Quasi-local probability averaging in the context of cutoff regularization

Quasi-lo cal probabilit y a v eraging in the con text of cutoff regularization A. V. Iv ano v † I. V. Korenev ⋆ † St. P etersburg Departmen t of Steklo v Mathematical Institute of Russian Academ y of Sciences, 27 F on tank a, St. Petersburg, 191023, Russia † Sain t Petersburg State Universit y , 7/9 Universitetsk ay a Emb., St. Petersburg, 199034, Russia ⋆ National Research Universit y ”Higher School of Economics”, F acult y of Mathematics, 6 Usachev a St., Moscow, 119048, Russia † E-mail: regul1@mail.ru ⋆ E-mail: jacepo ol332@gmail.com Abstract. In this paper, w e study the properties of av eraged fundamental solutions of a sp ecial t yp e for Laplace op erators in the Euclidean space of an arbitrary dimension. W e consider a class of k ernels suitable for probabilistic a veraging, and prop ose new representations for the deformed fundamental solutions and their v alues at zero. In addition, we giv e examples related to sp ecific quantum field mo dels in the context of studying renormalization prop erties. Key words and phrases: regularization, cutoff, av eraging, Green’s function, fundamen tal solution, Laplace op erator, smo othing, scalar mo del, sigma mo del. 1 Con ten ts 1 In tro duction 2 2 Main prop erties 4 3 Pro ofs 6 4 New examples 10 4.1 The case t = s = 1 / 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 The case n = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 The case n = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5 Conclusion 17 6 References 18 1 In tro duction Main ob jects. Consider the Euclidean space R n , where n ∈ N , and in tro duce the Laplace op erator in the Cartesian co ordinates A n ( x ) = − P n i =1 ∂ x i ∂ x i . Next, let r > 0, then we can define the set of functions G 2 ( r ) = − ln( r ) 2 π and G n ( r ) = r 2 − n ( n − 2) S n − 1 for n  = 2 , (1) where S n − 1 = 2 π n/ 2 / Γ( n/ 2) denotes the area of the unit sphere in R n . In particular, S 0 = 2. It is kno wn, see section 3.1 in [1], that G n ( · ) is a fundamental solution for the op erator A n ( x ), which should b e understo o d as a solution to the equation A n ( x ) G n ( | x | ) = δ n ( x ) in the sense of generalized functions on the Sch w artz class S ( R n ), see [2, 3], where δ n ( · ) is the corresp onding δ -functional. Motiv ation. The notable p osition of the presented functions in theoretical physics is due to the fact, see [4–6], that they are the main approximation for Green’s functions near the diagonal on smo oth compact Riemannian manifolds. In particular, it is for this reason that they often arise in the study of p erturbative decomp ositions for quan tum field mo dels [7–9]. As is known, within the framew ork of this approach, non-linear combinations of Green’s functions and their deriv atives can appear, therefore, the regularization of Green’s functions, whic h consists in deforming certain parts of it, plays a separate imp ortan t role. This pap er fo cuses on studying some imp ortan t prop erties from a physical p oin t of view for a sp ecial deformation of the functions G n ( | · | ), see [10–12], whic h can b e obtained by applying the a veraging op erator twice. Av eraging op erator. Let us introduce the main elemen ts of the av eraging op erator O . Let Λ > 0 and a function ω ( | · | ) ∈ C ∞ ( R n , R ) with the prop erties supp( ω ) ⊂ [0 , 1 / 2] , Z R n d n x ω ( | x | ) = 1 . (2) Then we will assume that the a veraging op erator O Λ x , where the index ” x ” denotes the v ariable for whic h a veraging is p erformed, and the parameter Λ is resp onsible for the av eraging radius, acts on the function ϕ ( · ) ∈ L 1 , loc ( R n ) according to the following rule O Λ x ϕ ( x ) = Z B 1 / 2 d n y ϕ ( x + y / Λ) ω ( | x | ) . 2 Here, B 1 / 2 ⊂ R n denotes a closed ball of radius 1 / 2 centered at the origin. In physics, the parameter Λ is called regularizing, and the pro cess of removing regularization is the transition to the limit of Λ → + ∞ . Indeed, in this case, O Λ x ϕ ( x ) → ϕ ( x ) due to the prop erties of (2). Regularization. Before defining the regularized (deformed) Green’s function, w e need to explain the app earance of double av eraging in applications. It is known, see [13], that the co efficien ts of formal series in the framew ork of p erturbative quantum field theory are obtained b y calculating Gaussian integrals from p olynomials, whic h ultimately boil down to rep eated applications of Wick’s theorem on pairings. Roughly speaking, the field functions resp onsible for fluctuations near a solution of the equation of motion are replaced in pairs by the corresponding Green’s functions. In the main order, this can be written as follo ws ϕ ( x ) ϕ ( y ) → G n ( x − y ) . One of the recent approaches to the regularization of quantum field systems is the quasi-lo cal probability a veraging of fluctuation fields, see [14], that is, the transition ϕ ( x ) → O Λ x ϕ ( x ). Although the introduction of regularization using a smo othing in tegral op erator is not original, see, for example, a cutoff in the momen tum represen tation [15–17] within the framework of the functional renormalization group or the metho d with higher co v ariant deriv ativ es [18–20], the approach with av eraging o v er a ”small” neighbor- ho od is new. Its systematic study w as initiated in the study of the cubic mo del [21] and significan tly impro ved in subsequent works, see [14, 22] and references therein. The w ord ”quasi-lo calit y” in this case is asso ciated with the presence of a small area (a ball of radius 1 / (2Λ)) near a fixed p oint x for av eraging. After such a transformation, the Wick’s theorem on pairing works according to the rule O Λ x ϕ ( x ) O Λ y ϕ ( y ) → O Λ x O Λ y G n ( x − y ) ≡ G Λ n,ω ( | x − y | ) and, thus, leads to the app earance of the double av eraging. Next, by quasi-lo cal av eraging of the funda- men tal solution G n ( | · | ), we name the transformation of the form G n ( | x | ) → G Λ n,ω ( | x | ) = Z R n d n y Z R n d n z G n ( | x + y / Λ + z / Λ | ) ω ( | y | ) ω ( | z | ) . (3) In this case, we call quasi-lo cal av eraging probabilistic if, in addition to the prop erties of (2), ω ( · ) ⩾ 0 is v alid almost everywhere. Ab out representations. Note that it is sufficient