Convergence of Singular Integral Operators in Weighted Lebesgue Spaces

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1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, ISSN Published by New York Business Globl Convergence of Singulr Inegrl Operors in Weighed Lebesgue Spces Mine Menekse Yilmz 1, Gumrh Uysl 2, 1 Deprmen of Mhemics, Fculy of Ars nd Science, Gzinep Universiy, Gzinep, Turkey 2 Deprmen of Compuer Technologies, Division of Technology of Informion Securiy, Krbuk Universiy, Krbuk, Turkey Absrc. In his pper, he poinwise pproximion o funcions f L 1,w, b by he convoluion ype singulr inegrl operors given in he following form: L λ (f; x) f () K λ ( x) d, x, b, λ Λ R 0 where, b snds for rbirry closed, semi closed or open bounded inervl in R or R iself, L 1,w, b denoes he spce of ll mesurble bu non-inegrble funcions f for which is inegrble on, b nd w : R R is corresponding weigh funcion, µ-generlized Lebesgue poin nd he re of convergence his poin re sudied Mhemics Subjec Clssificions: 41A35, 41A25, 45P05 Key Words nd Phrses: Generlized Lebesgue poin, Weighed poinwise convergence, Re of convergence f w 1. Inroducion In pper [16], Tberski nlyzed he poinwise convergence of inegrble funcions nd he pproximion properies of derivives of inegrble funcions in L 1 π, π by wo prmeer fmily of convoluion ype singulr inegrl operors of he form: T λ (f; x) π π f () K λ ( x) d, x π, π, λ Λ R 0, (1) where K λ () is he kernel fulfilling pproprie ssumpions nd λ Λ nd Λ is given se of non-negive numbers wih ccumulion poin λ 0. Ler on, he weighed Corresponding uhor. Emil ddresses: menekse@gnep.edu.r (M. Menekse Yilmz), guysl@krbuk.edu.r (G. Uysl) hp:// 335 c 2017 EJPAM All righs reserved.

2 M. M. Yilmz, G. Uysl, / Eur. J. Pure Appl. Mh, 10 (2) (2017), poinwise convergence nd re of convergence of fmily of m-singulr inegrl operors in f L p (R) were invesiged by Mmedov [14]. Then, Gdjiev [9] nd Rydzewsk [15] sudied he poinwise convergence heorems nd he order of poinwise convergence heorems for operors ype (1) generlized Lebesgue poin nd µ generlized Lebesgue poin of f L 1 ( π, π) bsed on Tberski s nlysis, respecively. Furher, in [10] nd [12] Krsli exended he resuls of Tberski [16], Gdjiev [9] nd Rydzewsk s [15] sudies by considering he more generl inegrl operors of he form: T λ (f; x) f () K λ ( x) d, x, b, λ Λ R 0 (2) for funcions in L 1, b, where, b is n rbirry inervl in R such s [, b], (, b), [, b) or (, b]. In [11, 13], Krsli nlyzed he poinwise convergence heorems nd he re of poinwise convergence heorems for fmily of nonliner singulr inegrl operors µ generlized Lebesgue poin nd generlized Lebesgue poin of f L 1, b, respecively. Brdro nd Cocchieri [2] evlued he degree of poinwise convergence of Fejer-Type singulr inegrls he generlized Lebesgue poins of he funcions f L 1 (R). In n noher sudy, Brdro [3] sudied similr convergence resuls bou momen ype operors. Besides, he poinwise convergence of fmily of nonliner Mellin ype convoluion operors Lebesgue poins ws lso nlyzed by Brdro nd Mnellini [4]. Brdro, Krsli nd Vini [5] obined some pproximion resuls reled o he poinwise convergence nd he re of poinwise convergence for non-convoluion ype liner operors Lebesgue poin bsed on Brdro nd Mnellini s sudy [4]. Afer h, hey go similr resuls for is nonliner counerpr in [6] while in n noher sudy [7], he poinwise convergence nd he re of poinwise convergence resuls for fmily of Mellin ype nonliner m-singulr inegrl operors m-lebesgue poins of f were invesiged. Almli [1] sudied he problem of poinwise convergence of non-convoluion ype inegrl operors Lebesgue poins of some clsses of mesurble funcions. In his pper, we lso invesiged he poinwise convergence nd he re of convergence of he operors similr o he sudy of Krsli [12]. However, he difference beween his sudy nd Krsli [12] is h while Krsli [12] considers he poinwise convergence of he inegrl operors o he funcions f L 1, b, our sudy covers he cse f / L 1, b. The min conribuion of his pper is obining he poinwise convergence of he convoluion ype singulr inegrl operors of he form: L λ (f; x) f () K λ ( x) d, x, b, λ Λ R 0, (3) where he symbol, b snds for n rbirry closed, semi closed or open bounded inervl in R or R iself, o he funcion f L 1,w, b where L 1,w, b is he spce of

