x ) | p d x n d x ′ ≤ C ∥ w ∥ p W 2 ,p ( R n + ) . (3.66) A d I 3 , 1 , 2 ε : Recalling the deﬁnition of R 2 (see ( 3.61 )) and ( 3.50 ), w e use H¨ older’s inequalit y and Lemma 2.4(a) to obtain I 3 , 1 , 2 ε = Z A     Z B J ε  D F γ ( x )( x − y )  R 2 ( x, y ) d F γ ( x ) d y     p d x ≤ C X | β | =2 Z A  Z 1 0 Z B ρ ε  D F γ ( x )( x − y )  | D β w  y + t ( x − y )  | d y d t  p d x ≤ C X | β | =2 Z A  Z B ρ ε  D F γ ( x )( x − y )  d y  p − 1 (3.67) ·  Z 1 0 Z B ρ ε  D F γ ( x )( x − y )  | D β w  y + t ( x − y )  | p d y d t  d x. 24 Using the c hange of v ariables y 7→ D F γ ( x )( x − y ) as w ell as Lemma 2.4(a) , we deduce that Z B ρ ε  D F γ ( x )( x − y )  d y ≤ Z R n ρ ε ( z ) d z = ∥ ρ ∥ L 1 ( R n ) . Consequen tly , we ha ve I 3 , 1 , 2 ε ≤ C X | β | =2 Z A Z 1 0 Z B ρ ε  D F γ ( x )( x − y )  | D β w  y + t ( x − y )  | p d y d t d x. W e now apply the change of v ariables R 2 n ∋  x y  7→  ξ η  : =  x − y y + t ( x − y )  ∈ R 2 n . (3.68) Note that det  D ( x,y )  ξ η   = det  I − I tI (1 − t ) I  = 1 . In this wa y , we obtain I 3 , 1 , 2 ε ≤ C Z 1 0 X | β | =2 Z R n  Z R n ρ ε  D F γ  η + (1 − t ) ξ  ξ  d ξ  | D β w ( η ) | p d η d t. (3.69) Applying ( 3.50 ), w e deduce that    D F γ  η + (1 − t ) ξ  ξ    ≥ 1 2 | ξ | (3.70) for all ξ ∈ R n and t ∈ (0 , 1). Inv oking Assumption (A2) and applying the c hange of v ariables ξ 7→ ζ = ξ /ε , w e infer that Z R n ρ ε  D F γ  η + (1 − t ) ξ  ξ  d ξ = Z R n ρ ε  D F γ  η + (1 − t ) ξ  ξ  d ξ ≤ C ε − n Z R n    ξ ε    2 − α − n  1 +    ξ ε     − N d ξ = C Z R n | ζ | 2 − α − n (1 + | ζ | ) − N d ζ ≤ C . (3.71) Hence, in view of ( 3.69 ), we obtain I 3 , 1 , 2 ε ≤ C X | β | =2 Z R n | D β w ( z ) | p d z ≤ C ∥ w ∥ p W 2 ,p ( R n ) . In summary , recalling ( 3.60 ), we ﬁnally conclude that I 3 , 1 ε ≤ C ∥ w ∥ p W 2 ,p ( R n ) . (3.72) A d I 3 , 2 ε : It follows from ( 3.40 ) that d F γ is Lipschitz con tinuous. Hence, using the 25 fundamen tal theorem of calculus and H¨ older’s inequality , we deduce that I 3 , 2 ε = Z A      Z B J ε  G γ ( x, y )( x − y )  ( w ( x ) − w ( y ))  d F γ ( y ) − d F γ ( x )  d y      p d x ≤ C Z A  Z B Z 1 0 ρ ε  G γ ( x, y )( x − y )    ∇ w  y + t ( x − y )    d t d y  p d x ≤ C Z A  Z B ρ ε  G γ ( x, y )( x − y )  d y  p − 1 (3.73) ·  Z 1 0 Z B ρ ε  G γ ( x, y )( x − y )    ∇ w  y + t ( x − y )    p d y d t  d x Applying the change of v ariables y 7→ z = x − y , w e obtain Z B ρ ε  G γ ( x, y )( x − y )  d y ≤ Z R n ρ ε  G γ ( x, x − z ) z  d z . (3.74) Hence, pro ceeding similarly to ( 3.71 ), w e infer that Z B ρ ε  G γ ( x, y )( x − y )  d y ≤ C. (3.75) This implies that I 3 , 2 ε ≤ C Z 1 0 Z R n Z R n ρ ε  G γ ( x, y )( x − y )    ∇ w  y + t ( x − y )    p d y d x d t Applying the c hange of v ariables ( 3.68 ) and pro ceeding similarly to ( 3.69 )–( 3.71 ), w e ﬁnally conclude that I 3 , 2 ε ≤ C ∥ w ∥ p W 2 ,p ( R n ) . (3.76) A d I 3 , 3 ε : In order to estimate I 3 , 3 ε , we ﬁrst notice that the fundamental theorem of calculus yields J ε  G γ ( x, y )( x − y )  − J ε ( G γ ( x, x )( x − y )) = Z 1 0 ∇ J ε ( z s ) · ( G γ ( x, y ) − G γ ( x, x ))( x − y ) d s, (3.77) for all x, y ∈ R n , where z s = z s ( x, y ) : = G γ ( x, x )( x − y ) + s  G γ ( x, y ) − G γ ( x, x )  ( x − y ) (3.78) for all s ∈ [0 , 1]. Recalling the deﬁnition of G γ in ( 3.45 ) and using the Lipschitz con tinuit y of D F γ , which follows from ( 3.40 ), we ﬁnd that   G γ ( x, y ) − G γ ( x, x )   ≤ C | x − y | (3.79) for all x, y ∈ R n . Plugging this estimate into ( 3.77 ), we infer that   J ε  G γ ( x, y )( x − y )  − J ε ( G γ ( x, x )( x − y ))   ≤ C Z 1 0 |∇ J ε ( z s ) | | x − y | 2 d s (3.80) 26 for all x, y ∈ R n . Combining ( 3.48 ) and ( 3.49 ), we further deduce that | z s | ≥      G γ ( x, x )( x − y )   − s    G γ ( x, y ) − G γ ( x, x )  ( x − y )      ≥ 5 6 | x − y | − 1 3 s | x − y | ≥ 1 2 | x − y | (3.81) for all x, y ∈ R n . Using the chain rule, w e derive the estimate |∇ J ε ( x ) | ≤ |∇ ρ ε ( x ) | | x | 2 + 2 | ρ ε ( x ) | | x | 3 for all x ∈ R n \ { 0 } . In view of ( 3.81 ), w e thus obtain |∇ J ε ( z s ) | ≤ C |∇ ρ ε ( z s ) | | x − y | 2 + C | ρ ε ( z s ) | | x − y | 3 (3.82) for all x, y ∈ R n \ { 0 } . In voking the fundamen tal theorem of calculus, we further ha ve w ( x ) − w ( y ) = Z 1 0 ∇ w  y + t ( x − y )  d t · ( x − y ) (3.83) for all x, y ∈ R n . Combining ( 3.80 ), ( 3.82 ) and ( 3.83 ), we no w conclude that I 3 , 3 ε ≤ C Z A      Z B Z 1 0 Z 1 0 |∇ J ε ( z s ) |   ∇ w  y + t ( x − y )    | x − y | 3 d s d t d y      p d x. ≤ C Z A Z B Z 1 0 Z 1 0  | ρ ε ( z s ) | + |∇ ρ ε ( z s ) | | x − y |  ·   ∇ w  y + t ( x − y )    d s d t d y ! p d x Hence, by means of H¨ older’s inequality , we obtain I 3 , 3 ε ≤ C Z A Z B Z 1 0  | ρ ε ( z s ) | + |∇ ρ ε ( z s ) | | x − y |  d s d y ! p − 1 · Z B Z 1 0 Z 1 0  | ρ ε ( z s ) | + |∇ ρ ε ( z s ) | | x − y |  ·   ∇ w  y + t ( x − y )    p d s d t d y ! d x (3.84) Due to ( 3.81 ) and Condition ( 2.6 ) from Assumption (A2) , we pro ceed as in the estimates for I 3 , 1 , 2 ε and I 3 , 2 ε to conclude that I 3 , 3 ε ≤ C ∥ w ∥ p W 2 ,p ( R n ) . (3.85) 27 No w, combining ( 3.72 ), ( 3.76 ) and ( 3.85 ), w e infer from ( 3.56 ) that I 3 ε ≤ C ∥ w ∥ p W 2 ,p ( R n ) . (3.86) Since I 1 ε = I 2 ε = 0, we conclude from ( 3.53 ) and ( 3.46 ) that ∥R ε ˜ u ∥ p L p ( R n γ ) ≤ C ∥ w ∥ p W 2 ,p ( R n ) ≤ C ∥ u ∥ p W 2 ,p ( R n γ ) . (3.87) In view of ( 3.37 ), this means that ( 3.35 ) is ﬁnally veriﬁed. Step 2: Pro of of Estimate ( 3.33 ) for u ∈ C 3 c ( R n γ ) ∩ W 3 ,p I ( R n γ ) . No w, let u ∈ C 3 c ( R n γ ) ∩ W 3 ,p I ( R n γ ). Our goal is to sho w that there exists a constant C γ > 0 dep ending only on R n γ , ρ and p suc h that ∥L R n γ ε u + ∆ u ∥ L p ( R n γ ) ≤ C γ p √ ε ∥ u ∥ W 3 ,p ( R n γ ) . Since R n γ is of class C 3 , we can ﬁnd an extension ˜ u ∈ C 3 c ( R n ) with u | R n γ = u . In particular, ˜ u can b e chosen in such a w a y that ∥ ˜ u ∥ W 3 ,p ( R n ) ≤ C ∥ u ∥ W 3 ,p ( R n γ ) . (3.88) By means of Theorem 3.3 , we derive the estimate ∥L R n γ ε u + ∆ u ∥ p L p ( R n γ ) ≤ C ∥L R n ε ˜ u + ∆ ˜ u ∥ p L p ( R n ) + ∥R ε ˜ u ∥ p L p ( R n γ ) ≤ C ε p ∥ ˜ u ∥ p W 3 ,p ( R n ) + ∥R ε ˜ u ∥ p L p ( R n γ ) ≤ C ε p ∥ u ∥ p W 3 ,p ( R n γ ) + ∥R ε ˜ u ∥ p L p ( R n γ ) , (3.89) where the error term is given by R ε ˜ u ( x ) : = Z R n γ c J ε ( x − y )  u ( x ) − ˜ u ( y )  d y for all x ∈ R n γ . Therefore, it remains to sho w that ∥R ε ˜ u ∥ p L p ( R n γ ) ≤ C ε ∥ u ∥ p W 3 ,p ( R n γ ) (3.90) since ( 3.35 ) then follo ws by combining ( 3.37 ) and ( 3.38 ). T o this end, w e deﬁne the radius R > 0, the functions F γ , G γ and w and the sets A and B as in Step 1. In particular, using the chain rule along with ( 3.88 ), ( 3.40 )and ( 3.41 ), we deduce ∥ w ∥ W 3 ,p ( R n ) ≤ C ∥ ˜ u ∥ W 3 ,p ( R n ) ≤ C ∥ u ∥ W 3 ,p ( R n γ ) . (3.91) Moreo ver, recalling ( 3.53 ), we hav e ∥R ε ˜ u ∥ p L p ( R n γ ) ≤ C ( I 1 ε + I 2 ε + I 3 ε ) , (3.92) where I 1 ε , I 2 ε , I 3 ε are the integral terms in tro duced in Step 1. Now, as N > 3 − α and higher regularit y of u is assumed, we intend to improv e the estimates from Step 1 for I 1 ε , I 2 ε and I 3 ε suc h that the desired rate with resp ect to ε is obtained. 28 A d I 1 ε : Recalling ( 3.54 ), it holds that   G γ ( x, y )( x − y )   ≥ 1 2 | x − y | ≥ 1 for all x ∈ R n + \ A and all y ∈ B . Let I 1 , 1 ε and I 1 , 2 ε b e deﬁned as in Step 1. Using Assumption (A2) , we deduce that Z B ρ ε  G γ ( x, y )( x − y )  d y ≤ C ε Z B ρ ε  G γ ( x, y )( x − y )     x − y ε    d y ≤ C ε − n +1 Z B    x − y ε    3 − α − n  1 +    x − y ε     − N d y = C ε Z R n | z | 3 − α − n (1 + | z | ) − N d z ≤ C ε, (3.93) since N > 3 − α . Consequently , w e hav e I 1 , 1 ε ≤ C Z R n + \A | w ( x ) | p  Z B ρ ε  G γ ( x, y )( x − y )  d y  p d x ≤ C ε p ∥ w ∥ p L p ( R n + ) . T o estimate I 1 , 2 ε , we ﬁrst use H¨ older’s inequality to obtain I 1 , 2 ε ≤ C Z R n + \A  Z B ρ ε  G γ ( x, y )( x − y )  d y  p − 1 ·  Z B ρ ε  G γ ( x, y )( x − y )  | w ( y ) | p d y  d x. Emplo ying ( 3.93 ), we infer that I 1 , 2 ε ≤ C ε 1 − 1 p Z R n + \A  Z B ρ ε  G γ ( x, y )( x − y )  | w ( y ) | p d y  d x. F urthermore, recalling N > 3 − α , Assumption (A2) , using F ubini’s theorem, and pro ceeding similarly to ( 3.93 ), we deduce that Z B ρ ε  G γ ( x, y )( x − y )  | w ( y ) | p d y ≤ C ε − n +1 Z R n + \A  Z B    x − y ε    3 − α − n  1 +    x − y ε     − N | w ( y ) | p d y  d x ≤ C ε − n +1 Z B | w ( y ) | p Z R n + \A    x − y ε    3 − α − n  1 +    x − y ε     − N d x ! d y ≤ C ε ∥ w ∥ p L p ( R n ) . Com bining these estimates, we arriv e at I 1 , 2 ε ≤ C ε ∥ w ∥ p L p ( R n ) 29 Altogether, this shows I 1 ε ≤ C ε ∥ w ∥ p L p ( R n ) . (3.94) A d I 2 ε : Arguing in a similar manner as for I 1 ε , we obtain I 2 ε ≤ C ε ∥ w ∥ p L p ( R n ) . (3.95) A d I 3 ε : T o derive a b ound on I 3 ε , we recall from Step 1 that I 3 ε ≤ C  I 3 , 1 , 1 ε + I 3 , 1 , 2 ε + I 3 , 2 ε + I 3 , 3 ε  . (3.96) Here, the integral terms on the right-hand side are deﬁned as in Step 1 and will b e handled separately . A d I 3 , 1 , 1 ε : According to ( 3.63 ), we ha ve I 3 , 1 , 1 ε = C Z A "  Z 1 0   ∂ 2 x n w ( x ′ , tx