Operators of Harmonic Analysis in Weighted Spaces with Non-standard Growth

Op erators of Harmonic Analysis in W eigh ted Spaces with Non-standard Growth b y V.M. Kokilashv ili, A.R azmadze Mathematic al Institute and Black Se a University, Tbilisi, Ge or gia kokil@rmi.acne t.ge and S.G.Samk o Universidade do A lgarve, Portugal ssamko@ualg.pt Abstracts Last years there w as increasing an interest to the so called function spaces with non-standar d growth, known also as v ariable exp onent Lebes gue spa ces. F or weigh ted such space s on homo- geneous spaces, w e develop a certain v arian t o f Rubio de F ra ncia’s extrapo lation theor em. This extrap olation theorem is a pplied to obtain the b oundedness in suc h spaces of v arious op erators of harmonic analys is, such a s ma ximal and singula r ope rators, p otential oper ators, F o urier mu ltipli- ers, dominants o f partial sums of trig onometric F ourier s eries and others , in weight ed Leb esgue spaces with v ariable expone n t. There ar e also g iven their vector-v alued analog ues. 1. In tro d uction During last y ears a signific ant progress w as made in the study of maximal and singular op era t ors and p oten tial t yp e o p erators in the generalized Leb esgue spaces L p ( · ) with v ariable exp onen t, kno wn also as the spaces with non- standard growth. A n um ber of mathematical problems leading to s uc h spaces with v ariable ex p onent arise in applications to partial differential equations, v ar iational problems and con- tin uum mec hanics (in particular, in the theory of the so called electrorheological fluids), see E. Acerbi and G.Mingione [1],[2], X.F an and D.Zhao [2 0], M.Ru ˇ z iˇ c k a [63], V.V. Z hik o v [76 ], [77]. These a pplications stipulated a significant in terest to suc h spaces in the last decade. The most adv ance in t he study of the classic al op erators of harmonic analy- sis in the case of v ariable exp onen t w as made in the Euclidean setting, includ- ing w eigh ted estimates. W e refer in particular to the surv eying articles L.D iening, P .H¨ ast¨ o and A.Nekvinda [16], V.Kokilash vili [33], S.Samko [74] and pap ers D.Cruz- Urib e, A.Fiorenza, J.M.Martell and C.P erez [10], D.Cruz-Urib e, A.Fiorenza a nd C.J.Neugebauer [1 1], L. D iening [1 3 ], [14], [15], L.Diening and M.Ru ˇ z i ˇ c k a [17], V. Kokilashv ili, N.Samk o and S.Samk o [38], V.Kokilash vili and S.Samk o [45], [4 1], [42], [43], A.Nekvinda [59 ], S.Samk o [71], [72], [73], S.Samko, E.Shargoro dsky and B.V akulo v [75] and references therein. Recen tly there also started the inv es tigatio n of these classical op erators in the spaces with v ariable exp onen t in the setting of metric measure spaces, t he case 1 of cons tant p in this setting ha ving a long histor y , w e refer, in particular to the pap ers A.P .Calder´ on [6], R.R.Coifman and G.W eiss [7], [8], R.Mac ´ ıas and C.Sego via [52], b o oks D.E.Edm unds and V.Kokilash vili and A.Meskhi [18] a nd I.Genebash vili, A.Gogatish vili, V.Kokilash vili and M.Krb ec [22 ], J.Heinonen [26] a nd references therein. The no n- w eigh ted b oundedness of the maximal op erator on homogeneous spaces w as prov ed b y P .Harjuleh to, P .H¨ ast¨ o and M.Pe re [25] and Sob olev embedding theorem with v ariable expo nen ts on ho mo g eneous spaces with v ariable dimension w as pro v ed in P .Harj uleh to, P .H¨ ast¨ o and V.Latv ala [24]. In the presen t pap er we giv e a dev elopmen t of w eigh ted estimations of v arious op erators of harmonic ana lysis in Leb esgue spaces with v ariable exp onen t p ( x ). W e first giv e theorems on the w eigh ted boundedness of the maximal op erato r on homogeneous spaces (Theorems 2.11 a nd 2.12). Next, in Section 3. w e giv e a certain p ( · ) → q ( · )-ve rsion of Rubio de F rancia’s extrap olation theorem [62] within the framew orks of w eigh ted spaces L p ( · )  on metric measure spaces. Pro ving this v ersion w e dev elop some ideas and approaches o f pap ers [10], [12]. By means of this extrap olation theorem and know n theorems on the b oundedness with Muc k enhoupt w eigh ts in the case of cons tant p , w e obtain res ults on w eigh ted p ( · ) → q ( · )- or p ( · ) → p ( · )-b oundedness - in the case of v ariable exp onen t p ( x ) - of the follo wing op erators 1) pot en tial t yp e op erators, 2) F ourier multipliers (we ighted Mikhlin, H¨ ormander and Lizorkin-type theorems, Subsection 4.2), 3) m ultipliers of trigonometric F ourier series (Subsection 4.3), 3) ma jora n ts of partial sums of F ourier series (Subse ction 4.4), 4) singular in tegral operat o rs on Carleson curv es and in Euclidean setting (Subsec- tions 4.6-4.9), 5) F effermann-Stein function (Subsection 4.10), 6) some v ector-v alued op erators (Subsection 4.11). 