to define the deformed function only for Λ = 1, since the following large-scale transformations are v alid for the initial functions (1) G 2 ( r / Λ) = G 2 ( r ) + ln(Λ) 2 π and G n ( r / Λ) = Λ 2 − n G n ( r ) for n  = 2 , (4) whic h are preserved when switching to the regularized ob jects. Therefore, further, assuming Λ = 1, we will omit the corresp onding index. It w as previously shown, see Section 3 in [14], that the deformed function G Λ n,ω ( | · | ) admits the following representation G n,ω ( | x | ) = G f n ( | x | ) = f n ( | x | 2 ) + ( G n (1) , | x | ⩽ 1; G n ( | x | ) , | x | > 1 , where the auxiliary function f n ( · ) has the set of prop erties f n ( · ) ∈ C ([0 , + ∞ ) , R ) , supp( f n ) ⊂ [0 , 1] , f n (1) = 0 . Structure of the w ork. This article consists of an introduction, three main sections and a conclusion. Here is a brief description of the sections and the corresp onding results. — Section 2 contains the basic prop erties of the function (3) and kernels of the av eraging op erator, which are often used in practice, for example, in the framew ork of p erturbativ e quan tum field theory in the 3 searc h for coefficients of renormalization constan ts. Theorem 1 includes an explicit represen tation, see (6), for double a v eraging of the function G n ( | · | ) ov er spheres of arbitrary radius, as well as its basic prop erties such as smoothness and monotonicity interv als. Note that some in termediate results related to the theory of double av eraging ha ve app eared earlier, see for example section 1.5 in [23], dev oted to con volutions of δ -functionals. Nevertheless, in the context of regularization, this topic is new, contin uing the series of papers [10, 11], therefore the theorem is useful from the point of view of further applications. Theorem 2 is devoted to an example of a suitable class of k ernels ω ( | · | ), as well as to the deriv ation of represen tations for some sp ecial functionals. — Section 3 provides pro ofs for the statements made in the previous section. — Section 4 examines new examples that are important from the point of view of applications in the framew ork of perturbative quantum field theory and renormalization theory . The section consists of three parts. ( i ) Section 4.1 pro vides an example with double av eraging ov er spheres of the same radius. This choice is very p opular in scalar mo dels, see [22] and the corresp onding references in the introduction. Moreo ver, it represents an extreme case, namely , it minimizes the maximum v alue of the deformed Green’s function, and thus leads to the extreme case of the renormalized mass. ( ii ) Section 4.2 provides an explicit calculation for a family of kernels in three-dimensional space. This result is imp ortan t in the con text of studying the prop erties of the sextic model [24] with the cutoff in the coordinate represen tation, see [25, 26], since the free parameter con tin uously connects the extreme case with the limiting situation of regularization remov al. This fact gives an additional degree of freedom when fixing the renormalization co efficients. ( iii ) Section 4.3 discusses a family of k ernels in the t w o-dimensional case, which are an example of mixed regularization, that is, com bining cutoffs in the coordinate and momen tum representations. Using the example of the t wo-dimensional non-linear sigma mo del [27] with the cutoff regularization, see [28–30], it is shown that this approach leads to additional freedom, allo wing us to make special- t yp e functionals (functions θ i from [30]) equal to zero. — Section 5 contains concluding remarks and op en questions. 2 Main prop erties Theorem 1. L et n ∈ N and x, y , z ∈ R n . L et us intr o duc e notation for the absolute values of r = | x | , s = | y | , and t = | z | . The symb ol S n − 1 denotes the unit ( n − 1) -dimensional spher e c enter e d at the origin. Consider an inte gr al of the form k n ( r , s, t ) = Z S n − 1 d n − 1 σ ( ˆ y ) S n − 1 Z S n − 1 d n − 1 σ ( ˆ z ) S n − 1 G n ( | x + s ˆ y + t ˆ z | ) , (5) wher e ˆ y = y / | y | , ˆ z = z / | z | , and also d n − 1 σ ( ˆ y ) denotes the standar d me asur e on the spher e, which is normalize d by the numb er S n − 1 and, at n = 1 , de gener ates into the summation over the set {± 1 } . T aking into ac c ount al l the ab ove, the r elation holds k n ( r , s, t ) =      G n (max( t, s )) , r < | t − s | ; G n ( t + s ) + g n ( r , s, t ) , | t − s | ⩽ r ⩽ | t + s | ; G n ( r ) , r > | t + s | , (6) wher e g n ( r , s, t ) = 2 n − 2 S n − 2 S 2 n − 1 Z t + s r d u u 1 − n Z ( u 2 − ( t − s ) 2 ) / (4 ts ) 0 d p p n − 3 2 (1 − p ) n − 3 2 (7) 4 for n > 1 and g 1 ( r , s, t ) = ( r − t − s ) / 2 . In this c ase, the fol lowing four sets of pr op erties ar e valid. 1) The function k n ( r , s, t ) is c ontinuous, symmetric, and de cr e asing with r esp e ct to e ach ar gument on the set R 3 = R 3 + \ { 0 , 0 , 0 } for al l n ∈ N . Mor e over, the r atio is c orr e ct k n ( r , s, t ) > 0 for n ⩾ 3 and ( r, s, t ) ∈ R 3 . 2) F or fixe d ( s, t ) ∈ R 2 + \ { 0 , 0 } and n ⩾ 3 the function k n ( · , s, t ) r e aches its maximum at R + at p oint 0 , which is k n (0 , s, t ) = G n (max( t, s )) . 3) L et ( s, t ) ∈ R 2 + \ { 0 , 0 } ar e fixe d and n ⩾ 2 , then, with r esp e ct to the variable r , the function ∂ r k n ( r , s, t ) is de cr e asing and c ontinuous on R + . In addition, the r elations ar e c orr e ct ∂ r k n ( r , s, t ) = 0 for r ⩽ | t − s | , ∂ r k n ( r , s, t ) < 0 for r > | t − s | . In the c ase of n = 1 , the function has disc ontinuities of the typ e ”jump” for r = | t − s | and r = | t + s | . 