n )   d t  ·  Z B ρ ε  U ( x )( x − y )  d y  # p d x. Recalling the deﬁnition of A and B , we infer that I 3 , 1 , 1 ε ≤ C Z R n − 1 ∥ ∂ 2 x n w ( x ′ , · ) ∥ p L ∞ ((0 , 2)) · Z 2 0  Z R n − 1 Z 0 − 2 ρ ε  U ( x )( x − y )  d y n d y ′  p d x n d x ′ . (3.97) Recalling the deﬁnition of ρ ε , we use the c hanges of v ariables y 7→ z = x − y , z 7→ εz , x n 7→ εx n and z 7→ y = x − z to compute Z 2 0  Z R n − 1 Z 0 − 2 ρ ε  U ( x )( x − y )  d y n d y ′  p d x n = Z 2 0  ε − n Z R n − 1 Z x n +2 x n ρ  U ( x ) z ε  d z n d z ′  p d x n = Z 2 0 Z R n − 1 Z ( x n +2) /ε x n /ε ρ  U ( x ) z  d z n d z ′ ! p d x n = ε Z 2 /ε 0 Z R n − 1 Z x n +(2 /ε ) x n ρ  U ( x ′ , εx n ) z  d z n d z ′ ! p d x n ≤ ε Z ∞ 0 Z R n − 1 Z x n +(2 /ε ) x n ρ  U ( x ′ , εx n ) z  d z n d z ′ ! p d x n = ε Z ∞ 0 Z R n − 1 Z 0 − 2 /ε ρ  U ( x ′ , εx n )( x − y ))  d y n d y ′ ! p d x n . (3.98) for all x ′ ∈ R n − 1 . Since U ( z ) ∈ SO( n ) for all z ∈ R n , we know that   U ( x ′ , εx n )( x − y )   = | x − y | for all x, y ∈ R n . 30 F urthermore, we notice that for any x, y ∈ R n with x n ≥ 0 and y n ≤ 0, it holds that | x − y | ≥ | x n − y n | = x n + | y n | ≥ x n . Moreo ver, since N > 3 − α , w e can ﬁnd a δ > 0 suc h that N > 3 − α + δ . Based on ( 3.98 ), we deduce that Z 2 0  Z R n − 1 Z 0 − 2 ρ ε  U ( x )( x − y )  d y n d y ′  p d x n ≤ ε Z ∞ 0  Z R n − 1 Z 0 −∞ ρ  U ( x ′ , εx n )( x − y ))  d y n d y ′  p d x n ≤ C ε Z ∞ 0  Z R n − 1 Z 0 −∞ | x − y | 2 − α − n (1 + | x − y | ) − N d y n d y ′  p d x n ≤ C ε Z ∞ 0 Z R n − 1 Z 0 −∞ | x − y | 2 − α − n (1 + | x − y | ) − N +1+ δ · (1 + x n ) − (1+ δ ) d y n d y ′ ! p d x n ≤ C ε  Z R n | y | 2 − α − n (1 + | y | ) − N +1+ δ d y  p Z ∞ 0 (1 + x n ) − (1+ δ ) p d x n ≤ C ε (3.99) for all x ′ ∈ R n − 1 . Here, the integral with respect to y exists since N > 3 − α + δ , and the integral with resp ect to x n exists since − (1 + δ ) p < − 1. Plugging this estimate in to ( 3.97 ), we infer that I 3 , 1 , 1 ε ≤ C ε Z R n − 1 ∥ ∂ 2 x n w ( x ′ , · ) ∥ p L ∞ ((0 , 2)) d x ′ . As the embedding W 1 ,p ((0 , 2))  → L ∞ ((0 , 2)) is contin uous, this implies that I 3 , 1 , 1 ε ≤ C ε Z R n − 1 ∥ ∂ 2 x n w ( x ′ , · ) ∥ p W 1 ,p ((0 , 2)) d x ′ ≤ C ε ∥ w ∥ p W 3 ,p ( R n ) . (3.100) A d I 3 , 1 , 2 ε : W e already know from ( 3.50 ) that   D F γ ( x ) z   ≥ 1 2 | z | for all x, z ∈ R n . (3.101) Moreo ver, for any x, y ∈ R n with y n < 0 < x n , we ha ve | x n − y n | = | y n | + | x n | . Hence a straightforw ard computation yields   ( x ′ − y ′ , y n , x n )   ≤ | x − y | and | x − y | ≤ √ 2   ( x ′ − y ′ , y n , x n )   (3.102) for all x, y ∈ R n with y n < 0 < x n . Thus, by deﬁnition of ρ ε in (A1) , ( 3.101 ) and Assumption (A2) , we infer that | ρ ε ( D F γ ( x ))( x − y ) | ≤ C ε n    x − y ε    2 − α − n  1 +    x − y ε     − N ≤ C ε n     x ′ − y ′ ε , y n ε , x n ε     2 − α − n  1 +     x ′ − y ′ ε , y n ε , x n ε      − N 31 for all x, y ∈ R n with y n < 0 < x n . T o simplify the notation, we now introduce the function g : R n +1 → R , g ( ζ ) : = | ζ | 2 − α − n (1 + | ζ | ) − N . Recalling the deﬁnitions of A and B , we then deduce that I 3 , 1 , 2 ε ≤ C X | β | =2 Z R n − 1 Z 2 0 ε − n Z 1 0 Z R n − 1 Z 0 − 2 g  x ′ − y ′ ε , y n ε , x n ε  · | D β w ( y + t ( x − y )) | d y n d y ′ d t ! p d x n d x ′ ≤ C X | β | =2 Z R n − 1 Z 2 0 ε − n Z 1 0 Z R n − 1 Z 0 − 2 g  x ′ − y ′ ε , y n ε , x n ε  · ∥ D β w ( y ′ + t ( x ′ − y ′ ) , . ) ∥ L ∞ (( − 2 , 2)) d y n d y ′ d t ! p d x n d x ′ . (3.103) Using the c hange of v ariables y ′ 7→ z ′ = x ′ − y ′ as well as z ′ 7→ εz ′ , x n 7→ εx n , and y n 7→ εz n , it follows that I 3 , 1 , 2 ε ≤ C ε X | β | =2 Z ∞ 0 Z R n − 1 Z 1 0 Z 0 −∞ Z R n − 1 g ( z ′ , z n , x n ) · ∥ D β w ( x ′ − εz ′ + tεz ′ , . ) ∥ L ∞ (( − 2 , 2)) d z ′ d z n d t ! p d x ′ d x n . No w, the change of v ariables x ′ 7→ x ′ − εz ′ + tεz ′ yields I 3 , 1 , 2 ε ≤ C ε X | β | =2 Z ∞ 0 Z R n − 1 Z R n g ( z ′ , z n , x n ) ∥ D β w ( x ′ , . ) ∥ L ∞ (( − 2 , 2)) d z ′ d z n ! p d x ′ d x n ≤ C ε ∥ w ∥ p W 3 ,p ( R n ) Z ∞ 0 Z 0 −∞ Z R n − 1 g ( z ′ , z n , x n ) d z ′ d z n ! p d x n . (3.104) Next, we introduce the auxiliary function G : R → R , G ( s ) : = Z 0 −∞ Z R n − 1 g ( z ′ , z n , s ) d z ′ d z n . Using interpolation, we obtain Z ∞ 0 Z 0 −∞ Z R n − 1 g ( z ′ , z n , x n ) d z ′ d z n ! p d x n = ∥ G ∥ L p ((0 , ∞ ) ) p ≤ ∥ G ∥ L 1 ((0 , ∞ )) ∥ G ∥ p − 1 L ∞ ((0 , ∞ )) . 