2. Definiti o ns and preli minaries 2.1 On v ariable dimensions in metric measure spaces In the sequel, ( X , d, µ ) denotes a metric space with the (quasi)metric d and non- negativ e measure µ . W e refer to [18], [22], [26] fo r the basics on metric measure spaces. By B ( x, r ) = { y ∈ X : d ( x, y ) < r } w e denote a ball in X . The following standard conditions will be assumed to be satisfied: 1) all the balls B ( x, r ) = { y ∈ X : d ( x, y ) < r } are meas urable, 2) the space C ( X ) of unifor mly con tin uous functions on X is dense in L 1 ( µ ). In most of the statemen ts w e also supp o se that 3) the measure µ satisfies the doubling condition: µB ( x, 2 r ) ≤ C µB ( x, r ) , where C > 0 do es not dep end on r > 0 and x ∈ X . A measure satisfy ing this condition will be called doubling measure. 2 F or a lo cally µ -integrable function f : X → R 1 w e consider the Hardy-Litt lewoo d maximal function M f ( x ) = sup r > 0 1 µ ( B ( x, r )) Z B ( x,r ) | f ( y ) | d µ ( y ) . By A s = A s ( X ), where 1 ≤ s < ∞ , we denote the class of weigh ts (lo cally al- most ev erywhe re p ositiv e µ -integrable functions) w : X → R 1 whic h satisfy the Muc k enhoupt condition sup B   1 µB Z B w ( y ) dµ ( y )     1 µB Z B w − 1 s − 1 ( y ) d µ ( y )   s − 1 < ∞ in the case 1 < s < ∞ , and the condition M w ( x ) ≤ C w ( x ) for almost all x ∈ X , with a constant C > 0, not depending on x ∈ X , in t he case s = 1. O b viously , A 1 ⊂ A s , 1 < s < ∞ . As is kno wn, see [6], [52], the weigh ted b oundedness Z X ( M f ( x )) s w ( x ) dµ ( x ) ≤ C Z X | f ( x ) | s w ( x ) dµ ( x ) , holds, if and only if w ∈ A s . Let Ω be an op en set in X . Definition 2.1. By P (Ω) w e denote the class of µ -measurable functions on Ω, suc h that 1 < p − ≤ p + < ∞ , (2.1) where p − = p − (Ω) = ess inf x ∈ Ω p ( x ) and p + = p + (Ω) = ess sup x ∈ Ω p ( x ) . Definition 2.2. By L p ( · )  (Ω) w e denote t he w eigh ted Banach function space of µ -measurable functions f : Ω → R + 1 , suc h that k f k L p ( · )  := k f k p ( · ) = inf    λ > 0 : Z Ω      ( x ) f ( x ) λ     p ( x ) dµ ( x ) ≤ 1    < ∞ . (2.2) Definition 2.3. We say that a weight  b elo n gs to the class A p ( · ) (Ω) , if the maximal op er a tor M is b ounde d in the sp ac e L p ( · )  (Ω) . 3 Definition 2.4. A function p : Ω → R 1 is said to b elong to the class W L (Ω) (we ak Lipshitz), if | p ( x ) − p ( y ) | ≤ A ln 1 d ( x,y ) , d ( x, y ) ≤ 1 2 , x, y ∈ Ω , (2.3) wher e A > 0 do es not dep end on x and y . The notion of low er and upp er lo cal dimension of X at a point x introduced as dim X ( x ) = lim r → 0 ln µB ( x, r ) ln r , dim X ( x ) = lim r → 0 ln µB ( x, r ) ln r are known, see e.g. [19]. W e will use differen t notions of lo cal lo w er a nd upper dimensions, inspired by the notion of the so called index num b ers m ( w ) , M ( w ) of almost monotonic f unctions w , see their definition in (2.17). These indices studied in [64], [66], [65], are v ersions of Matuzew sk a- Orlicz index n umbers used in the theory of Orlicz spaces, see [53], [54]. The idea t o intro duce lo cal dimensions in terms of these indices b y the following definition w as b orrow ed from the pap ers [67], [68]. Definition 2.5. The n um b ers dim ( X ; x ) = sup r > 1 ln  lim h → 0 µB ( x,r h ) µB ( x,h )  ln r , dim ( X ; x ) = inf r > 1 ln  lim h → 0 µB ( x,r h ) µB ( x,h )  ln r (2.4) will be referred to as lo cal lo w er and upper dimensions. Observ e that the ”dimension” dim ( X ; x ) may b e also rewritten in terms of the upp er limit as well: dim ( X ; x ) = sup 0 0 c 2 > 0, not dep ending on x ∈ X and r > 0, the inequalities in (2.12), (2 .13) and (2.19) turn in to − s p ( x k ) < β k < s p ′ ( x k ) , − s p ∞ < β ∞ + N X k =1 β k < s p ′ ∞ (2.14) and − s p ( x k ) < m ( w ) ≤ M ( w ) < s p ′ ( x k ) , k = 1 , 2 , ..., N , (2.15) resp ectiv ely . In fact, w e may a dmit a more general class of w eigh ts  ( x ) = w 0 [1 + d ( x 0 , x )] N Y k =1 w k [ d ( x, x k )] (2.16) with ” radial” weigh ts, whe re the functions w 0 and w k , k = 1 , ..., N , b elong to a class of Zygm und-Bary-Stec hkin t yp e, whic h admits an oscillation b et w een t w o p ow er functions with differen t expo nents. By U = U ([0 , ℓ ]) w e denote the class o f functions u ∈ C ([0 , ℓ ]) , 0 < ℓ ≤ ∞ , su ch that u (0) = 0 , u ( t ) > 0 for t > 0 and u is an almost increasing function on [0 , ℓ ]. (W e recall that a function u is called almost incr e asing on [0 , ℓ ], if there exists a constan t C ( ≥ 1) suc h that u ( t 1 ) ≤ u ( t 2 ) fo r all 0 ≤ t 1 ≤ t 2 ≤ ℓ ). By e U w e denote the class of function u , suc h that t a u ( t ) ∈ U for some a ∈ R 1 . Definition 2.8. ([4]) A function v is said to b elong to the Zygmund-Bary- Ste chkin class Φ 0 δ , if Z h 0 v ( t ) t dt ≤ cv ( h ) and Z ℓ h v ( t ) t 1+ δ dt ≤ c v ( h ) h δ , wher e c = c ( v ) > 0 do es not dep end on h ∈ (0 , ℓ ] . It is kno wn that v ∈ Φ 0 δ , if a nd only if 0 < m ( v ) ≤ M ( v ) < δ , where m ( w ) = sup t> 1 ln  lim h → 0 w ( ht ) w ( h )  ln t and M ( w ) = sup t> 1 ln  lim h → 0 w ( ht ) w ( h )  ln t (2.17) (see [64], [66], [29]). F or f unctions w defined in the neigh b orho o d of infinit y and suc h that w  1 r  ∈ e U ([0 , δ ]) f o r some δ > 0, we in tro duce also m ∞ ( w ) = sup x> 1 ln h lim h →∞ w ( xh ) w ( h ) i ln x , M ∞ ( w ) = inf x> 1 ln h lim h →∞ w ( xh ) w ( h ) i ln x . (2.18) Generalizing Definition 2.7, w e in tro duce also the fo llo wing notion. 