4) The function A n ( x ) k n ( r , s, t ) is symmetric on R 3 . At the same time, for n ⩾ 4 the function is c ontinuous on R 3 , and for n = 3 , 2 and al l fixe d ( s, t ) ∈ R 2 + \ { 0 , 0 } the function has br e ak p oints 1 in r ∈ {| t − s | , | t + s |} , of the first kind for n = 3 and of the se c ond kind for n = 2 , and is c ontinuous on the set R + \ {| t − s | , | t + s |} . In addition, for n ⩾ 2 , the r elations ar e c orr e ct A n ( x ) k n ( r , s, t ) = 0 for r ∈ R + \ [ | t − s | , | t + s | ] , A n ( x ) k n ( r , s, t ) < 0 for r ∈ ( | t − s | , | t + s | ) . Theorem 2. L et the assumptions fr om The or em 1 b e fulfil le d, Ω = (0 , 1 / 2) , n ∈ N , and a function 2 is define d as ω ( t ) = t − ( n − δ 1 n ) / 2+ ν n w ( t ) , wher e the p ar ameter ν n > 0 and w ∈ L 1 , + (Ω) = { u ∈ L 1 (Ω) : u ⩾ 0 almost everywher e } . Then the deforme d function (3) al lows a r epr esentation of the form G n,ω ( r ) = S 2 n − 1 Z 1 / 2 0 d s Z 1 / 2 0 d t s n − 1 t n − 1 ω ( s ) ω ( t ) k n ( r , s, t ) , (8) wher e x ∈ R n and | x | = r , and the fol lowing four statements ar e true. 1) The function G n,ω ( · ) is c ontinuous and de cr e asing on R + . At the same time, for n ⩾ 3 , it is b ounde d fr om b elow by zer o, that is, 0 < G n,ω ( r ) for al l r ∈ R + . In addition, for n  = 2 , formula (8) is valid for the extr eme c ase ν n → +0 . The maximum is r e ache d at the p oint r = 0 and is e qual to G n,ω (0) = 2 S 2 n − 1 Z 1 / 2 0 d s s n − 1 G n ( s ) ω ( s ) Z s 0 d t t n − 1 ω ( t ) ≡ I 1 [ w ] . (9) 2) F or ν n > 0 the value of G n,ω (0) c an b e r ewritten in the e quivalent form G n,ω (0) = G n (1 / 2) + S n − 1 Z 1 / 2 0 d s s 1 − n  Z s 0 d t t n − 1 ω ( t )  2 ≡ I 2 [ w ] . (10) Mor e over, in the c ase of n = 1 , the formula is also true for ν 1 = 0 . 1 See Paragraph 2 of Chapter 2 in [33]. 2 Note that the multiplier t − ( n − δ 1 n ) / 2+ ν n originally app eared for conv enience reasons when describing a class of v alid kernels for whic h the transition from (9) to (10) is possible. 5 3) If ν n ⩾ 1 / 2 for al l n ⩾ 2 or ν 1 ⩾ 0 , then the function ∂ r G n,ω ( r ) is c ontinuous on R + and has two pr op erties ∂ r G n,ω ( r )    r =0 = 0 , ∂ r G n,ω ( r )    r ⩾ 1 = − r 1 − n S n − 1 . 4) If ν n ⩾ 1 / 2 for al l n ⩾ 2 or ν 1 ⩾ 0 , and also w ∈ L 2 (Ω) , then the function A n ( x ) G n,ω ( r ) is c ontinuous on R + and has two pr op erties A n ( x ) G n,ω ( r )    r =0 =     | · | ν n − (1 − δ 1 n ) / 2 w ( | · | )     2 L 2 (Ω) , A n ( x ) G n,ω ( r )    r ⩾ 1 = 0 . (11) Corollary 1. L et w ∈ L 1 , + (Ω) fr om The or em 2. L et ther e b e such a numb er 1 / 2 > ϵ > 0 , that w ( r ) = 0 for almost al l r ∈ [0 , ϵ ] . Then the first thr e e p oints of The or em 2 ar e true for al l ν n ∈ R . Besides, the fol lowing estimate is valid G n (1 / 2) ⩽ G n,ω (0) ⩽ G n ( ϵ ) . (12) 3 Pro ofs Pro of of Theorem 1. Let us first consider the case of n > 2. Let us apply the Laplace op erator A n ( x ) to the kernel k n ( r , s, t ), then we get A n ( x ) k n ( r , s, t ) = Z S n − 1 d n − 1 σ ( ˆ y ) S n − 1 Z S n − 1 d n − 1 σ ( ˆ z ) S n − 1 δ n ( x + s ˆ y + t ˆ z ) , (13) where the equation A n ( x ) G n ( | x | ) = δ n ( x ) was used. Recall that δ n ( · ) is the corresp onding δ -functional on the class S ( R n ), see [2, 3]. Note that this function has been studied b efore, see pages 381 and 404 in the monograph [31] and page 585 in [32]. Nevertheless, a short summary of the conclusion is useful from the p oin t of view of general understanding. Let us use the relation for the in ternal integral, see formula (14) in [11], Z S n − 1 d n − 1 σ ( ˆ z ) δ n ( x + s ˆ y + t ˆ z ) = t 1 − n δ 1 ( | x + s ˆ y | − t ) , and let us mo v e to h yp erspherical co ordinates, then writing out the explicit formula for d n − 1 σ ( ˆ z ) using the angular co ordinates and additionally calculating the in tegral o ver the ( n − 2)-dimensional h yp ersphere, w e obtain a representation for the right side of (13) in the form t 1 − n S n − 2 S 2 n − 1 Z π 0 d ϕ sin n − 2 ( ϕ ) δ 1  v ( ϕ )  , where the auxiliary function was defined v ( ϕ ) =  r 2 + 2 r s cos( ϕ ) + s 2  1 / 2 − t. The search for zeros for this function reduces to solving the equation cos( ϕ c ) = t 2 − r 2 − s 2 2 r s . Considering the parameters t and s fixed, the equation has a nonzero solution in the interv al [0 , π ] only if the condition is fulfilled | t − s | ⩽ r ⩽ | t + s | . Th us, calculating the v alue of the first deriv ativ e ˙ v ( ϕ c ) = − r s sin( ϕ c ) v ( ϕ c ) + t = − r s t s 1 −  t 2 − r 2 − s 2 2 r s  2 6 and replacing the v ariables in the functional δ 1  v ( ϕ )  = δ 1 ( ϕ − ϕ c ) | ˙ v ( ϕ c ) | , w e obtain the following result for integral (13) θ ( r − | t − s | ) θ ( | t + s | − r ) 2 S n − 2 S 2 n − 1 (2 tsr ) 2 − n   r 2 − ( t − s ) 2  ( t + s ) 2 − r 2 )   n − 3 2 , (14) where θ ( · ) denotes the Heaviside (step) function. Next, represen ting the Laplace op erator as follows − r 1 − n ∂ r r n − 1 ∂ r and in tegrating o ver the v ariable r , taking into account contin uous gluing at p oin ts r = | t − s | and r = | t + s | and equalities k n ( r , s, t ) = G n ( r ) for r > | t + s | , w e obtain formula (6). Note that, kno wing the answer, w e can simply c hec k the fulfillmen t of all the prop erties and conditions presen ted in the formulation of the theorem. Indeed, k n ( r , s, t ) is contin uous, since the function from (7) satisfies the conditions g n ( | t − s | , s, t ) = G n (max( t, s )) − G n ( t + s ) , g n ( | t + s | , s, t ) = 0 . (15) The second follo ws from the direct substitution of r = t + s , while the first is obtained in several steps. T o chec k, select r = | t − s | in (7) and replace the v ariable with u 2 = 4 tsv + ( t − s ) 2 , then we get 2 n − 1 S n − 2 ts S 2 n − 1 Z 1 0 d v  4 tsv + ( t − s ) 2  − n 2 Z v 0 d p p n − 3 2 (1 − p ) n − 3 2 . Next, we c hange the order of in tegration and explicitly calculate the inner integral. When calculating, for conv enience, w e can assume that s  = t , understanding the situation s = t , which was studied in [11], as limit case. W e hav e − 2 n − 2 S n − 2 S 2 n − 1 | t + s | 2 − n n − 2 B  n − 1 2 , n − 1 2  + 2 n − 2 S n − 2 | t − s | 2 − n S 2 n − 1 ( n − 2) P n ( t, s ) , where B( · , · ) is the Euler’s b eta-function and P n ( t, s ) is defined by the equality P n ( t, s ) = Z 1 0 d p p n − 3 2 (1 − p ) n − 3 2  1 − p  − 4 ts ( t − s ) 2  2 − n 2 . Then we use the equalities, see paragraphs 8.383, 9.121, and 9.131 in [34], 2 n − 2 S n − 2 S n − 1 B  n − 1 2 , n − 1 2  = 1 , P n ( t, s ) = Γ 2  ( n − 1) / 2  Γ( n − 1) 2 F 1  n − 2 2 , n − 1 2 , n − 1; − 4 ts ( t − s ) 2  , 2 F 1  n − 2 2 , n − 1 2 , n − 1; − z  = 2 n − 2  1 + √ 1 − z  2 − n , | t − s | + t + s = 2 max( t, s ) , where z > 0. Finally , after substituting the men tioned relations and reducing iden tical terms, w e come to the first relation from (15). Thus, the contin uity of k n ( r , s, t ) o v er the v ariable r is sho wn. In this