32 Recalling the deﬁnitions of G and g , we deduce that ∥ G ∥ L 1 ((0 , ∞ )) ≤ Z R n +1 | ζ | 2 − α − n (1 + | ζ | ) − N d ζ ≤ C since N > 3 − α . Moreov er, recalling ( 3.102 ) and applying the changes of v ariables z ′ 7→ x ′ − z ′ and z 7→ x − z , we get ∥ G ∥ L ∞ ((0 , ∞ )) = sup x n ∈ (0 , ∞ ) Z 0 −∞ Z R n − 1 | ( x ′ − z ′ , z n , x n ) | 2 − α − n (1 + | ( x ′ − z ′ , z n , x n ) | ) − N d z ′ d z n ≤ sup x n ∈ (0 , ∞ ) C Z R n | x − z | 2 − α − n (1 + | x − z | ) − N d z = C Z R n | z | 2 − α − n (1 + | z | ) − N d z ≤ C. Finally , we conclude that I 3 , 1 , 2 ε ≤ C ε ∥ w ∥ p W 3 ,p ( R n ) . (3.105) A d I 3 , 2 ε : W e hav e already shown in ( 3.73 ) that I 3 , 2 ε ≤ C Z A  Z B ρ ε  G γ ( x, y )( x − y )  d y  p − 1 ·  Z 1 0 Z B ρ ε  G γ ( x, y )( x − y )    ∇ w  y + t ( x − y )    p d y d t  d x. As we know from ( 3.50 ) that   D F γ ( x ) z   ≥ 1 2 | z | for all x, z ∈ R n , the in tegral term I 3 , 2 ε can be estimated analogously to I 3 , 1 , 2 ε . In this wa y , w e obtain I 3 , 2 ε ≤ C ε ∥ w ∥ p W 3 ,p ( R n ) . (3.106) A d I 3 , 3 ε : According to ( 3.84 ), we ha ve I 3 , 3 ε ≤ C Z A Z B Z 1 0  | ρ ε ( z s ) | + |∇ ρ ε ( z s ) | | x − y |  d s d y ! p − 1 · Z B Z 1 0 Z 1 0  | ρ ε ( z s ) | + |∇ ρ ε ( z s ) | | x − y |  ·   ∇ w  y + t ( x − y )    p d s d t d y ! d x, 33 where z s is given by ( 3.78 ), that is, z s = z s ( x, y ) : = ˜ G s ( x, y )( x − y ) with ˜ G s ( x, y ) : = G γ ( x, x ) + s  G γ ( x, y ) − G γ ( x, x )  Moreo ver, we ha v e shown in ( 3.81 ) that   ˜ G s ( x, y )( x − y )   = | z s | ≥ 1 2 | x − y | (3.107) for all x, y ∈ R n . Hence, recalling the deﬁnitions of A and B as w ell as (A2) , w e use the change of v ariables y 7→ z = x − y to deduce that I 3 , 3 ε ≤ C Z R n − 1 Z 2 0 Z R n − 1 Z 0 − 2 Z 1 0  | ρ ε ( z s ) | + |∇ ρ ε ( z s ) | | x − y |  d s d y n d y ′ ! p − 1 d x n · Z R n − 1 Z 4 0    z ε    2 − α − n  1 +    z ε     − N (3.108) · Z 1 0   ∇ w  x ′ − z ′ + tz ′ , ·    p L ∞ (( − 2 , 2)) d t d z n d z ′ ! d x ′ No w, we ﬁx an arbitrary x ′ ∈ R n − 1 and pro ceed similarly to ( 3.98 ) and ( 3.99 ). Recalling ε ∈ (0 , 1] and using the growth assumptions from (A2) as w ell as the c hanges of v ariables y 7→ x − z and z 7→ εz , we obtain Z 2 0 Z R n − 1 Z 0 − 2 Z 1 0  | ρ ε ( z s ) | + |∇ ρ ε ( z s ) | | x − y |  d s d y n d y ′ ! p − 1 d x n ≤ C Z 2 0 ε − n Z R n − 1 Z x n +2 x n Z 1 0    ˜ G s ( x, x − z ) z ε    2 − α − n  1 +    ˜ G s ( x, x − z ) z ε     − N +    ˜ G s ( x, x − z ) z ε    1 − α − n  1 +    ˜ G s ( x, x − z ) z ε     − N   z ε   d s d z n d z ′ ! p − 1 d x n ≤ C Z 2 0 Z R n − 1 Z ( x n +2) /ε x n /ε Z 1 0    ˜ G s ( x, x − εz ) z    2 − α − n  1 +    ˜ G s ( x, x − εz ) z     − N +    ˜ G s ( x, x − εz ) z    1 − α − n  1 +    ˜ G s ( x, x − εz ) z     − N | z | d s d z n d z ′ ! p − 1 d x n . Next, emplo ying the changes of v ariables x n 7→ εx n and z 7→ y = x − z , using the abbreviation x ε = ( x ′ , εx n ), and inv oking ( 3.107 ), we infer that Z 2 0 Z R n − 1 Z 0 − 2 Z 1 0  | ρ ε ( z s ) | + |∇ ρ ε ( z s ) | | x − y |  d s d y n d y ′ ! p − 1 d x n 34 ≤ C ε Z 2 0 Z R n − 1 Z x n +(2 /ε ) x n Z 1 0    ˜ G s ( x ε , x ε − εz ) z    2 − α − n  1 +    ˜ G s ( x ε , x ε − εz ) z     − N +    ˜ G s ( x ε , x ε − εz ) z    1 − α − n  1 +    ˜ G s ( x ε , x ε − εz ) z     − N | z | d s d z n d z ′ ! p − 1 d x n . ≤ C ε Z 2 0 Z R n − 1 Z x n +(2 /ε ) x n | z | 2 − α − n (1 + | z | ) − N d z n d z ′ ! p − 1 d x n . ≤ C ε Z ∞ 0 Z R n − 1 Z 0 −∞ | x − y | 2 − α − n (1 + | x − y | ) − N d y n d y ′ ! p − 1 d x n . Arguing as in ( 3.99 ), we deduce that the in tegral term in the last line is ﬁnite. Hence, we conclude that Z 2 0 Z R n − 1 Z 0 − 2 Z 1 0  | ρ ε ( z s ) | + |∇ ρ ε ( z s ) | | x − y |  d s d y n d y ′ ! p − 1 d x n ≤ C ε for all x ′ ∈ R n − 1 . Using this inequalit y to estimate the right-hand side of ( 3.108 ), w e infer I 3 , 3 ε ≤ C ε Z R n − 1 Z R n − 1 Z 4 0    z ε    2 − α − n  1 +    z ε     − N · Z 1 0   ∇ w  x ′ − z ′ + tz ′ , ·    p L ∞ (( − 2 , 2)) d t d z n d z ′ d x ′ This integral term can b e handled analogously to ( 3.103 ) and ( 3.104 ). In this w ay , w e ﬁnally conclude the estimate I 3 , 3 ε ≤ C ε ∥ w ∥ p W 3 ,p ( R n ) . (3.109) In summary , collecting ( 3.100 ), ( 3.105 ), ( 3.106 ) and ( 3.109 ) we arriv e at I 3 ε ≤ C ε ∥ w ∥ p W 3 ,p ( R n ) . (3.110) Finally , plugging the estimates ( 3.94 ), ( 3.95 ), and ( 3.110 ) into ( 3.92 ), we ha ve ∥R ε ˜ u ∥ p L p ( R n γ ) ≤ C ε ∥ w ∥ p W 3 ,p ( R n ) , whic h veriﬁes ( 3.90 ). Combining this estimate with ( 3.89 ) and ( 3.91 ), w e ﬁnally conclude that ∥R ε ˜ u ∥ p L p ( R n γ ) ≤ C ε ∥ u ∥ p W 3 ,p ( R n γ ) . This means that ( 3.33 ) is established for u ∈ C 3 c ( R n γ ) ∩ W 3 ,p I ( R n γ ). Step 