6 Definition 2.9. A weight function  of form (2.16) is said to b elong to the class V osc p ( · ) (Ω , Π) , wher e p ( · ) ∈ C (Ω) , if w k ( r ) ∈ e U ([0 , ℓ ]) , ℓ = diam Ω and − dim (Ω) p ( x k ) < m ( w k ) ≤ M ( w k ) < dim (Ω) p ′ ( x k ) , (2.19) k = 1 , 2 , ..., N , and (in the c as e Ω is infinite) w 0  ℓ 2 r  ∈ e U ([0 , ℓ ]) and − dim ∞ (Ω) p ∞ < N X k =0 m ∞ ( w k ) ≤ N X k =0 M ∞ ( w k ) < dim ∞ (Ω) p ′ ∞ − ∆ p ∞ , (2.20) wher e ∆ p ∞ = d i m ∞ (Ω) − d i m ∞ (Ω) p ∞ . Observ e that in the case Ω = X = R n conditions ( 2 .19) and (2.20) tak e the form w k ( r ) ∈ e U ( R 1 + ) :=  w : w ( r ) , w  1 r  ∈ e U ([0 , 1])  (2.21) and − n p ( x k ) < m ( w k ) ≤ M ( w k ) < n p ′ ( x k ) , − n p ∞ < N X k =0 m ∞ ( w k ) ≤ N X k =0 M ∞ ( w k ) < n p ′ ∞ . (2.22) Remark 2.10. F o r every p 0 ∈ (1 , p − ) ther e hol d the impli c ations  ∈ V p ( · ) (Ω , Π) = ⇒  − p 0 ∈ V ( e p ) ′ ( · ) (Ω , Π) and  ∈ V osc p ( · ) (Ω , Π) = ⇒  − p 0 ∈ V osc ( e p ) ′ ( · ) (Ω , Π) , wher e e p ( x ) = p ( x ) p 0 . 2.3 The b oun d edness of the Hardy-Littlew o o d maximal op- erator on metric spaces with doubling measure, in w eigh ted Leb esgue sp aces with v ariable exp onen t The follo wing statemen ts are v alid. Theorem 2.11. L et X b e a m etric sp ac e wi th doublin g me asur e a n d let Ω b e b ounde d. If p ∈ P (Ω) ∩ W L (Ω) and  ∈ V osc p ( · ) (Ω , Π) , then M is b ounde d in the sp ac e L p ( · )  (Ω) . 7 Theorem 2.12. L et X b e a m etric sp ac e wi th doublin g me asur e a n d let Ω b e unb ounde d. L et p ∈ P ( Ω) ∩ W L (Ω) an d let ther e exist R > 0 such that p ( x ) ≡ p ∞ = const fo r x ∈ Ω \ B ( x 0 , R ) . If  ∈ V osc p ( · ) (Ω , Π) , then M is b ounde d in the sp ac e L p ( · )  (Ω) . The Euclidean v ersion of Theorems 2.11 and 2.12 w as pro v ed in [13] in the non- w eigh ted case and in [38], [40] in the weigh ted case; in [40] there w ere a lso prov ed the corresponding v ersions of Theorems 2.11 and 2.12 for the maximal op erator o n Carleson curv es (a typic al example of metric measure spaces with constant dimen- sion). The pro o f of Theorems 2.11 and 2 .12 in the general case in main is similar, b eing based on the approaches used in the pro o fs fo r the case of Carles on curve s. Theorem 2.13. L e t Ω b e a b o unde d op en set in a doubling me asur e metric sp ac e X , let the exp onent p ( x ) s a tisfy c onditions (2.1), (2. 3). Then the op er a tor M is b ounde d in L p ( · )  (Ω) , if [  ( x )] p ( x ) ∈ A p − (Ω) . W e refer to [44] for Theorem 2.12, its detailed pro of for the case where X is a Carleson curve is giv en in [40], the pro o f for a doubling measure metric space b eing in fact the same. 3. Extrap o lation theorem on met ric measu r e spaces In the s equel F = F (Ω) denotes a family of ordered pairs ( f , g ) of non-negative µ -measurable functions f , g , defined on an op en set Ω ⊂ X . When say ing that there holds an inequalit y of ty p e (3.3) for all pairs ( f , g ) ∈ F and weigh ts w ∈ A 1 , w e alw a ys mean that it is v alid for all the pairs, fo r which the left-hand side is finite, and that the constant c dep ends only on p 0 , q 0 and the A 1 -constan t of the w eight. In what follo ws, b y p 0 and q 0 w e denote po sitiv e n um b ers suc h that 0 < p 0 ≤ q 0 < ∞ , p 0 < p − and 1 p 0 − 1 p + < 1 q 0 (3.1) and use the notatio n e p ( x ) = p ( x ) p 0 , e q ( x ) = q ( x ) q 0 . (3.2) Remark 3.1. The extrap o la tion Theorem 3.2 with v ariable exp onen ts in t he non-w eigh ted case  ( x ) ≡ 1 a nd in the Euclidean setting w as pro v ed in [10]. F or extrap olation theorems in the case of constan t expo nen ts w e refer to [62 ], [23]. Observ e that the measure µ in Theorem 3.2 is not assumed to b e doubling. Theorem 3.2. L et X b e a metric me asur e sp ac e a nd Ω an op en set in X . Assume that for some p 0 and q 0 , satisfying c onditions (3.1) and every weig h t w ∈ 8 A 1 (Ω) ther e holds the ine quality   Z Ω f q 0 ( x ) w ( x ) dµ ( x )   1 q 0 ≤ c 0   Z Ω g p 0 ( x )[ w ( x )] p 0 q 0 dµ ( x )   1 p 0 (3.3) for al l f , g in a given family F . L e t the va riable exp onent q ( x ) b e define d by 1 q ( x ) = 1 p ( x ) −  1 p 0 − 1 q 0  , (3.4) let the exp on ent p ( x ) and the weigh t  ( x ) satisfy the c onditions p ∈ P (Ω) and  − q 0 ∈ A ( e q ) ′ (Ω) . (3.5) Then