case, the function is strictly p ositiv e. Calculate the first deriv ativ e for (6) ∂ r k n ( r , s, t ) = − r 1 − n S n − 1      0 , r < | t − s | ; h n ( r , s, t ) , | t − s | ⩽ r ⩽ | t + s | ; 1 , r > | t + s | , (16) 7 where h n ( r ; s, t ) = 2 n − 2 S n − 2 S n − 1 Z ( r 2 − ( t − s ) 2 ) / (4 ts ) 0 d p p n − 3 2 (1 − p ) n − 3 2 . (17) Let us verify the con tin uity of the function. F or r = | t − s | the equality h n ( | t − s | ; s, t ) = 0 is v alid. F or r = t + s , using the ab o ve mentioned relations, w e obtain in a similar wa y h n ( t + s ; s, t ) = 1. Therefore, the first deriv ativ e is a con tin uous non-positive decreasing function. At the same time, it is zero only when r ⩽ | t − s | . Next, applying the op erator − r 1 − n ∂ r r n − 1 , we mak e sure that it is consisten t with the presen tation (14). Note that the symmetry of the functions k n ( r , s, t ) and A n ( x ) k n ( r , s, t ) follows from the fact that they dep end on the absolute v alue of | x | . Indeed, in this case, we can add av eraging o ver the v ariable ˆ x = x/r , whic h lead to a symmetric form in form ulas (5) and (13). Th us, taking into accoun t the mentioned part of the proof, we obtain prop erties 1)–4) for n > 2. F ollowing similar steps, w e v erify the v alidit y of relation (6) for n = 1 , 2 and the corresp onding properties. The theorem is prov ed. Pro of of Theorem 2. The first p oint. Let t ∈ Ω and ω ( t ) = t − ( n − δ 1 n ) / 2+ ν n w ( t ), where w ∈ L 1 , + (Ω) and ν n ⩾ 0. Using prop ert y 1 from Theorem 1 and the non-negativit y of the k ernel ω ( · ) for almost all v alues of the argumen t, w e obtain decreasing of the function G n,ω ( · ) on R + . In this case, the low er b oundedness for n ⩾ 3 follows from the b oundedness of the function k n . Next, using prop erty 2 of Theorem 1, w e obtain explicit form ula (9) for G n,ω (0). In this case, the integral is finite, since the estimates are v alid Z s 0 d t t n − 1 ω ( t ) = Z s 0 d t t ( n + δ 1 n ) / 2 − 1+ ν n w ( t ) ⩽ s ( n + δ 1 n ) / 2 − 1+ ν n || w || L 1 ,    I 1 [ w ]    ⩽ 2 S 2 n − 1 || w || 2 L 1 max s ∈ [0 , 1 / 2]  s n + δ 1 n − 2+2 ν n G n ( s )  . A t the same time, if n  = 2, then the transition ν n → +0 is v alid, since the function in paren theses remains con tinuous at [0 , 1 / 2]. The first p oint is prov ed. The second p oint. Note that if ω ∈ C ∞ c (Ω) then in I 1 [ w ] we can integrate in parts, even tually getting the stated answer (10). In this case, the restriction of ν n > 0 on the parameter leads to the fact that the function t 2 ν n − 1 b elongs to L 1 (Ω). T urning to generalization, let us use the density of the set C ∞ c (Ω) in L 1 (Ω), see Corollary 4.23 in [35]. Let the sequence { w i ∈ C ∞ c (Ω) } + ∞ i =1 tend to w in L 1 (Ω), that is, for any ε > 0 there is such a n um b er N ∈ N , that || w i − w || L 1 < ε for all i > N . Moreo ver, we can additionally assume that || w i || L 1 < (1 + i − 1 ) || w || L 1 . In this case, the estimates are v alid for n ⩾ 3 and i > N   I 1 [ w ] − I 2 [ w ]   ⩽   I 1 [ w ] − I 1 [ w i ]   +   I 1 [ w i ] − I 2 [ w ]   =   I 1 [ w ] − I 1 [ w i ]   +   I 2 [ w i ] − I 2 [ w ]   ,   I 1 [ w ] − I 1 [ w i ]   ⩽ 2 S n − 1 n − 2  || w || L 1 + || w i || L 1  || w i − w || L 1 ⩽ ε 6 S n − 1 n − 2 || w || L 1 ,   I 2 [ w ] − I 2 [ w i ]   ⩽ 2 − 2 ν n S n − 1 2 ν n  || w || L 1 + || w i || L 1  || w i − w || L 1 ⩽ ε 3 S n − 1 2 ν n || w || L 1 . Hence, we get   I 1 [ w ] − I 2 [ w ]   ⩽ 3 ε || w || L 1 S n − 1 4 ν n − 2 + n 2( n − 2) ν n . Since the left part do es not dep end on ε and the index i , we obtain the desired equality of integrals. The cases of n = 1 , 2 are treated similarly . In this case, for n = 1, the case of ν 1 = 0 is v alid as well. The second p oint is pro ved. The third p oint. Let n ⩾ 2. Let us consider the first deriv ativ e ∂ r G n,ω ( r ). W e immediately note the equalit y ∂ r G n,ω ( r ) = ∂ r G n ( r ) in the region r ⩾ 1, which follows from the explicit form ula (6). Next, let us use representation (16) and study the function ˆ h n ( r ) = r 1 − n h n ( r , s, t ) in more detail with fixed 8 ( s, t ) ∈ Ω × Ω \ { u × u : u ∈ Ω } in the interv al | t − s | ⩽ r ⩽ t + s . T o do this, first, in integral (17), we scale the v ariable p → p κ , where κ = ( r 2 − ( t − s ) 2 )) / (4 ts ), then we get ˆ h n ( r ; s, t ) = 2 n − 2 S n − 2 S n − 1 ( κ /r 2 ) n − 1 2 Z 1 0 d p p n − 3 2 (1 − p κ ) n − 3 2 . Note that for the specified range of v alues of r , the parameter κ ∈ [0 , 1]. Therefore, the estimates are v alid κ /r 2 = 1 4 ts  1 − ( t − s ) 2 r 2  ⩽ 1 ( t + s ) 2 , 1 − p ⩽ 1 − p κ ⩽ 1 . Next, consider the case of n ⩾ 3. Using the b oth upp er b ounds, we obtain ˆ h n ( r ; s, t ) ⩽ 2 n − 2 S n − 2 S n − 1 ( t + s ) n − 1 Z 1 0 d p p n − 3 2 ⩽ 2 n − 2 S n − 2 S n − 1 ( t + s ) n − 1 . In the case n = 2, the low er b ound should b e used for the integral expression, then we get ˆ h 2 ( r ; s, t ) ⩽ S 0 S 1 ( t + s ) Z 1 0 d p p − 1 2 (1 − p ) − 1 2 = S 0 π S 1 ( t + s ) . Th us, defining the auxiliary v alue C n = π 2 n − 2 S n − 2 S n − 1 ⩾ 1 , and noticing that in the range r ∈ [ t + s, 1] the inequalit y C n | t + s | 1 − n ⩾ r 1 − n holds, we get an estimate of the form | ∂ r k n ( r , s, t ) | ⩽ C n S n − 1 ( 0 , | t − s | > r ⩾ 0; | t + s | 1 − n , | t − s | ⩽ r < 1 , ! ⩽ C n | t + s | 1 − n S n − 1 . (18) Using the second inequality and integrating, for r ∈ [0 , 1), we obtain an inequality of the form    ∂ r G n,ω ( r )    ⩽ C n S n − 1 Z 1 / 2 0 d s Z 1 / 2 0 d t ( st ) n/ 2 − 1+ ν n ( s + t ) n − 1 w ( s ) w ( t ) ⩽ 2 (1 − n ) / 2 C n S n − 1 || w || 2 L 1 , where the inequalities 2 st < ( s + t ) 2 and ( st ) ν n − 1 / 2 ⩽ 1 were used. Th us, the function ∂ r G n,ω ( r ) is b ounded and con tinuous for ν n ⩾ 1 / 2. Let us show that it is zero at r = 0. Note that w hen r → +0, after integrating the first inequality from (18), the estimate follows    ∂ r G n,ω ( r )    ⩽ 2 C n S n − 1  Z 1 / 2 r Z s s − r + Z r 0 Z s 0  d s d t ( st ) n/ 2 − 1+ ν n ( s + t ) n − 1 w ( s ) w ( t ) ⩽ 2 (5 − n ) / 2 C n S n − 1 || w || L 1 max s ∈ [ r, 1 / 2]  Z s s − r d t w ( t )  , the righ t-hand side of which tends to zero when r → +0. Hence, we get ∂ r G n,ω ( r )   r =0 = 0. Similarly , taking into account explicit form (7), the case n = 1 can b e analyzed. The third p oint is prov ed. The fourth p oint. Let us mov e on to the consideration of A n ( x ) G n,ω ( r ). In this case, it is conv enien t to use the represen tation of (3), then for ω ( | · | ) ∈ L 2 (B 1 / 2 ), where B 1 / 2 = { x ∈ R n : | x | < 1 / 2 } , the equality holds A n ( x ) G n,ω ( x ) = Z R n d n y ω ( | x − y | ) ω ( | y | ) , In this case, L 2 (B 1 / 2 ) is understoo d as a natural subset of L 2 ( R n ), that is, ω ( t ) = 0 for all t > 1 / 2. Next, w e observ e that the norm in L 2 (B 1 / 2 ), taking into accoun t the spherical symmetry of the function, can b e rewritten as follows     ω ( | · | )     2 L 2 (B 1 / 2 ) = S n − 1 Z 1 / 2 0 d s  s ν − (1 − δ 1 n ) / 2 w ( s )  2 = S n − 1     | · | ν − (1 − δ 1 n ) / 2 w ( | · | )     2 L 2 (Ω) . 