3: Construction of the op erator and completion of the pro of. As the inclusion C 2 c ( R n γ ) ∩ W 2 ,p I ( R n γ ) ⊂ W 2 ,p I ( R n γ ) is dense, we can use the result established in Step 1 to extend L R n γ ε : C 2 c ( R n γ ) ∩ W 2 ,p I ( R n γ ) → L p loc ( R n γ ) 35 to a b ounded linear op erator L R n γ ε : W 2 ,p I ( R n γ ) → L p ( R n γ ) (3.111) with ∥L R n γ ε u ∥ L p ( R n γ ) ≤ C γ ∥ u ∥ W 2 ,p ( R n γ ) (3.112) for a suitable constant C γ > 0 that dep ends only on R n γ , ρ and p . In particular, the extension ( 3.111 ) is unique. W e further kno w from Step 2 that ( 3.33 ) holds for all u ∈ C 3 c ( R n γ ) ∩ W 3 ,p I ( R n γ ). Because of density , it is clear that ( 3.33 ) holds true for all u ∈ W 3 ,p I ( R n γ ). Finally , pro ceeding similarly to the pro of of Theorem 3.3(a) , we use the Banach–Steinhaus theorem to conclude the conv ergence ( 3.34 ). This means that all claims are established and thus, the pro of is complete. Pro of of Corollary 3.5 . First, let u ∈ C 2 c ( Q R n γ ) ∩ W 2 ,p Q ( Q R n γ ) b e arbitrary . W e no w introduce the function w : R n γ → R , w ( x ) = u ( Qx ) . As the transformation x 7→ Qx is linear, it is clear that w ∈ W 3 ,p ( R n γ ). In particular, since Q ∈ SO( n ), we obtain ∥ w ∥ W 2 ,p ( R n γ ) = ∥ u ∥ W 2 ,p ( Q R n γ ) . (3.113) Moreo ver, using the c hain rule, we obtain that ∇ w ( x ) · e n = Q ∇ u ( Qx ) · e n = 0 for all x ∈ ∂ R n γ since Qx ∈ ∂ Q R n γ . Thus, w e hav e w ∈ W 3 ,p I ( R n γ ). Moreov er, as the Laplace op erator is inv ariant under rotations, we ha v e ∆ w ( x ) = ∆ u ( Qx ) for all x ∈ R n γ . This can also be easily v eriﬁed b y means of the c hain rule. W e no w deﬁne b ρ : R n → R with b ρ ( x ) = ρ ( Qx ) for all x ∈ R n \ { 0 } . Hence, its asso ciated kernel is giv en b y b J = J ( Q · ). Accordingly , we write b J ε = J ε ( Q · ) for all ε > 0. Since ρ and J satisfy (A1) and (A2) and QQ T = I , it is straigh tforward to chec k that b ρ and b J fulﬁll (A1) and (A2) with c M : = 1 2 Z R n b J ( z ) z ⊗ z d z = 1 2 Q Z R n J ( Qz ) ( Qz ) ⊗ ( Qz ) d z Q T = 1 2 Q Z R n J ( Qz ) ( Qz ) ⊗ ( Qz ) d z Q T = 1 2 Q Z R n J ( z ) z ⊗ z d z Q T = I , b A : = I in place of M and A . In particular, according to Theorem 3.4 and Corollary 3.2 , 36 the nonlo cal op erators b L R n γ ε : W 2 ,p ( R n γ ) → L p ( R n γ ) , b L R n γ ε v ( x ) = P . V . Z R n γ b J ε ( x − y )  v ( x ) − v ( y )  d y and L Q R n γ ε : C 2 c ( Q R n γ ) ∩ W 2 ,p Q ( Q R n γ ) → L p loc ( Q R n γ ) , L Q R n γ ε v ( x ) = P . V . Z Q R n γ J ε ( x − y )  v ( x ) − v ( y )  d y are w ell-deﬁned for every ε > 0. Emplo ying the c hanges of v ariables x 7→ Qx and y 7→ Qy , w e deduce that   L Q R n γ ε u + ∆ u   p L p ( Q R n γ ) = Z Q R n γ      P . V . Z Q R n γ J ε ( x − y )  u ( x ) − u ( y )  d y + ∆ u ( x )      p d x = Z R n γ      P . V . Z R n γ J ε  Q ( x − y )  w ( x ) − w ( y )  d y + ∆ w ( x )      p d x =   b L R n γ ε w + ∆ w   p L p ( R n γ ) . (3.114) In particular, inv oking the estimate ( 3.32 ) from Theorem 3.4 and the identit y ( 3.113 ), this entails   L Q R n γ ε u   L p ( Q R n γ ) ≤   L Q R n γ ε u + ∆ u   L p ( Q R n γ ) +   ∆ u   L p ( Q R n γ ) =   b L R n γ ε w + ∆ w   L p ( R n γ ) +   ∆ w   L p ( R n γ ) ≤   b L R n γ ε w   L p ( R n γ ) + 2   ∆ w   L p ( R n γ ) ≤ C ∥ w ∥ W 2 ,p ( R n γ ) = C ∥ u ∥ W 2 ,p ( Q R n γ ) . (3.115) Consequen tly , L Q R n γ ε can b e extended to a b ounded linear op erator L Q R n γ ε : W 2 ,p Q ( Q R n γ ) → L p ( Q R n γ ) . No w, we additionally assume that u ∈ W 3 ,p Q ( Q R n γ ). Hence, u ∈ W 3 ,p I ( R n γ ) with ∥ w ∥ W 3 ,p ( R n γ ) = ∥ u ∥ W 3 ,p ( Q R n γ ) . (3.116) Using the inequality ( 3.33 ) from Theorem 3.4 to estimate the righ t-hand side of ( 3.114 ), we obtain   L Q R n γ ε u + ∆ u   L p ( Q R n γ ) =   b L R n γ ε w + ∆ w   L p ( R n γ ) ≤ C γ p √ ε ∥ w ∥ W 3 ,p ( R n γ ) = C γ p √ ε ∥ u ∥ W 3 ,p ( Q R n γ ) 37 as desired. Pro ceeding as in Step 3 of the pro of of Theorem 3.4 , we apply the Banac h–Steinhaus theorem to conclude that L Q R n γ ε u → − ∆ u for all u ∈ W 2 ,p ( Q R n γ ). Therefore, the pro of of Corollary 3.5 is complete. 