for al l ( f , g ) ∈ F with f ∈ L p ( · )  (Ω) the ine quality k f k L q ( · )  ≤ C k g k L p ( · )  (3.6) is va li d with a c onstant C > 0 , not dep ending on f and g . P r o o f . By the Riesz theorem, v a lid fo r the spaces with v a riable exp onent in the case 1 < p − ≤ p + < ∞ , (see [46 ], [70]), w e hav e k f k q 0 L q ( · )  = k f q 0  q 0 k L e q ( · ) ≤ sup Z Ω f p 0 ( x ) h ( x ) dµ ( x ) , where we assume that f is non- negativ e and sup is taken with r esp ect to all non- negativ e h suc h that k h − q 0 k L ( e q ) ′ ( · ) ≤ 1 . W e fix any suc h a function h . L et us sho w that Z Ω f q 0 ( x ) h ( x ) dµ ( x ) ≤ C k g  k q 0 L q ( · ) (3.7) for an arbitrary pair ( f , g ) fro m the giv en family F with a constan t C > 0, not dep ending on h, f a nd g . By the assumption  − q 0 ∈ A ( e q ) ′ (Ω) w e ha ve k  − q 0 M ϕ k L e q ′ ( · ) (Ω) ≤ C 0 k  − q 0 ϕ k L e p ′ ( · ) (Ω) (3.8) where the constan t C 0 > 0 do es not dep end on ϕ . W e mak e use of the follo wing construction wh ich is due to Rubio de F rancia [62] S ϕ ( x ) = ∞ X k =0 (2 C 0 ) − k M k ϕ ( x ) , (3.9) where M k is the k -iterated maximal op erator a nd C 0 is the constant f rom (3.8) (one ma y tak e C 0 ≥ 1). T he follow ing statemen ts are ob vious: 1) ϕ ( x ) ≤ S ϕ ( x ) , x ∈ Ω f o r a n y non-negativ e function ϕ ; 2) k  − q 0 S ϕ k L ( e q ) ′ (Ω) ≤ 2 k  − q 0 ϕ k L ( e q ) ′ (Ω) , (3.10) 9 3) M ( S ϕ )( x ) ≤ 2 C 0 S ϕ ( x ) , x ∈ Ω , so that S ϕ ∈ A 1 (Ω) with the A 1 -constan t not depending on ϕ . Therefore S ϕ ∈ A q 0 (Ω). By 1), for ϕ = h w e ha v e Z Ω f q 0 ( x ) h ( x ) dµ ( x ) ≤ Z Ω f q 0 ( x ) S h ( x ) dµ ( x ) . (3.11) By t he H¨ older inequalit y for v ariable exponent, prop ert y 2) and the condition f ∈ L q ( · )  , w e ha v e Z Ω f q 0 ( x ) S h ( x ) dµ ( x ) ≤ k k f q 0  q 0 k L e q ( · ) · k  − q 0 S h k L ( e q ) ′ ( · ) ≤ C k f  k q 0 L q ( · ) · k h − q 0 k L ( e q ) ′ ( · ) ≤ C k f  k q 0 L q ( · ) < ∞ . Consequen tly , the in tegral R Ω f q 0 ( x ) S h ( x ) dµ ( x ) is finite, whic h enables us to mak e use of condition (3.3) with respect to the righ t-hand side of (3.11). Condition (3.3) b eing a ssumed t o b e v alid with an arbitrary w eigh t w ∈ A 1 , is in particular v alid for w = S h . Th erefore, Z Ω f q 0 ( x ) S h ( x ) dµ ( x ) ≤ C   Z Ω g p 0 ( x )[ S h ( x )] p 0 q 0 dµ ( x )   q 0 p 0 . Applying the H¨ older inequalit y on the rig h t-hand side, w e get Z Ω f q 0 ( x ) S h ( x ) dµ ( x ) ≤ C  k g p 0  p 0 k L p ( · ) p 0    ( S h ) p 0 q 0  − p 0    L ( e p ) ′  q 0 p 0 . Th us Z Ω f q 0 ( x ) S h ( x ) dµ ( x ) ≤ C k g k q 0 L p ( · )     − p 0 ( S h ) p 0 q 0    q 0 p 0 L ( e p ) ′ ( · ) . (3.12) F rom (3.4) w e easily obtain that ( e p ) ′ ( x ) = q 0 p 0 ( e q ) ′ ( x ) and then     − p 0 ( S h ) p 0 q 0    q 0 p 0 L ( e p ) ′ ( · ) =    − q 0 S h   L e q ′ ( · ) . Consequen tly , Z Ω f q 0 ( x ) S h ( x ) dµ ( x ) ≤ C k g k q 0 L p ( · )    − q 0 S h   L e q ′ ( · ) . (3.13) 10 T o pro v e (3.7), in view of (3.13 ) it suffices to sho w that k  − q 0 S h k L e q ′ ( · ) ma y b e estimated by a constan t not dep ending on h . This follows from (3.10) and the condition k h − q 0 k L ( e q ) ′ ( · ) ≤ 1 and prov es the t heorem. ✷ Remark 3.3. It is easy to c hec k that in view of T heorem 2.13 the condition [  ( y )] q 1 ( y ) ∈ A s , where q 1 ( y ) = q ( y )( q + − q 0 ) q ( y ) − q 0 and s = q + q 0 , (3.14) is sufficie nt for the v alidit y of the condition  − q 0 ∈ A ( e q ) ′ (Ω) of Theorem 3.2. By means of Theorems 2.1 1 and 2.12, w e obtain the follo wing statemen t as an immediate consequen ce of Theorem 3.2 in whic h w e denote γ = 1 p 0 − 1 q 0 . Theorem 3.4. L et X b e a metric sp ac e with doubling m e asur e and Ω an o p en set in X . L et also the fol lowing b e s a tisfi e d 1) p ∈ P (Ω) ∩ W L (Ω) , and in the c ase Ω is an unb ounde d set, let p ( x ) ≡ p ∞ = const for x ∈ Ω \ B ( x 0 , R ) with s o me x 0 ∈ Ω and R > 0 ; 2) ther e holds ine quality (3.3) for some p 0 and q 0 satisfying the assumptions in (3.1) and al l ( f , g ) ∈ F fr om some family F and every weigh t w ∈ A 1 (Ω) . Then I) ther e holds ine quality (3.6) for al l p airs ( f , g ) fr om the same fam ily F , such that f ∈ L p ( · )  (Ω) and weights  of form (2.16) wh e r e  γ − 1 p ( x k )  dim (Ω) < m ( w k ) ≤ M ( w k ) <  1 p ′ ( x k ) − 1 p ′ 0  dim (Ω) (3.15) and, in c ase Ω is unb ounde d, δ +  γ − 1 p ∞  dim (Ω) < N X k =0 m ( w k ) ≤ N X k =0 M ( w k ) <  1 p ′ ∞ − 1 p ′ 0  dim (Ω) , (3.16) wher e δ =  dim ∞ (Ω) − dim ∞ (Ω)   1 p 0 − 1 p ∞  ; II) in c ase ine q uality (3.3) holds for a l l p 0 ∈ (1 , p − ) , the term 1 p ′ 0 in (3.15) and ( 3 .16) may b e omitte d and δ may b e taken in t he form δ =  dim ∞ (Ω) − dim ∞ (Ω)   1 p − − 1 p ∞  . 