9 Then, taking in to accoun t the restrictions for the supports, w e get the properties from (11). Consequen tly , the fourth p oint has also b een verified, and the theorem is completely prov ed. Pro of of Corollary 1. Since ω ( r ) = 0 for almost all r ∈ [0 , ϵ ], then t a ω ( t ) b elongs to the class L 1 , + (Ω). Therefore, by choosing the parameter a appropriately , it is p ossible to satisfy the co nditions of the first three p oin ts of Theorem 2. Relation (12) follows from the application of form ula (10) using an estimate of the form S n − 1 Z 1 / 2 0 d s s 1 − n  Z s 0 d t t n − 1 ω ( t )  2 ⩽ || ω || 2 L 1 (B 1 / 2 ) S n − 1 Z 1 / 2 ϵ d s s 1 − n = G n ( ϵ ) − G n (1 / 2) , where definition (1) and condition (2) w ere taken in to account. Corollary 1 is prov ed. 4 New examples 4.1 The case t = s = 1 / 2 . Let the notations from Theorem 1 b e true. Substitute the parameter v alues in to form ulas (6) and (7), then we get k n ( r , 1 / 2 , 1 / 2) = ( G n (1) + g n ( r , 1 / 2 , 1 / 2) , r ⩽ 1; G n ( r ) , r > 1 , where g n ( r , 1 / 2 , 1 / 2) = 2 n − 2 S n − 2 S 2 n − 1 Z 1 r d u u 1 − n Z u 2 0 d p p n − 3 2 (1 − p ) n − 3 2 for n > 1 and g 1 ( r , 1 / 2 , 1 / 2) = ( r − 1) / 2. This c hoice of parameters corresp onds to the k ernel (2) of a sp ecial t yp e ω ( | x | ) → ˆ ω ( | x | ) = 2 n − 1 δ 1 ( | x | − 1 / 2 + ϵ ) /S n − 1 when ϵ → +0, which should b e understoo d in the sense of generalized functions, see [2, 3]. Normalization can b e chec k ed by direct substitution and switc hing to spherical co ordinates. Here, the auxiliary regularization with the parameter ϵ reveals the am biguity at the b oundary of the domain, since in tegration occurs ov er | x | ∈ [0 , 1 / 2]. Note that this c hoice of kernel does not fit into Theorem 2, since the generalized function ˆ ω ( | · | ) is not contained in the set L 1 , + (Ω). Nevertheless, suc h a v arian t exists and can be obtained as a limit of functions from L 1 , + (Ω). Indeed, let us choose 3 a sequence of non-negative functions of the form ˆ ω k ( r ) = r 1 − n S n − 1      0 , 0 ⩽ r < 1 / 2 − 1 /k ; k , 1 / 2 − 1 /k ⩽ r ⩽ 1 / 2; 0 , 1 / 2 < r. It is clear that ˆ ω k ( | · | ) → ˆ ω ( | · | ) for k → + ∞ in the sense of generalized functions on C ( R + ), and, moreo ver, the normalization is v alid Z R n d n x ˆ ω k ( r ) = 1 . This example shows that the class of k ernels is broader than the one described in Theorem 2. How ev er, the found representativ e of a broader class can b e obtained by appropriate limit transition. Thus, we get G n, ˆ ω ( | x | ) = Z R + d s Z R + d t s n − 1 t n − 1 2 2 n − 4 δ 1 ( t − 1 / 2) δ 1 ( s − 1 / 2) k n ( r , s, t ) = k n ( r , 1 / 2 , 1 / 2) and f n ( | x | 2 ) = G n, ˆ ω ( | x | ) − ( G n (1) , r ⩽ 1; G n ( r ) , r > 1 , = g n ( r , 1 / 2 , 1 / 2) θ (1 − r ) . 3 Note that this is not the only choice. As an example, w e can tak e any δ -shap ed sequence on the segment [0 , 1 / 2] conv erging to the functional ˆ ω ( | x | ). 10 Suc h an example of regularization has b een discussed in recen t pap ers. In particular, see [12], it w as sho wn that f n ( | x | 2 ) can b e represen ted b y a finite sum of elementary functions for an y n ∈ N . Explicit examples for n ∈ { 3 , . . . , 6 } are given in Corollary 1 of [11]. In conclusion, we note that the kernel ˆ ω ( | · | ) giv es the minimum v alue for the regularized function G n,ω ( · ) from (10) at zero, whic h can b e achiev ed, for example, by limit transition. Indeed, let ω = ˆ ω k , then G n, ˆ ω k (0) = G n (1 / 2) + 1 S n − 1 Z 1 / 2 1 / 2 − 1 /k d s s 1 − n  s − 1 / 2 + 1 /k  2 k → + ∞ − − − − − → G n, ˆ ω (0) = G n (1 / 2) . In tuitively , this result is highly expected. Indeed, the function G n ( · ) is decreasing in the sp ecified di- mensions. Therefore, the minim um v alue of the a v eraged function should be achiev ed on k ernels whose densit y is concen trated at the upp er b oundary of the segment [0 , 1 / 2]. This is exactly the option that is ac hieved by a veraging o ver the sphere. 4.2 The case n = 3 . Substitute the index v alue, then S 2 = 4 π , S 1 = 2 π . In this case, form ula (7) can b e written out explicitly , but it is not conv enien t to calculate the integral with suc h a function, so we use (14), then we get A 3 ( x ) k 3 ( r , s, t ) = θ ( r − | t − s | ) θ ( | t + s | − r ) 1 8 π tsr , and, as a consequence, we get the formula A 3 ( x ) G 3 ,ω ( r ) = 16 π 2 Z 1 / 2 0 d s Z 1 / 2 0 d t s 2 ω ( s ) t 2 ω ( t ) A 3 ( x ) k 3 ( r , s, t ) . Note that for each fixed r > 0, the integration domain is defined by the following conditions | t − s | ⩽ r , | t + s | ⩾ r , s, t ∈ [0 , 1 / 2] , therefore, the integral can b e represented as follows A 3 ( x ) G 3 ,ω ( r ) = 1 4 π      A 3 ( x ) R 1 ( r ) , r < 1 / 2; A 3 ( x ) R 2 ( r ) , 1 / 2 ⩽ r ⩽ 1; 0 , r > 1 , where A 3 ( x ) R 1 ( r ) = 8 π 2 r  Z 1 / 2 0 Z 1 / 2 0 − 2 Z 1 / 2 r Z s − r 0 − Z r 0 Z r − s 0  d s d t sω ( s ) tω ( t ) , A 3 ( x ) R 2 ( r ) = 8 π 2 r Z 1 / 2 r − 1 / 2 Z 1 / 2 r − s d s d t sω ( s ) tω ( t ) . F or example, consider a kernel of the form ω α ( t ) = ρ α e αt / (2 π t ), where α ∈ R and ρ α is a normalization constan t, see (2), and is equal to ρ α = α 2 ( α − 2) e α/ 2 + 2 . Note that the denominator turns to zero only at α = 0. Moreo ver, the asymptotic expansion near zero has the form α 2 (1 + o (1)) / 4, and, therefore, the function ρ α > 0 for all α ∈ R . Substituting the function ω α ( · ) into the integrals, we obtain A 3 ( x ) R α, 1 ( r ) = 2 ρ 2 α α 2 r  e α − e α (1 − r ) − αr e αr  , A 3 ( x ) R α, 2 ( r ) = 2 ρ 2 α α 2 r  e α − e αr − α (1 − r ) e αr  . 