3.4 Con vergence on a domain with compact boundary In this section, we ﬁnally consider the case where Ω is a domain with compact C 3 -b oundary . Theorem 3.6. L et Ω ⊂ R n b e a domain with c omp act b oundary of class C 3 , p ∈ [1 , ∞ ) , ε ∈ (0 , 1] , and supp ose that (A1) and (A2) hold. Then, the op er a- tor intr o duc e d in ( 3.7 )  r estricte d to C 2 c (Ω) ∩ W 2 ,p M (Ω)  c an b e extende d to a b ounde d line ar op er ator L Ω ε : W 2 ,p M (Ω) → L p (Ω) (3.117) with ∥L Ω ε u ∥ L p (Ω) ≤ c Ω ∥ u ∥ W 2 ,p (Ω) (3.118) for al l u ∈ W 2 ,p M (Ω) , wher e c Ω is a p ositive c onstant dep ending only on Ω , ρ and p . Mor e over, ther e exists a p ositive c onstant C Ω dep ending only on Ω , ρ and p such that for al l ε > 0 and al l u ∈ W 3 ,p M (Ω) , it holds   L Ω ε u − L Ω u   L p (Ω) ≤ C Ω p √ ε ∥ u ∥ W 3 ,p (Ω) . (3.119) F urthermor e, for al l u ∈ W 2 ,p M (Ω) , it holds L Ω ε u → L Ω u in L p (Ω) as ε → 0 . (3.120) Pro of. T o pro vide a cleaner presen tation, w e will usually refrain from indicating the principal v alue b y the sym b ol P . V . whenever the meaning is clear. In the follo wing, the letter C will denote generic p ositiv e constan ts dep ending only on Ω, ρ and p . The concrete v alue of C may v ary throughout the pro of. The pro of is split into three steps. Step 1. Reduction to a simpler setting on a reference domain. W e already know from Corollary 3.2 that the op erator L Ω ε : C 2 c (Ω) ∩ W 2 ,p M (Ω) → L p loc (Ω) , L Ω ε v ( x ) = P . V . Z Ω J ε ( x − y )  v ( x ) − v ( y )  d y (3.121) is w ell-deﬁned and linear. Let now u ∈ C 2 c (Ω) ∩ W 2 ,p M (Ω) b e arbitrary and let A b e the matrix introduced in ( 2.8 ). This means that det A = det √ M = (det M ) 1 2 . 38 Using the change of v ariables z 7→ Az , w e observe that AA T = M = 1 2 Z Ω J ( z ) z ⊗ z d z = 1 2 A Z A − 1 Ω J ( Az ) z ⊗ z d z ! A T det A. This implies that 1 2 Z R n J ( Az ) z ⊗ z d z = I (det A ) − 1 , (3.122) where I ∈ R n × n denotes the iden tit y matrix. In the following, w e choose b Ω : = A − 1 Ω as our reference domain. As Ω is a domain of class C 3 , so is b Ω. Thanks to the chain rule, the normal spaces satisfy the relation N ˆ x ( ∂ b Ω) = A T N A ˆ x ( ∂ Ω) . In particular, since A T = A , this implies that n ∂ b Ω ( ˆ x ) = A n ∂ Ω ( A ˆ x ) for all x ∈ ∂ b Ω . W e now deﬁne the function w : b Ω → R , v ( ˆ x ) = u ( A ˆ x ) . By means of the c hain rule, it is straigh tforw ard to chec k that w ∈ C 2 c  b Ω  ∩ W 2 ,p ( b Ω) with ∥ w ∥ W 2 ,p ( b Ω) ≤ C ∥ u ∥ W 2 ,p (Ω) . (3.123) Recalling the deﬁnition of W 2 ,p M (Ω) (see ( 3.4 )) and using the abov e identities along with the chain rule, we observe that ∇ w ( ˆ x ) · n ∂ b Ω ( ˆ x ) = A ∇ u ( A ˆ x ) · A n ∂ Ω ( A ˆ x ) = A 2 ∇ u ( A ˆ x ) · n ∂ Ω ( A ˆ x ) = M ∇ u ( A ˆ x ) · n ∂ Ω ( A ˆ x ) = 0 for all x ∈ ∂ b Ω. This means that w ∈ W 2 ,p I ( b Ω). Moreov er, b y means of the chain rule, we further deduce that ∆ w ( ˆ x ) = div( ∇ w ( ˆ x )) = div( A ∇ u ( A ˆ x )) for all ˆ x ∈ b Ω. W e now deﬁne b ρ : R n → R with b ρ ( x ) = ρ ( Ax ) | x | 2 | Ax | 2 det A for all x ∈ R n \ { 0 } , (3.124) and we write b J to denote its asso ciated kernel. Since ρ and J satisfy (A1) and (A2) , it is straightforw ard to chec k that b ρ and b J fulﬁll (A1) and (A2) with c M : = 1 2 Z R n b J ( z ) z ⊗ z d z = I , (3.125) 39 b A : = I (3.126) in place of M and A , resp ectiv ely . In particular, for all ε > 0, we hav e b J ( x ) = J ( Ax ) det A and b J ε ( x ) = J ε ( Ax ) det A for all x ∈ R n \ { 0 } , and in view of assumption ( 2.7 ), we hav e Z R n − 1 b J  Q  z ′ z n  z ′ d z ′ = 0 for almost all z n ∈ R and all Q ∈ O( n ). (3.127) This condition will b e crucial in Step 2 of this pro of and it is the reason, why ( 2.7 ) w as demanded in Assumption (A2) in the ﬁrst place. No w, we further in tro duce the nonlo cal op erator b L b Ω ε : C 2 c  b Ω  ∩ W 2 ,p ( b Ω) → L p loc ( b Ω) , b L b Ω ε v ( x ) = P . V . Z b Ω b J ε ( x − y )  v ( x ) − v ( y )  d y . (3.128) Since b ρ and b J satisfy (A1) and (A2) , we already know from Corollary 3.2 that this op erator is well-deﬁned and linear. In the remainder of this pro of, w e need to establish the following k ey estimates: ∥L b Ω ε v ∥ L p ( b Ω) ≤ C ∥ v ∥ W 2 ,p ( b Ω) for all v ∈ C 2 c  b Ω  ∩ W 2 ,p I ( b Ω) , (3.129) ∥L b Ω ε v + ∆ v ∥ L p ( b Ω) ≤ C p √ ε ∥ v ∥ W 3 ,p ( b Ω) for all v ∈ C 3 c  b Ω  ∩ W 3 ,p I ( b Ω) . (3.130) Once this is achiev ed, we use the c hanges of v ariables x 7→ ˆ x = A − 1 x and y 7→ ˆ y = A − 1 y to derive the identit y ∥L Ω ε u − L Ω u ∥ p L p (Ω) = Z A b Ω     P . V . Z A b Ω J ε ( x − y )  u ( x ) − u ( y )  d y + div  A ∇ u ( x )      p d x = det A Z b Ω     P . V . Z b Ω J ε  A ( ˆ x − ˆ y )  det A  w ( ˆ x ) − w ( ˆ y )  d ˆ y + ∆ w ( ˆ x )     p d ˆ x = det A ∥ b L b Ω ε w + ∆ w ∥ p L p ( b Ω) . (3.131) In particular, using ( 3.129 ) and ( 3.123 ), we infer that ∥L Ω ε u ∥ L p (Ω) ≤ ∥L Ω ε u − L Ω u ∥ L p (Ω) + ∥L Ω u ∥ L p (Ω) ≤ ∥ b L b Ω ε w + ∆ w ∥ L p ( b Ω) + ∥ ∆ w ∥ L p ( b Ω) ≤ ∥ b L b Ω ε w ∥ L p ( b Ω) + 2 ∥ ∆ w ∥ L p ( b Ω) ≤ C ∥ w ∥ W 2 ,p ( b Ω) ≤ C ∥ u ∥ W 2 ,p (Ω) . Consequen tly , L Ω ε can b e extended to a b ounded linear op erator L Ω ε : W 2 ,p M (Ω) → L p (Ω) , 40 whic h fulﬁlls ( 3.118 ). Now, w e take an arbitrary u ∈ C 3 c (Ω) ∩ W 3 ,p (Ω). This means that w ∈ C 3 c  b Ω  ∩ W 3 ,p ( b Ω) with ∥ w ∥ W 3 ,p ( b Ω) ≤ C ∥ u ∥ W 3 ,p (Ω) . (3.132) Com bining ( 3.130 ), ( 3.131 ) and ( 3.123 ), we further ha v e ∥L Ω ε u − L Ω u ∥ L p (Ω) = ∥ b L b Ω ε w + ∆ w ∥ p L p ( b Ω) ≤ C p √ ε ∥ w ∥ W 3 ,p ( b Ω) ≤ C p √ ε ∥ u ∥ W 3 ,p (Ω) . Because of densit y , it follows that ( 3.119 ) holds. Pro ceeding as in Step 3 of the pro of of Theorem 3.4 , w e apply the Banac h–Steinhaus theorem to conclude that ( 3.120 ) is fulﬁlled. This means that all the claims are established. Therefore, in order to complete the pro of, it remains to establish the key esti- mates ( 3.129 ) and ( 3.130 ). Step 2. Lo calizat ion of the reference domain. In order to verify the key estimates ( 3.129 ) and ( 3.130 ), we need to lo calize the reference domain b Ω. F or this purp ose, w e need to construct a suitable partition of unit y . W e start b y ﬁxing an arbitrary z ∈ ∂ b Ω. Then, since ∂ b Ω is of class C 3 , there exists an open set U z ⊆ R n , a function γ z ∈ C 3 b ( R n − 1 ) and a matrix Q z ∈ SO( n ) suc h that b Ω ∩ U z = Q z R n γ z ∩ U z . Since γ z ∈ C 3 b ( R n − 1 ), w e infer from [ 30 , Lemma 2.1] that there exists a C 2 , 1 - diﬀeomorphism F z : R n → R n with F γ z ( R n + ) = R n γ z , F z ( x ′ , 0) =  x ′ , γ ( x ′ )  and  D F z ( x ′ , 0)  − 1 n ∂ R n γ z  F z ( x ′ , 0)  ∈ span { e n } (3.133) for all x ′ ∈ R n − 1 . F or an y δ > 0, w e now intr o duce the functions φ z ∈ C ∞ c ( R n − 1 ; [0 , 1]) and ψ z ∈ C ∞ c ( R ; [0 , 1]) with φ z = 1 on B δ / 2 ( z ′ ) , supp φ z ⊂ B δ ( z ′ ) ⊂ R n − 1 , ψ z = 1 on  − δ 2 , δ 2  , supp ψ z ⊂ ( − δ, δ ) . Next, we deﬁne b ϕ z : R n → R , b ϕ z ( y ) = φ z ( y ′ ) ψ z ( y n ) and ˜ ϕ z : = b ϕ z ◦  Q z F z  − 1 . Since ( Q z F z ) − 1 ( U ) ⊂ R n is op en, we can ensure that supp b ϕ z ⊂ ( Q z F z ) − 1 ( U z ) ⇔ supp ˜ ϕ z ⊂ U z b y choosing δ z suﬃcien tly small. Moreov er, as F − 1 z ∈ C 2 , 1 ( R n ), we infer that ˜ ϕ z ∈ C 2 c ( R n ; [0 , 1]) ∩ W 3 , ∞ ( R n ) . Let no w y ∈ ∂ b Ω ∩ U z = ∂  Q z R n γ z  ∩ U z b e arbitrary . Hence, there exists x ′ ∈ R n − 1 41 suc h that Q z F z ( x ′ , 0) = y . By the construction of F z , we hav e D  ( Q z F z ) − 1  ( y ) =  D ( Q z F z )( x ′ , 0)  − 1 =  D F z ( x ′ , 0)  − 1 Q T z (3.134) Moreo ver, we ha v e Q T z n ∂ ( Q z R n γ z ) ( y ) = n ∂ ( R n γ z )  Q T z y  = n ∂ R n γ z  F z ( x ′ , 0)  (3.135) No w, using the chain rule as well as ( 3.134 ) and ( 3.135 ), w e derive the iden tit y ∇ ˜ ϕ z ( y ) · n ∂ b Ω ( y ) = D ˜ ϕ z ( y ) n ∂ b Ω ( y ) = D ˜ ϕ z ( y ) n ∂ ( Q z R n γ z ) ( y ) = D b ϕ z  ( Q z F z ) − 1 ( y )   D F z ( x ′ , 0)  − 1 Q T z n ∂ ( Q z R n γ z ) ( y ) = D b ϕ z  ( Q z F z ) − 1 ( y )   D F z ( x ′ , 0)  − 1 n ∂ R n γ z  F z ( x ′ , 0)  (3.136) By the construction of b ϕ z , we hav e D b ϕ z ( y ) =      0 D φ z ( y ′ ) . . . 0 0 . . . 0 ψ ′ z ( y n )      for all y ∈ R n (3.137) Consequen tly , since ψ ′ z (0) = 0, it holds that D b ϕ z  ( Q z F z ) − 1 ( y )  e n = D b ϕ z ( x ′ , 0) e n = 0 Th us, in view of ( 3.133 ), we conclude from ( 3.136 ) that ∇ ˜ ϕ z ( y ) · n ∂ b Ω ( y ) = 0 for all y ∈ ∂ b Ω ∩ U z . By the ab o ve construction, it is clear that the sets supp ˜ ϕ z , z ∈ ∂ b Ω, are an op en co ver of ∂ b Ω. Hence, since ∂ b Ω is compact, we can select z 1 , . . . , z N suc h that ∂ b Ω ⊂ N [ j =1 { ˜ ϕ z j > 0 } ⊂ N [ j =1 U z j . Therefore, in the following, w e simply write U j : = U z j , Q j : = Q z j , γ j : = γ z j , ˜ ϕ j : = ˜ ϕ z j for all j ∈ { 1 , . . . , N } . W e further ﬁnd an op en set U 0 ⊂ R n with U 0 ⊂ b Ω and a function ˜ ϕ 0 ∈ C ∞ c ( R n ; [0 , 1]) with supp ˜ ϕ 0 ⊂ U 0 suc h that b Ω ⊂ N [ j =0 { ˜ ϕ j > 0 } ⊂ N [ j =0 U j . 