11 4. Applicatio n to p r o blems of the b oun dedness in L p ( · )  of classical op erators of harmonic analysis 4.1 P oten tials op erators and fractional maximal fu nction W e first apply Theorem 3.2 to p oten tial op erators I γ X f ( x ) = Z X f ( y ) dµ ( y ) µB ( x, d ( x, y )) 1 − γ (4.1) where 0 < γ < 1. W e a ssume that µX = ∞ and the measure µ satisfies the doubling condition. W e also additionally suppo se the follo wing conditions to b e fulfilled: there exis ts a point x 0 ∈ X such that µ ( x 0 ) = 0 (4.2) and µ ( B ( x 0 , R ) \ B ( x 0 , r ) ) > 0 for all 0 < r < R < ∞ . (4.3) The follo wing statemen t is v alid, see f o r instance [18 ], p. 412. Theorem 4.1. L et X b e a metric me asur e sp ac e with doubli n g me as ur e satisfying c ondi tion s ( 4 .2)-(4.3), µX = ∞ , le t 0 < γ < 1 , 1 < p 0 < 1 γ and 1 q 0 = 1 p 0 − γ . The op er ator I γ X admits the estimate   Z X | v ( x ) I γ X f ( x ) | q 0 dµ   1 q 0 ≤   Z X | v ( x ) f ( x ) | p 0 dµ   1 p 0 , (4.4) if the weight v ( x ) satisfies the c ondition sup B   1 µB Z B v q 0 ( x ) dµ   1 q 0   1 µB Z B v − p ′ 0 ( x ) dµ   1 p ′ 0 < ∞ (4.5) wher e B stands for a b al l in X . By means of Theorem 4.1 and extrap olation Theorem 3.2 w e a rriv e at the fol- lo wing statemen t. Theorem 4.2. L et X satisfy the assumptions of Th e or e m 4.1, let p ∈ P , 0 < γ < 1 and p + < 1 γ . The weighte d estimate k I γ X f k L q ( · ) ρ ≤ C k f k L p ( · ) ρ (4.6) with the limiting exp onent q ( · ) define d by 1 q ( x ) = 1 p ( x ) − γ , holds if  − q 0 ∈ A “ q ( · ) q 0 ” ′ ( X ) (4.7) 12 under any choic e of q 0 > p − 1 − γ p − . P r o o f . By Theorem 4.1, inequality (4.4) holds under condition (4.5). As is kno wn, inequalit y (3.3) with f = I α g holds for ev ery we ight w satisfying the 1 < p 0 < ∞ and 1 q 0 = 1 p 0 − γ . Condition (4.5) is satisfied if v q 0 ∈ A 1 . Consequen tly , inequalit y (3.3) with f = I α g holds f or eve ry w ∈ A 1 . Then (4.6) follo ws f rom Theorem 3.2. ✷ F rom Theorem 4.2 w e derive the following corollary for the Riesz p oten tial op- erators I α f ( x ) = Z R n f ( y ) dy | x − y | n − α . (4.8) Corollary 4.3. L et p ∈ P , let 0 < α < n and p + < n α . The w e ighte d Sob olev the or em k I α f k L q ( · ) ρ ≤ C k f k L p ( · ) ρ (4.9) with the limiting exp onent q ( · ) define d by 1 q ( x ) = 1 p ( x ) − α n , hold s if  − q 0 ∈ A “ q ( · ) q 0 ” ′ ( R n ) (4.10) under any choic e of q 0 > np − n − αp − . Remark 4.4. Since Theorems 2.11 a nd 2 .12 pro vide sufficien t conditions for the we ight  to satisfy assumption (4.1 0), we could write do wn t he corresp onding statemen ts on the v a lidit y of (4 .9) in terms o f the w eights used in Theorems 2.11 and 2.12. In the sequel we giv e results of suc h a kind for other op erators. F o r p oten tial op erators in the case Ω = R n w e refer to [75] and [6 9], where for p ow er w eights of the class V p ( · ) ( R n , Π) and f o r radia l oscillating w eigh ts of the class V osc p ( · ) ( R n , Π), resp ectiv ely , there w ere obtained estimates (4.9) under assumptions mor e general than should be imp osed b y the usage of Theorem 2.12. 4.2 F ourier m u ltipliers A measurable function R n → R 1 is said to b e a F ourier m ultiplier in the space L p ( · )  ( R n ), if t he operato r T m , defined on the Sc h w artz space S ( R n ) b y d T m f = m b f , admits an exte nsion to the b ounded op erator in L p ( · )  ( R n ). W e give b elow a generalization of the classical Mikhlin theorem ([55], see also [56]) on F ourier m ultipliers to the case of Leb esgue spaces with v ariable exponent. Theorem 4.5. L et a function m ( x ) b e c ontinuous everywher e in R n , exc ept for pr ob ably the origin, have the mixe d distributional deriv a tive ∂ n m ∂ x 1 x 2 ··· x n and the 13 derivatives D α m = ∂ | α | m ∂ x α 1 1 x α 2 2 ··· x α n n , α = ( α 1 , ..., α n ) of or ders | α | = α 1 + · · · + α n ≤ n − 1 c ontinuous b eyond the origin and | x | | α | | D α m ( x ) | ≤ C , | α | ≤ n − 1 , wher e the c onstant C > 0 do es not dep e nd on x . Then under c onditions (3.5) and (3.1) with Ω = R n , m is a F ourier multiplier in L p ( · )  ( R n ) . P r o o f . Theorem 4.5 follows from Theorem 3.2 under the choice Ω = X = R n and F = { T m g , g } with g ∈ S ( R n ), if w e ta ke in to accoun t t hat in the case of constan t p 0 > 1 and w eigh t  ∈ A p 0 ( ⊃ A 1 ), a function m , satisfying the assumptions of Theorem 4.5, is a F ourier multiplie r in L p 0  ( R n ). The latter was prov ed in [49], see also [34]. ✷ Corollary 4.6. L et m satisfy the assumptions of The or em 4.5 and let the exp o- nent p and the weight  satisfy the assumptions i) p ∈ P ( R n ) ∩ W L ( R n ) and p ( x ) = p ∞ = const fo r | x | ≥ R with so m e R > 0 , ii)  ∈ V osc p ( · ) ( R n , Π) , Π = { x 1 , ...x N } ⊂ R n . Then m is a F ourier multiplier in L p ( · )  ( R n ) . In p articular, assumption ii) holds for weights  of form  ( x ) = (1 + | x | ) β ∞ N Y k =1 | x − x k | β k , x k ∈ R n , (4.11) wher e − n p ( x k ) < β k < n p ′ ( x k ) , k = 1 , 2 , ..., N , (4.12) − n p ∞ < β ∞ + N X k =1 β k < n p ′ ∞ . (4.13) P r o o f . It suffices to o bserv e that conditions on the weigh t  imp osed in Theorem 4.5, a re fulfilled f or  ∈ V osc p ( · ) ( R n , Π) which follow s from Remark 2.10 and Theorem 2.12. In the case of p o w er weigh ts, conditions defining the class V osc p ( · ) ( R n , Π) turn in to (4.12)-(4.13). ✷ The statemen t of Theorem 4.5 also holds in a more general form of Mikhlin/H¨ orman- der theorem. Theorem 4.7. L e t a function m : R n → R 1 have distributional derivatives up to or der ℓ > n p − satisfying the c ondition sup R> 0    R s | α |− n Z R< | x | < 2 R | D α m ( x ) | s dx    1 s < ∞ 14 for so me s, 1 < s ≤ 2 a n d al l α with | α | ≤ ℓ. If c ondition s (3.5), (3.1) with Ω = X = R n on p and  ar e satisfie d, then m is a F ourier multiplier in L p ( · )  ( R n ) . P r o o f . Theorem 4.7 is similarly deriv ed from fro m Theorems 3.2 , if w e tak e in to accoun t that in the case of constan t p 0 the statemen t o f the theorem for Muc k enhoupt w eigh ts w as prov ed in [50]. ✷ Corollary 4.8. L et a function m : R n → R 1 satisfy the assumptions of The o r em 4.7 and le t p and ρ sa tisfy c ondition s i ) and ii ) of Cor ol lary 4.6. Th en m is a F ourier multiplier in L p ( · )  ( R n ) . P r o o f . F ollows fro m Theorem 4.7 since conditions on the weigh t  imp o sed in Theorem 4.5, are fulfilled fo r  ∈ V osc p ( · ) ( R n , Π) by Theorem 2.12 and Remark 2.10 . ✷ In the next theorem b y ∆ j w e denote the in terv al of the form ∆ j = [2 j , 2 j +1 ] or ∆ j = [ − 2 j +1 , − 2 j ] , j ∈ Z . Theorem 4.9. L et a function m : R 1 → R 1 b e r epr esentable in e ach interval ∆ j as m ( λ ) = λ Z −∞ dµ ∆ j , λ ∈ ∆ j , wher e µ ∆ j ar e finite me asur es such that sup j v ar µ ∆ j < ∞ . If c onditions (3.5), (3.1) with Ω = X = R n on p and  ar e satisfie d , then m is a F ourier multiplier in L p ( · )  ( R 1 ) . P r o o f . T o deriv e Theorem 4.9 from Theorem 3 .4, it suffices to refer to the b oundedness of the maximal op erator in the space L p ( · )  ( R 1 ) b y Theorem 2.12 a nd the fact that in the case of constant p the theorem was pro v ed in [51] (for  ≡ 1) and [34], [35] (for  ∈ A p ). ✷ Corollary 4.10. L et m satisfy the assumptions of T he or em 4.9 and the exp onen t p and weight  fulfil l c onditions i) and ii) of Cor ol lary 4.6 with n = 1 . Then m is a F ourier multiplier in L p ( · )  ( R 1 ) . The ”off-dia gonal” L p ( · )  → L q ( · )  -v ersion o f Theorem 4.9 in the case q ( x ) > p ( x ) is co v ered b y the follo wing theorem. Theorem 4.11. L et p ∈ P ( R 1 ) ∩ W L ( R 1 ) and p ( x ) ≡ p ∞ = const fo r lar ge | x | > R, and let a function m : R 1 → R 1 b e r epr esentable in e ach interval ∆ j as m ( λ ) = λ Z −∞ dµ ∆ j ( t ) ( λ − t ) α , λ ∈ ∆ j , wher e 0 < α < 1 p + and µ ∆ j ar e the sa m e as in The or em 4.9. Then T m is a b ounde d 15 op er ator fr om L p ( · )  ( R 1 ) to L q ( · )  ( R 1 ) , wh e r e 1 q ( x ) = 1 p ( x ) − α and  is a weight of form (4.11) whose exp onents satisfy the c onditions α − 1 p ( x k ) < β k < 1 p ′ ( x k ) , k = 1 , 2 , ..., N , and α − 1 p ∞ < β ∞ + N X k =1 β k < 1 p ′ ∞ . (4.14) P r o o f . In [36] there w as pro v ed that the op erator T m is b ounded from L p 0 v ( R 1 ) in to L q 0 v ( R 1 ) for ev ery p 0 ∈ (1 , ∞ ), 0 < α < 1 p 0 , 1 q 0 = 1 p 0 − α , and an arbitrary w eigh t v satisfying the condition sup I   1 | I | Z I v q 0 ( x ) dx   1 q 0   1 | I | Z I v − p ′ 0 ( x ) dx   1 p 0 , (4 .1 5) where the suprem um is tak en with resp ect to all one-dimensional in terv als. Condi- tion (4.1 5) is satisfied if v q 0 ∈ A 1 . Then inequalit y (3.3) with f = T m g holds fo r ev ery w ∈ A 1 . Then the statement of the theorem f ollo ws immediately from Part II of The- orem 3.4, conditions (3.15)-(3.16) turning in to (4.14) since dim (Ω) = di m ∞ (Ω) = 1, m ( w k ) = M ( w k ) = β k , k = 1 , . . . , N , and m ( w 0 ) = M ( w 0 ) = β ∞ . ✷ All the statemen ts in the following subsections are also similar direct conse- quences of the general statemen t of Theorem 3.4 and Theorems 2.11 and 2.12 on the maximal op erato r in the spaces L p ( · )  , so that in the sequel for the proof s we only ma ke references t o where t hese statemen ts w ere pro v ed in the case of constan t p and Muck enhoupt w eigh ts. 4.3 Multipliers of trigonometric F ourier series With the help of Theorem 3.4 a nd kno wn results for constant exp onents , w e are no w able to giv e a generalization of theorems on Marcinkiewic z multipliers and Littlew o o d-Paley decomp ositions for trigonometric F o urier series to the case of w eigh ted spaces with v ariable exp onen t. Let T = [ π , π ] and let f b e a 2 π -p erio dic function