11 F urther, integrating and taking in to account the contin uity conditions R α, 1 ( r )   r =1 / 2 − 0 = R α, 2 ( r )   r =1 / 2+0 , ∂ r R α, 1 ( r )   r =1 / 2 − 0 = ∂ r R α, 2 ( r )   r =1 / 2+0 , R α, 2 ( r )   r =1 − 0 = r − 1   r =1+0 = 1 , ∂ r R α, 2 ( r )   r =1 − 0 = ∂ r r − 1   r =1+0 = − 1 , w e get an answer in the form G 3 ,ω α ( r ) = 1 4 π      R α, 1 ( r ) , r < 1 / 2; R α, 2 ( r ) , 1 / 2 ⩽ r ⩽ 1; r − 1 , r > 1 , where auxiliary functions are defined as follo ws R α, 1 ( r ) = 1 + 2 ρ 2 α α 2  e α 2 (1 − r ) + e α α 2  2 − 2 e − α/ 2 − 1 − e − αr r  − 2 α 2  2 − 2 e α/ 2 − 1 − e αr r  − 1 α  e α/ 2 − e αr  + (3 + α ) α 2  − 1 − e α + 2 e α/ 2  + 1 α  e α − e α/ 2  − 1 α 2  e α − e α/ 2 (2 α − 4) − α − 5)   , R α, 2 ( r ) = 1 + 2 ρ 2 α α 2  e α 2 (1 − r ) + (3 + α ) α 2  1 − e α − 1 − e αr r  + 1 α  e α − e αr  + 1 − r − 1 α 2  e α − e α/ 2 (2 α − 4) − α − 5)   . The following equalities are of particular in terest || ω || L 1 = 1 , G 3 ,ω α (0) = α 2 π e α ( α − 3) + 4 e α/ 2 − 1 ( e α/ 2 ( α − 2) + 2) 2 ≡ φ α , A 3 G 3 ,ω α   r =0 = α 3 2 π e α − 1 ( e α/ 2 ( α − 2) + 2) 2 = || ω || L 2 ≡ ψ α . Let us tak e a closer look at prop erties of φ α and ψ α . The function φ α is strictly decreasing on R . At the same time, we hav e φ α   α →−∞ = + ∞ , φ α   α → + ∞ = 1 2 π = G 3 (1 / 2) . The function ψ α > 0, for all v alues, has unique p oint α c ≈ 3 . 72 > 0, in which the first deriv ativ e ˙ ψ α c = 0 is equal to zero, strictly decreases on ( −∞ , α c ), and strictly increases on ( α c , + ∞ ). At the same time, the limits hav e the form ψ α   α →±∞ = + ∞ . This b ehavior is easily in terpretable. Indeed, in the limit α → −∞ , the density of ω α ( t ) concentrates around t = 0, which corresponds to the remov al of regularization and, as a result, the transition G 3 ,ω α → G 3 to a function with a singularity at zero. In turn, the limit of α → + ∞ corresp onds to a density concen tration near t = 1 / 2. Indeed, for every fixed t ∈ [0 , 1 / 2) the kernel of the av eraging op erator decreases exp onentially at α → + ∞ according to the rule ω α ( t ) = αe α ( t − 1 / 2)  1 + o (1)  , 12 while at the p oint t = 1 / 2, the function ω α (1 / 2) increases proportionally to α in the main order, thereby leading to a δ -shap ed sequence, since the normalization remains fixed. This situation symbolizes the transition to the δ -functional, which leads to the minim um of G 3 ,ω α (0). How ev er, the δ -functional is not quadratically integrable, so this regularization is not applicable for all mo dels equally . As an example of the application of the limiting case, w e can give a sextic mo del, see [25, 26]. 4.3 The case n = 2 . Substitute the index v alue, then S 1 = 2 π , S 0 = 2, and formula (7) can b e rewritten as g 2 ( r , s, t ) = 1 2 π 2 Z t + s r d u u − 1 Z ( u 2 − ( t − s ) 2 ) / (4 ts ) 0 d p p − 1 / 2 (1 − p ) − 1 / 2 . Next, using the v alue for the incomplete b eta function B r (1 / 2 , 1 / 2) = 2 arcsin( √ r ), see Paragraph 8.391 and formula 13 in Paragraph 9.121 of [34], we get g 2 ( r , s, t ) = 1 π 2 Z t + s r d u u − 1 arcsin  p ( u 2 − ( t − s ) 2 ) / (4 ts )  . Calculating the integral with such a density is a separate task, and, as in the case of n = 3, only some sp ecial quantities are of particular in terest. As a part of the study of a tw o-dimensional non-linear sigma mo del [30], the following v alues app eared θ j = Z R 2 d 2 k | k | − 2  υ 2 j +2 ( | k | ) − υ 2 j ( | k | )  , (19) where j ⩾ 1 and the auxiliary function is defined by the equalit y υ ( | k | ) = Z R 2 d 2 x e ikx ω ( | x | ) . Here, the density has properties from Theorem 2 . Let us study a special case. T o do this, w e calculate the F ourier transform for the sharp cutoff function in the momentum represen tation, which, after the scale transformation from (4), is the c haracteristic function of the ball B 1 , 1 2 π Z B 1 d 2 k e ikx = 1 2 π Z 1 0 d t Z 2 π 0 d ϕ te ist cos( ϕ ) = J 1 ( s ) s , (20) where | x | = s , | k | = t , and J i ( · ) is the Bessel function of the first kind. Thus, if the densit y of ω corresp onded to the cutoff in the momentum representation, then it w ould b e proportional to (20). Ho wev er, suc h a function has a non-compact supp ort and, therefore, is not suitable as an av eraging k ernel for a cutoff in the co ordinate represen tation. Let us cut it additionally manually . W e define new k ernel of the av eraging op erator by the equality ω ( s ) = α 2 π κ α J 1 ( αs ) s χ ( s < 1 / 2) , (21) where α > 0, χ ( · ) is the characteristic function of a set of R + , satisfying the equality , and also κ α = 1 − J 0 ( α/ 2) is a normalization factor, see (2). It is clear that the kernel from (21) leads to a mixed regu- larization com bining the prop erties of cutoffs b oth in the co ordinate represen tation and in the momen tum represen tation. Therefore, after substitution, we get υ ( αt ) = κ − 1 α Z α/ 2 0 d s J 0 ( st ) J 1 ( s ) ≡ ˆ υ ( α, t ) . F unction (19) with the new kernel will then b e lab eled with the index α . Then, after scaling the in tegration v ariable, we get θ α j = 2 π Z R + d t t  ˆ υ 2 j +2 ( α, t ) − ˆ υ 2 j ( α, t )  . 13 W e show that during in the limit α → + ∞ , the v alues θ α j tend to zero for all j ∈ N . This fact can b e pro ved in several steps. Let us define three auxiliary integrals for this θ α j, 1 = 2 π Z + ∞ 1 d t t  ˆ υ 2 j +2 ( α, t ) − ˆ υ 2 j ( α, t )  , θ α j, 2 = 2 π Z 1 1 / 2 d t t  ˆ υ 2 j +2 ( α, t ) − ˆ υ 2 j ( α, t )  , θ α j, 3 = 2 π Z 1 / 2 0 d t t  ˆ υ 2 j +2 ( α, t ) − ˆ υ 2 j ( α, t )  . It is clear that θ α j = θ α j, 1 + θ α j, 2 + θ α j, 3 . Moreov er, each comp onen t, as well as their sum, can b e estimated according to the following lemma. Lemma 1. F or e ach fixe d j ∈ N , ther e ar e such numb ers Θ j,i > 0 , wher e i ∈ { 1 , 2 , 3 } , and such a N > 1 that for al l α > N the ine qualities ar e fulfil le d | θ α j, 1 | ⩽ Θ j, 1 ln( α ) α , | θ α j, 2 | ⩽ Θ j, 2 α 1 / 2 , | θ α j, 3 | ⩽ Θ j, 3 ln( α ) α 1 / 2 , | θ α j | ⩽  Θ j, 1 + Θ j, 2 + Θ j, 3  ln( α ) α 1 / 2 . The pro of of this statemen t consists of five steps. Let us describ e them in more detail. Step 1. Let us first show that there exist such N > 0 and C > 0 that for all α > N and t ⩾ 1 / 2 the estimate is v alid | ˆ υ (+ ∞ , t ) − κ α ˆ υ ( α, t ) | ⩽ 1 π √ t  π C α +   si( α | 1 − t | / 2)   +   ci( α (1 + t ) / 2)    , (22) where definitions were used for the in tegral cosine functions si( · ) and integral sine ci( · ), see Paragraph 8.230 in [34]. In particular, using the boundedness of sp ecial functions on the in terv al under consideration, it can b e argued that there exists such a V 1 > 0 that | ˆ υ (+ ∞ , t ) − κ α ˆ υ ( α, t ) | ⩽ V 1 . (23) T o derive the relations, w e use form ula 2 of P aragraph 6.512 from [34] Z R + d s J 0 ( st ) J 1 ( s ) =      1 , t ∈ [0 , 1); 1 / 2 , t = 1; 0 , t > 1 . = ˆ υ (+ ∞ , t ) , where the v alue κ α | α → + ∞ = 1 was additionally used in the last equation. Then we get the equality ˆ υ (+ ∞ , t ) − κ α ˆ υ ( α, t ) = Z + ∞ α/ 2 d s J 0 ( st ) J 1 ( s ) . (24) Next, we apply formula 1 of Paragraph 8.451 from [34], according to which there exist such N 1 > 1 and C 1 > 0 that for all r > N 1 and l ∈ { 0 , 1 } an equation of the form holds J l ( r ) − p 2 /π r cos( r − π / 4 − l π / 2) = ϕ l ( r ) r − 3 / 2 , (25) where the absolute v alue of the con tin uous function ϕ l ( r ) for all r > N 1 and l ∈ { 0 , 1 } is b ounded by the num ber C 1 . F or con v enience, we will also assume that for all r > N 1 , the inequality | J 0 ( r ) | ⩽ 1 / √ r holds. In this case, applying equality to J 1 ( s ), the difference from (24) for all t ∈ R + and α > 2 N 1 can b e written out as ˆ υ (+ ∞ , t ) − κ α ˆ υ ( α, t ) = r 2 π Z + ∞ α/ 2 d s √ s J 0 ( st ) cos( s − 3 π / 4) + Z + ∞ α/ 2 d s s 3 / 2 J 0 ( st ) ϕ 1 ( s ) . (26) 14 Let us strengthen the constrain ts on the parameter α . F urther, let α > 4 N 1 , then the ratio from (25) can b e applied to J 0 ( st ). Then the right-hand side of the last equation is transformed to the sum of four terms ˆ υ (+ ∞ , t ) − κ α ˆ υ ( α, t ) = 2 π √ t Z + ∞ α/ 2 d s s cos( st − π / 4) cos( s − 3 π / 4) + 1 t 3 / 2 Z + ∞ α/ 2 d s s 3 ϕ 0 ( st ) ϕ 1 ( s ) r 2 π t 3 Z + ∞ α/ 2 d s s 2 ϕ 0 ( st ) cos( s − 3 π / 4) + r 2 π t Z + ∞ α/ 2 d s s 2 ϕ 1 ( s ) cos( st − π / 4) . Note that there is such a C > 0 that the absolute v alue of the sum of the last three terms after ev alu- ation mo dulo in tegral expressions do es not exceed the v alue of C / ( α √ t ). Applying b oundedness to the functions, we note that the following com bination can b e chosen as suc h a constant C = C 2 1 / N 2 1 + 6 C 1 p 2 /π . (27) Additionally transforming the pro duct of cosines 2 cos( st − π / 4) cos( s − 3 π / 4) = sin( s (1 − t )) − cos( s (1 + t )) and using definitions for the functions si( · ) and ci( · ), w e get the stated estimate (22), taking in to account the found num b ers N = 4 N 1 and C from (27). Step 2. Consider the integral for θ α j, 1 . Let the constraints on the α parameter stated in the previous step b e true. Note that the absolute v alue of the integral expression, taking in to account (23) and the equalit y ˆ υ (+ ∞ , t ) = 0 for t > 1, is estimated as follows 2 π Z + ∞ 1 d t t   ˆ υ 2 j +2 ( α, t ) − ˆ υ 2 j ( α, t )   ⩽ κ − 2 j − 2 α V 2 j − 1 1 ( V 1 + 1) 2 × × 2 Z + ∞ 1 d t t 3 / 2  π C α +   si( α | 1 − t | / 2)   +   ci( α (1 + t ) / 2)    . The first part of the integral can b e calculated explicitly . The integral with the function si( · ) must first b e divided into tw o parts by regions, for example, [1 , 3] and [3 , + ∞ ], and then tak e adv antage of the fact, see P aragraph 8.235 in [34], that there are suc h N 2 > 0 and C 2 > 0, that the following estimates are correct for all r > N 2   ci( r )   ⩽ C 2 /r and   si( r )   ⩽ C 2 /r . The third part of the in tegral is calculated without additional division. Next, we assume that in addition, α > N 2 is executed. Then the following inequalities are true Z + ∞ 3 d t t 3 / 2   si( α | 1 − t | / 2)   ⩽ 2 C 2 α Z + ∞ 2 d t t ( t + 1) 3 / 2 ⩽ C 2 α , Z 3 1 d t t 3 / 2   si( α | 1 − t | / 2)   ⩽ 2 α Z α 0 d t   si( t )   ⩽ 2 α  N 2 + C 2 ln( α/ N 2 )  , Z + ∞ 1 d t t 3 / 2   ci( α | 1 + t | / 2)   ⩽ 2 C 2 α Z + ∞ 2 d t t ( t − 1) 3 / 2 ⩽ C 2 α . Finally , collecting all the inequalities, we come to the conclusion that there exists suc h a Θ j, 1 > 0, dep ending only on the index of j , that for α > max { 4 N 1 , N 2 } an estimate of the form holds | θ α j, 1 | ⩽ Θ j, 1 ln( α ) α . 15 Step 3. Consider the integral for θ α j, 2 . In this case, it is necessary to take in to accoun t the fact that ˆ υ (+ ∞ , t ) = 1 for t ∈ [0 , 1). Using the maxim um v alue of the functions, w e can verify the v alidit y of the inequalit y 2 π Z 1 1 / 2 d t t   ˆ υ 2 j +2 ( α, t ) − ˆ υ 2 j ( α, t )   ⩽ κ − 2 j − 2 α ( V 1 + 1) 2 j ( V 1 + 1 + κ α ) × × 2 Z 1 1 / 2 d t t 3 / 2  π C α +   si( α | 1 − t | / 2)   +   ci( α (1 + t ) / 2)   +   1 − κ α    . Considering the intergal, it is clear that the first three parts can be studied in the same wa y as it was p erformed in the previous step. The fourth term is proportional to | 1 − κ α | = | J 0 ( α/ 2) | and therefore b eha v es like α − 1 / 2 , taking into account the expansion form (25). F urther, summing up all the relations, w e come to the conclusion that there exists suc h a Θ j, 2 > 0, depending only on the index of j , that for α > max { 4 N 1 , 4 N 2 } an estimate of the form is p erformed | θ α j, 2 | ⩽ Θ j, 2 α 1 / 2 . During the deriv ation, it w as taken in to accoun t that α − 1 / 2 tends to zero more slowly than ln( α ) /α , when α → + ∞ . Step 4. Let us sho w the b oundedness of the difference from (24) in the range t ∈ [0 , 1 / 2]. T o do this, w e use decomp osition (26) and the auxiliary representation for the Bessel function, see formula 1 in P aragraph 8.411 of [34], in the form J m ( st ) = 1 π Z π 0 d θ cos( mθ − st sin( θ )) . (28) Here, the index m ∈ N ∪ { 0 } . Decomp osing the cosine pro duct − 2 cos( st sin( θ )) cos( s − 3 π / 4) = cos( s (1 − t sin( θ )) + π / 4) + cos( s (1 + t sin( θ )) + π / 4) and moving on to integration o ver [ − π , π ], we come to the inequality    ˆ υ (+ ∞ , t ) − κ α ˆ υ ( α, t )    ⩽ 2 − 1 / 2 π 3 / 2 Z π − π d θ     Z + ∞ α/ 2 d s √ s cos( s (1 + t sin( θ )) + π / 4)     + 4 C 1 α 1 / 2 . Notice that the replacement of the integration order was possible due to the fact that the integral expression alwa ys contains an oscillating exp onen t, since t | sin( θ ) | ⩽ 1 / 2 < 1 is