42 T o simplify the notation, we further set Q 0 : = I and R n γ 0 = R n . Finally , we deﬁne ϕ j : R n → R n , ϕ j ( x ) =    ˜ ϕ j ( x ) S ( x ) if S ( x ) > 0 , 0 if S ( x ) = 0 for all j ∈ { 0 , . . . , N } , where S ( x ) = N X j =0 ˜ ϕ j ( x ) for all x ∈ R n . This ensures that N X j =0 ϕ j = 1 on b Ω. By the ab ov e construction, the family { ϕ j } j =0 ,...,N ⊂ C 2 c ( R n ; [0 , 1]) ∩ W 3 , ∞ ( R n ) is a partition of unit y of b Ω, which satisﬁes b Ω ∩ U j = Q j R n γ j ∩ U j , supp ϕ j ⊂ U j , ∇ ϕ j · n ∂ b Ω = 0 on ∂ b Ω (3.138) for all j ∈ { 0 , . . . , N } . Step 3. Pro of of the k ey estimates ( 3.129 ) and ( 3.130 ) . Let v ∈ C 2 c  b Ω  ∩ W 2 ,p I ( b Ω) b e arbitrary . F or every j ∈ { 0 , . . . , N } , we further choose a function ψ j ∈ C ∞ 0 ( R n ; [0 , 1]) with supp ψ j ⊆ U j and ψ j = 1 on supp ϕ j . No w, we set v j : = v ϕ j for j = 0 , . . . , N . Using the pro duct rule along with ( 3.138 ), w e ﬁnd that v j ∈ C 2 c  Q j R n γ j  ∩ W 2 ,p I ( Q j R n γ j ) for all j ∈ { 0 , . . . , N } . Recalling that b L b Ω ε is linear, we use the decomp osition v = P N j =0 v j , to obtain b L b Ω ε v = N X j =0 b L b Ω ε v j = N X j =0  J ε j, 1 + J ε j, 2  . (3.139) with J ε j, 1 : = ψ j b L b Ω ε v j , J ε j, 2 : = (1 − ψ j ) b L b Ω ε v j (3.140) for j = 0 , . . . , N . W e will now estimate these summands separately . 43 A d J ε j, 1 : Let j ∈ { 0 , . . . , N } b e arbitrary . W e obtain J ε j, 1 ( x ) = ψ j ( x ) Z b Ω b J ε ( x − y )  v j ( x ) − v j ( y )  d y = J ε j, 1 , 1 + J ε j, 1 , 2 − J ε j, 1 , 3 with J ε j, 1 , 1 ( x ) : = ψ j ( x ) Z Q j R n γ j b J ε ( x − y )  v j ( x ) − v j ( y )  d y , J ε j, 1 , 2 ( x ) : = ψ j ( x ) Z b Ω \ U j b J ε ( x − y ) v j ( x ) d y , J ε j, 1 , 3 ( x ) : = ψ j ( x ) Z Q j R n γ j \ U j b J ε ( x − y ) v j ( x ) d y for all x ∈ b Ω. Since b ρ and b J satisfy (A1) and (A2) , Theorem 3.3 (for j = 0) and Corollary 3.5 (for j > 0) imply that ∥ J ε j, 1 , 1 ∥ L p ( b Ω) ≤ C ∥ v j ∥ W 2 ,p ( b Ω) ≤ C ∥ v ∥ W 2 ,p ( b Ω) . (3.141) W e further recall that supp v j ⊆ supp ϕ j ⊂ U j . Thus, for x ∈ supp ϕ j and y ∈ R n \ U j , it holds that | x − y | ≥ δ j with δ j : = dist (supp ϕ j , R n \ U j ) . Moreo ver, in voking the deﬁnition of b ρ (see ( 3.124 )) and Lemma 2.4(b) , a straigh t- forw ard computation yields | J ε j, 1 , 2 ( x ) | ≤ δ − 3 j | v j ( x ) | Z R n b ρ ε ( x − y ) | x − y | d y = ε δ − 3 j | v j ( x ) | Z R n b ρ ( x − y ) | x − y | d y ≤ C ε δ − 3 j | v j ( x ) | . Consequen tly , we obtain ∥ J ε j, 1 , 2 ∥ L p ( b Ω) ≤ C ε ∥ v j ∥ L p ( b Ω) ≤ C ε ∥ v ∥ L p ( b Ω) . (3.142) The term J ε j, 1 , 3 can b e estimated completely analogously and we end up with ∥ J ε j, 1 , 3 ∥ L p ( b Ω) ≤ C ε ∥ v ∥ L p ( b Ω) . (3.143) Com bining ( 3.144 ) and ( 3.145 ) and recalling that ε ∈ (0 , 1], we conclude that ∥ J ε j, 1 ∥ L p ( b Ω) ≤ C ∥ v ∥ W 2 ,p ( b Ω) . (3.144) for all j ∈ { 0 , . . . , N } . 44 A d J ε j, 2 : Let j ∈ { 0 , . . . , N } b e arbitrary . W e ﬁrst notice that 1 − ψ j ( x ) = 0 if x ∈ supp ϕ j . If x ∈ b Ω \ supp ψ j and y ∈ supp ϕ j , we hav e | x − y | ≥ ˜ δ j with ˜ δ j : = dist ( R n \ supp ψ j , supp ϕ j ) . Hence, for all x ∈ b Ω, it holds that | J ε j, 2 ( x ) | ≤  1 − ψ j ( x )  ˜ δ − 3 j Z b Ω b ρ ε ( x − y ) | x − y || v j ( y ) | d y . This directly implies ∥ J ε j, 2 ∥ p L p ( b Ω) ≤ C ˜ δ − 3 p j Z R n Z b Ω b ρ ε ( x − y ) | x − y | d y ! p − 1 · Z b Ω b ρ ε ( x − y ) | x − y || v j ( y ) | p d y ! d x ≤ C ˜ δ − 3 p j ε p − 1 Z b Ω  Z R n b ρ ε ( x − y ) | x − y | d x  | v j ( y ) | p d y ≤ C ε p ∥ v j ∥ p L p ( b Ω) ≤ C ε p ∥ v ∥ p L p ( b Ω) . (3.145) In summary , combining ( 3.144 ) and ( 3.145 ) to b ound the righ t-hand side of ( 3.139 ), we infer the estimate ∥ b L b Ω ε v j ∥ L p ( b Ω) ≤ C ∥ v ∥ W 2 ,p ( b Ω) for all v ∈ C 2 c  b Ω  ∩ W 2 ,p I ( b Ω). This means that key estimate ( 3.129 ) is established. In order to v e rify key estimate ( 3.130 ), let no w v ∈ C 3 c  b Ω  ∩ W 3 ,p I ( b Ω) b e arbitrary . Using the decomp osition ( 3.139 ) along with the linearity of the Laplacian, we deriv e the identit y b L b Ω ε v + ∆ v = N X j =0   J ε j, 1 , 1 + ∆ v j  + J ε j, 1 , 2 + J ε j, 1 , 3 + J ε j, 2  , (3.146) where J ε j, 1 , 1 , J ε j, 1 , 2 , J ε j, 1 , 3 and J ε j, 2 , j = 0 , . . . , N , are deﬁned as ab ov e. Since ψ j = 1 on supp v j , we hav e J ε j, 1 , 1 + ∆ v j = ψ j ( x ) Z Q j R n γ j b J ε ( x − y )  v j ( x ) − v j ( y )  d y + ∆ v j ! . Hence, applying Theorem 3.3 (for j = 0) and Corollary 3.5 (for j > 0), we deduce ∥ J ε j, 1 , 1 + ∆ v j ∥ L p ( b Ω) ≤ C p √ ε ∥ v j ∥ W 3 ,p ( b Ω) ≤ C p √ ε ∥ v ∥ W 3 ,p ( b Ω) . (3.147) 45 In combination with ( 3.142 ), ( 3.142 ) and ( 3.145 ), we conclude from ( 3.146 ) that ∥ b L b Ω ε v + ∆ v ∥ L p ( b Ω) ≤ C p √ ε ∥ v ∥ W 3 ,p ( b Ω) . This means that key estimate ( 3.130 ) is established and th us, the pro of of Theo- rem 3.6 is complete. References [1] H. Abels and C. Hurm. 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Nonlocal-to-local $L^p$-convergence of convolution operators with singular, anisotropic kernels

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