and f ( x ) ∼ a 0 2 + ∞ X k =0 ( a k cos k x + b k sin k x ) . (4.16) Theorem 4.12. L et a se quenc e λ k satisfy the c ond itions | λ k | ≤ A and 2 j − 1 X k =2 j − 1 | λ k − λ k +1 | ≤ A, (4.17) 16 wher e A > 0 do es not dep end on k and j . Supp ose that p ∈ P ( T ) and  − p 0 ∈ A ( e p ) ′ ( T ) , whe r e e p ( · ) = p ( · ) p 0 (4.18) with some p 0 ∈ (1 , p − ( T )) . T hen ther e exists a func tion F ( x ) ∈ L p ( · )  ( T ) such that the series λ 0 a 0 2 + ∞ P k =0 λ k ( a k cos k x + b k sink x ) is F ourier seri e s for F a nd k F k L p ( · )  ≤ cA k f k L p ( · )  wher e c > 0 do es not dep end on f ∈ L p ( · )  ( T ) . Corollary 4.13. The statement of The or em 4.12 r ema ins valid if c on d i tion (4.18) is r e p lac e d by the assumption, sufficie nt fo r (4.18), that p ∈ P ( T ) ∩ W L ( T ) and the weight  has form  ( x ) = N Y k =1 w k ( | x − x k | ) , x k ∈ T (4.19) wher e w k ∈ e U ([0 , 2 π ]) and − 1 p ( x k ) < m ( w k ) ≤ M ( w k ) < 1 p ′ ( x k ) . (4.20) Theorem 4.14. L et A k ( x ) = a k cos k x + b k sin k x, k = 0 , 1 , 2 , ..., A 2 − 1 = 0 . (4.21) Under c onditions (4. 1 8) ther e exist c onstants c 1 > 0 and c 2 > 0 such that c 1 k f k L p ( · )  ≤          ∞ X j =0       2 j − 1 X k =2 j − 1 A k ( x )       2   1 2        L p ( · )  ≤ c 2 k f k L p ( · )  (4.22) for al l f ∈ L p ( · )  ( T ) . In the cas e of constan t p and  ∈ A p this theorem was pro v ed in [49]. Corollary 4.15. I ne qualities (4.22) hold for p ∈ P ( T ) ∩ W L ( T ) a n d weights  of form (4.19)-(4.20). 4.4 Ma joran ts of partial sums of F ourier series Let S ∗ ( f ) = S ∗ ( f , x ) = sup k ≥ 0 | S k ( f , x ) | , 17 where S k ( f , x ) = k P j =0 A j ( x ) is a partial sum of F ourier series (4.16). Theorem 4.16. Unde r c ondi tion s (4.18) k S ∗ ( f ) k L p ( · )  ≤ c k f k L p ( · )  , (4.23) for al l f ∈ L p ( · )  ( T ) , whe r e the c onstant c > 0 do es not dep e n d on f . In the cas e of constan t p and  ∈ A p , Theorem 4.16 was prov ed in [27]. Corollary 4.17. Ine quality (4.23) is va l i d fo r p ∈ P ( T ) ∩ W L ( T ) and wei g h ts  of form (4.19)-(4.20). 4.5 Zygm u nd and Cesaro summabilit y for trigonometric se- ries in L p ( · )  ( T ) Under notation (4.16) and (4.21) w e in tro duce the Zygm und and Cesaro means of summabilit y Z (2) n ( f , x ) = n X k =0 " 1 −  k n + 1  2 # A k ( x ) and σ n ( f , x ) = 1 n + 1 n X k =0 S k ( f , x ) , resp ectiv ely . By Ω p, ( f , δ ) = sup 0 n It is easy to c hec k that the multiplie r λ k ,n satisfies assumptions (4.17) of Theorem 4.12 with t he constan t A in (4.17) not dep ending on n . Therefore, by Theorem 4.12 w e get k S n ( f , · ) − Z (2) n ( f , · ) k L p ( · )  ≤ C      ∞ X k =1 1 − sin k n k n ! A k ( · )      L p ( · )  = C k f − τ h f k L p ( · )  b y (4.28). Th en in view of (4.26) es timate (4.24) follow s. Estimate (4.25) is s imilarly obta ined, with the m ultiplier λ k ,n of the f o rm    k n +1 n „ 1 − sin k n k n « , k ≤ n 0 , k > n . ✷ Corollary 4.19. Estimates (4 . 2 4),(4.25) ar e valid for p ∈ P ( T ) ∩ W L ( T ) and weights  of form (4.19)-(4.20). Remark 4.20. When p > 1 is constan t , estimates (4.24),(4.25) in the non- w eigh ted case w ere obtained in [32]. 4.6 Cauc h y singular in tegral W e consider the singular integral o p erator S Γ f ( t ) = 1 π i Z Γ f ( τ ) dν ( τ ) τ − t , 19 where Γ is a simple finite Carleson curv e and ν is an arc length. Theorem 4.21. L et p ∈ P (Γ) and  − p 0 ∈ A ( e p ) ′ (Γ) (4.29) for some p 0 ∈ (1 , p − ) , wher e e p ( · ) = p ( · ) p 0 . Then the op er ator S Γ is b ounde d in the sp ac e L p ( · )  (Γ) . F or the case of constan t p and  p ∈ A p (Γ), Theorem 4.21 b y differen t metho ds w as prov ed in [3 1] and [5]. (As is kno wn,  − p 0 ∈ A ( e p ) ′ (Γ)) ⇐ ⇒  p ∈ A p p 0 (Γ) for an arbitrary Carleson curv e in the case of constan t p , see [31 ] and [5], so that t he conditions  − p 0 ∈ A ( e p ) ′ (Γ)) and  p ∈ A p (Γ) are eq uiv alent in the sense that the former alw a ys yields the latter for ev ery p 0 > 1 and the latter yields the former for some p 0 > 1). Corollary 4.22. The o p er ator S Γ is b o und e d in the sp ac e L p ( · )  (Γ) , if p ∈ P (Γ) ∩ W L (Γ) and the weight  h a s the form  ( t ) = N Y k =1 w k ( | t − t k | ) , t k ∈ Γ , (4.30) wher e w k ∈ e U ([0 , ν (Γ)]) and − 1 p ( t k ) < m ( w k ) ≤ M ( w k ) < 1 p ′ ( t k ) . (4.31) In the case of p ow er w eights, the statemen t of Corollary 4.22 w as pro v ed in [37], where the case of an infinite Carles on curve w as also dealt with. 4.7 Multidimensional singular op erators W e consider a multidime nsional singular op erator T f ( x ) = lim ε → 0 Z y ∈ Ω : | x − y | >ε K ( x, y ) f ( y ) dy , x ∈ Ω ⊆ R n , (4.32) where w e assu me that the singular k ernel K ( x, y ) s atisfies the assumptions: | K ( x, y ) | ≤ C | x − y | − n , (4.33) | K ( x ′ , y ) − K ( x, y ) | ≤ C | x ′ − x | α | x − y | n + α , | x ′ − x | < 1 2 | x − y | , (4.34) | K ( x, y ′ ) − K ( x, y ) | ≤ C | y ′ − y | α | x − y | n + α , | y ′ − y | < 1 2 | x − y | , (4.35) 20 where α is an arbitra ry p o sitive ex p onent, there exis ts lim ε → 0 Z y ∈ Ω : | x − y | >ε K ( x, y ) dy , (4.36) op erator (4.32) is b ounded in L 2 (Ω). (4.37) Theorem 4.23. L et the kernel K ( x, y ) fulfil l c onditions ( 4.33)-(4.37). Then under the c on ditions p ∈ P (Ω) an d  − p 0 ∈ A ( e p ) ′ (Ω) w i th e p ( · ) = p ( · ) p 0 (4.38) the op er ator T is b ounde d in the sp ac e L p ( · )  (Ω) . In the cas e of constan t p and  ∈ A p ( R n ), Theorem 4.23 w as pro ve d in [9]. Corollary 4.24. L et p ∈ P (Ω) ∩ W L (Ω) and let p ( x ) ≡ p ∞ = const outside some b al l | x | < R in c ase Ω is unb ounde d. The o p er a tor T with the kernel satisfying c ondi tion s (4.33)-(4.37) is b ounde d in the sp ac e L p ( · )  (Ω) with a weight  of the form  ( x ) = N Y k =1 w k ( | x − x k | ) , x k ∈ Ω , (4.39) wher e w k ∈ e U ( R 1 + ) and − 1 p ( x k ) < m ( w k ) ≤ M ( w k ) < 1 p ′ ( x k ) and − n p ∞ < N X k =1 m ∞ ( w k ) ≤ N X k =1 M ∞ ( w k ) < n p ′ ∞ . In t he case o f v ar ia ble p ( · ) , the statement of Corollary 4.2 4 w as pro v ed in [17] in the non-w eigh ted case, and in [39 ] in w eigh ted case (4.41) for bounded sets Ω. 4.8 Comm u tators Let us cons ider the comm utators [ b, T ] f ( x ) = b ( x ) T f ( x ) − T ( bf )( x ) , x ∈ R n generated b y the operat o r ( 4 .32) with Ω = R n and a f unction b ∈ B M O ( R n ). Theorem 4.25. L et the kernel K ( x, y ) fulfil l assumptions (4.33)-(4.37) and let b ∈ B M O ( R n ) . Then under the c onditions p ∈ P ( R n ) and  − p 0 ∈ A ( e p ) ′ ( R n ) with e p ( · ) = p ( · ) p 0 (4.40) the c ommutator [ b, T ] is b ounde d i n the sp ac e L p ( · )  ( R n ) . 21 In the case of constant p and  ∈ A p ( R n ) , 1 < p < ∞ , Theorem 4.25 w as pro v ed in [60 ]. In the case of v aria ble p ( · ), the non- w eigh ted case of Theorem 4.25 w as pro v ed in [30] under t he assumption that 1 ∈ A p ( · ) ( R n ). Corollary 4.26. L et the kern e l K ( x, y ) fulfil l c o n ditions ( 4 . 33)-(4.37) a n d let b ∈ B M O ( R n ) . Then the c omm utator [ b, T ] is b ounde d in the sp ac e L p ( · )  ( R n ) if i) p ∈ P ( R n ) ∩ W L ( R n ) and p ( x ) ≡ p ∞ = const outside some b al l | x | < R , 2) the weight  has the form  ( x ) = w 0 (1 + | x | ) N Y k =1 w k ( | x − x k | ) , x k ∈ R n , (4.4 1) with the factors w k , k = 0 , 1 , ..., N , satisfying c onditions (2.21)-(2.22). 4.9 Pseudo-differen tial op erators W e consider a pseudo-differen tial op erator σ ( x, D ) defined by σ ( x, D ) f ( x ) = Z R n σ ( x, ξ ) e 2 π i ( x,ξ ) ˆ f ( ξ ) dξ . Theorem 4.27. L et the symb o l σ ( x, ξ ) satisfy the c on dition   ∂ α ξ ∂ β x σ ( x, ξ )   ≤ c αβ (1 + | ξ | ) −| α | for al l the multiindic es α and β . T h e n under c ond ition (4.40) the op er ator σ ( x, D ) admits a c ontinuous extension to the sp ac e L p ( · )  ( R n ) . In the cas e of constan t p and  ∈ A p Theorem 4.27 w as prov ed in [57]. Corollary 4.28. L et p ∈ P ( R n ) ∩ W L ( R n ) a nd p ( x ) ≡ p ∞ = const outside some b al l | x | < R and le t  ∈ V osc p ( · ) ( R n , Π) . F or v ariable p ( · ) the statemen t of Coro lla ry 4.28 by a differen t metho d w as prov ed in the non-w eighted case in [61]. 4.10 F effermann-Stein function Let f b e a measurable lo cally in tegrable function on R n , B a n arbitrary ball in R n , f B = 1 | B | R B f ( x ) dx and M # f ( x ) = sup B ∈ X 1 | B | Z B | f ( x ) − f B | d x b e the F efferman-Stein maximal function. 22 Theorem 4.29. Unde r c ondi tion (4.40), the ine q uali ty kM f k L p ( · )  ( R n ) ≤ C kM # f k L p ( · )  ( R n ) (4.42) is va li d, wher e C > 0 do es not dep end on f . In the cas e of constan t p and  ∈ A p inequalit y (4.42) w as pro v ed in [21]. Corollary 4.30. In e quality (4.42) is vali d under the c on d itions: i) p ∈ P ( R n ) ∩ W L ( R n ) and p ( x ) ≡ p ∞ = const outside some b al l | x | < R , ii)  ∈ V osc p ( · ) ( R n , Π) . 4.11 V ector-v alued op erators Let f = ( f 1 , · · · , f k , · · · ), where f i : R n → R 1 are lo cally in tegrable functions. Theorem 4.31. L et 0 < θ < ∞ . Under c onditions (4.40), the ine quality       ∞ X j =1 ( M f j ) θ ! 1 θ       L p ( · )  ( R n ) ≤ C       ∞ X j =1 | f j | θ ! 1 θ       L p ( · )  ( R n ) (4.43) is va li d, wher e c > 0 do es not dep en d on f . In the case of constan t p and  ∈ A p w eigh ted inequalities for v ector-v alued functions w ere pro v ed in [34], [35], [3 6], se e also [3]. Corollary 4.32. In e quality (4.43) is vali d under the c on d itions i) p ∈ P ( R n ) ∩ W L ( R n ) and p ( x ) ≡ p ∞ = const outside some b al l | x | < R , ii)  ∈ V osc p ( · ) (Ω , Π) . Remark 4.33. 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