executed for all v alues of θ ∈ [ − π , π ] and 0 ⩽ t ⩽ 1 / 2. Indeed, w e note that the integral Z + ∞ α/ 2 d s √ s cos( s (1 + t sin( θ )) + π / 4) = 1 √ β  S  p αβ  − C  p αβ   , (29) where β = (1 + t sin( θ )) /π , after scaling of the v ariable can be calculated explicitly and expressed in terms of the F resnel in tegrals C ( · ) and S ( · ), see P aragraph 8.250 in [34]. F rom this representation, it can be seen that the integral from (29) conv erges uniformly with respect to the parameter θ in the interv al [ − π , π ], and, therefore, the form ula for replacing the order of in tegration is v alid, see Theorem 4 of Paragraph 521 in [36]. T aking into accoun t additionally the fact that the F resnel integrals are b ounded functions on the real axis, w e obtain the final statement. There exists a V 2 > 0 suc h that for all t ∈ [0 , 1 / 2] and α > max { 4 N 1 , 4 N 2 } the inequality holds    ˆ υ (+ ∞ , t ) − κ α ˆ υ ( α, t )    ⩽ V 2 . 16 Step 5. Consider the integral for θ α j, 3 . Using the equality ˆ υ (+ ∞ , t ) = 1 for t ∈ [0 , 1) and using the maxim um v alue of the functions, we can verify the v alidit y of the inequality 2 π Z 1 / 2 0 d t t   ˆ υ 2 j +2 ( α, t ) − ˆ υ 2 j ( α, t )   ⩽ 2 π κ − 2 j − 2 α ( V 2 + 1) 2 j ( V 2 + 1 + κ α ) × × Z 1 / 2 0 d t t     Z + ∞ α/ 2 d s J 0 ( st ) J 1 ( s ) − J 0 ( α/ 2)     . Let us integrate it in parts, then the density inside the mo dule is transformed to the form J 0 ( α/ 2)  J 0 ( αt/ 2) − 1  − t Z + ∞ α/ 2 d s J 1 ( st ) J 0 ( s ) . (30) It is clear that to complete the pro of, it is sufficient to consider only t wo in tegrals. In the case of the first term, the main estimate follows from the following chain of inequalities Z 1 / 2 0 d t t     J 0 ( αt/ 2) − 1     ⩽ Z α/ 4 0 d t t    J 0 ( t ) − 1    ⩽  Z 1 0 d t t    J 0 ( t ) − 1    + 2 ln( α/ 4)  ⩽ 2 ln( α ) , whic h are true for all α ⩾ 4. W e assume that α satisfies the conditions from the previous steps. Consid- ering that | J 0 ( α/ 2) | is dominated b y 2 1 / 2 α − 1 / 2 , we see that the first term in (30) leads to the dep endence ln( α ) / √ α . Next, consider the integral of the second term. Using (25), we ha ve     Z + ∞ α/ 2 d s J 1 ( st ) J 0 ( s )     ⩽ r 2 π     Z + ∞ α/ 2 d s √ s J 1 ( st ) cos( s − π / 4)     + 4 C 1 √ α . (31) Then using representation (28) for the Bessel function and performing transformations similar to those in step 4, we make sure that the integral on the righ t side of (31) has an estimate of the form r 2 π     Z + ∞ α/ 2 d s √ s J 1 ( st ) cos( s − π / 4)     ⩽ max r ⩾ √ α/ (2 π )    S ( r ) − C ( r )   +   S ( r ) + C ( r ) − 1    . With the help of prop erties of F resnel integrals for large v alues of the argument, we we see that there is a b eha vior of the form α − 1 / 2 . Indeed, there are such N 3 > 0 and C 3 > 0 that for all r > p N 3 / (2 π ) the relations are fulfilled   C ( r ) − 1 / 2   ⩽ C 3 /r ,   S ( r ) − 1 / 2   ⩽ C 3 /r , where the formulas of Paragraph 8.255 from [34] w ere used. Thus, collecting all the inequalities, w e make sure that under the conditions imp osed on α , that is, under α > max { 4 N 1 , 4 N 2 , N 3 } , there exists such a Θ j, 3 > 0, dep ending only on the index of j , for which the inequality holds | θ α j, 3 | ⩽ Θ j, 3 ln( α ) α 1 / 2 . 5 Conclusion The av eraging op erator w as considered in the con text of problems related to the regularization of fun- damen tal solutions. Several statemen ts were pro ved, and three sp ecial cases that previously arose when studying the structure of singularities in quantum field mo dels w ere examined. In particular, in Theo- rem 1 , representations were obtained for fundamental solutions in a flat Euclidean space of arbitrary dimension, a v eraged twice ov er spheres of non-fixed radius, and the properties of smo othness and areas of monotonicit y w ere studied. In turn, Theorem 2 presen ted acceptable classes of kernels for a v eraging op erators, dep ending on additional conditions. Section 4 dealt with sp ecial cases. The first of them concerns the situation when av eraging o v er a ball turns into av eraging ov er a sphere by concentrating density at the b oundary . This leads to extreme 17 v alues of the deformed fundamental solution at zero. The second case is related to a three-dimensional situation that o ccurs in the sextic mo del. In this case, a family of kernels and corresponding smo othed fundamen tal solutions were explicitly presented. The third option is devoted to the t wo-dimensional case that arises in the mo del of the principal c hiral field. It w as sho wn that the co efficien ts of renormalization (functionals of a sp ecial kind) in the case of using sharp cutoff in the momentum represen tation can also b e obtained by limit transition from a cutoff in the co ordinate representation. Ab out other examples. One of the in teresting op en problems is the study of sp ecial functionals for a mixed type of regularization (by analogy with Section 4.3) in the four-dimensional case. Sets of suc h functionals are given in the app endix of [22], devoted to the study of three-lo op corrections. Ab out extending the kernel class. In this paper, w e studied the case exclusively with probabilistic a veraging, that is, when the k ernel density is almost everywhere non-negativ e. Such a statement is more meaningful from a physical p oin t of view. Nevertheless, mathematically , this limitation can b e dispensed with. In this case, it would b e interesting to study the properties (for example, monotonicity in terv als) of the function f ( · ), which may differ significan tly due to negative v alues of the kernels. Ab out the inv erse problem. In the course of the work, the prop erties of the function f ( · ) w ere studied, which was explicitly built using the kernel ω ( · ) of the av eraging op erator. In this regard, a fair question arises: ”Under what conditions can the k ernel be restored using the function f ( · ), and what prop erties should b e fulfilled?”. The answer to this question is directly related to the p ositiv e definite- ness of functions and Boas–Kac ro ots, see for example [37, 38]. This issue was not cov ered in this pap er. Nev ertheless, it is imp ortant and deserves atten tion. Ac kno wledgments. A.V. 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