3 M. M. Yilmz, G. Uysl, / Eur. J. Pure Appl. Mh, 10 (2) (2017), ll mesurble nd non-inegrble funcions f for which f w is inegrble on, b nd w : R R is corresponding weigh funcion, common µ-generlized Lebesgue poin of f w nd w. The pper is orgnized s follows: In Secion 2, we inroduce he fundmenl definiions. In Secion 3, we prove he exisence of he operors ype (3). In Secion 4, we presen wo heorems concerning he poinwise convergence of L λ (f; x, y) o f (x 0, y 0 ) whenever (x 0, y 0 ) is common µ generlized Lebesgue poin of f w nd w. In secion 5, we give wo heorems concerning he re of poinwise convergence. 2. Preliminries Definiion 1. A poin x 0, b is clled µ generlized Lebesgue poin of he funcion f L 1, b, if h 1 lim f(x 0 ) f(x 0 ) d 0, h 0µ (h) 0 where he funcion µ : R R is incresing nd bsoluely coninuous on [0, b ] nd µ(0) 0 [15, 12]. Exmple 1. Consider he funcion f L 1 (R) defined by { e f(), if (0, 1] 0, if R\(0, 1]. One cn compue h x 0 0 is no Lebesgue poin of f. On he oher hnd, by king µ() we see h x 0 0 is µ generlized Lebesgue poin of f. Now, we define new clss for he weighed pproximion. Definiion 2. (Clss A w ) Le Λ R 0 be n index se nd λ 0 is n ccumulion poin of i. Furher, le K λ ]: R R be n inegrble funcion for ech λ Λ nd ϕ(),, b. sup x,b [ w(x) w(x) I is sid h K λ () belongs o clss A w, if i sisfies he following condiions:. K λ ϕ L1 (R) M <, λ Λ. [ ] b. lim λ λ 0 c. lim λ λ 0 sup K λ () >ξ [ K λ () d >ξ 0, ξ > 0. ] 0, ξ > 0. d. For given δ 0 > 0, K λ () is non-decresing funcion wih respec o on δ 0, 0] nd non-incresing funcion wih respec o on [0, δ 0.

4 M. M. Yilmz, G. Uysl, / Eur. J. Pure Appl. Mh, 10 (2) (2017), e. A some x R, K λ (x) ends o infiniy s λ ends o λ 0. f. lim K λ () d 1 0. λ λ 0 R Throughou his pper K λ belongs o clss A w. 3. Exisence of he Operors Min resuls in his work re bsed on he following heorem. Theorem 1. Suppose h f L 1,w, b. Then he operor L λ (f; x) defines coninuous rnsformion cing on L 1,w, b. Proof. Le, b be n rbirry closed, semi closed or open bounded inervl in R. By he lineriy of he operor L λ (f; x), i is sufficien o show h he expression: L λ (f; x) L1,w,b L λ 1,w sup f 0 f L1,w,b remins finie. Here, he norm of f L 1,w, b is given by he following equliy (see, for exmple [14]): b f L1,w,b f(x) w(x) dx. Le us define new funcion g by g() : { f(), if, b 0, if R\, b. Now, using Fubini s Theorem [8] we cn wrie 1 b L λ (f; x) L1,w,b w(x) f()k λ ( x)d dx 1 w( x) w(x) g( x) w( x) K λ()d dx 1 g( x) w(x) w( x) w( x) K λ()d dx 1 w(x) w( x) g( x) w( x) K λ() d dx

5 M. M. Yilmz, G. Uysl, / Eur. J. Pure Appl. Mh, 10 (2) (2017), K λ () M f L1,w,b. w( x) w(x) g( x) w( x) dx d Now, he proof of he indiced cse is compleed. The sserion cn be proved wih he bove mehod for he cse, b R. Thus he proof is compleed. 4. Convergence Chrcerisic Poins In his secion, wo heorems concerning poinwise convergence of he operors of ype (3) will be presened. In he following heorem we suppose h, b is n rbirry closed, semi closed or open bounded inervl in R. Theorem 2. Suppose h w() nd K λ ( x) re lmos everywhere differenible funcions on R wih respec o vrible such h he following inequliy: d d w() d d K λ( x) > 0, for ny fixed x, b (4) holds. If x 0, b is common µ generlized Lebesgue poin of funcions f L 1,w, b nd w L 1, b, hen on ny plnr se Z on which he funcion lim L λ(f; x) f(x 0 ) (x,λ) (x 0,λ 0 ) K λ ( x) w() {µ( x 0 )} d 2 K λ (0) w (x) µ ( x 0 x ), 0 < δ < δ 0, where δ 0 is fixed posiive rel number, is bounded s (x, λ) ends o (x 0, λ 0 ). Proof. Le x 0 δ < b, x 0 δ > nd x 0 x < δ 2 for given δ > 0. The proof will be sed for he cse 0 < x 0 x < δ 2. The proof of he reverse cse is similr. Se I L λ (f; x) f (x). Using Theorem 2 in [12] we my wrie he inegrl I s follows: I f () K λ ( x) d f(x 0 ) [ f () w () f (x ] 0) w () K λ ( x) d f (x 0) b w () K λ ( x) d w (x 0 ) w (x 0 ) w (x 0 )

6 M. M. Yilmz, G. Uysl, / Eur. J. Pure Appl. Mh, 10 (2) (2017), f () w () f (x 0) w (x 0 ) w () K λ ( x) d w (x 0 ) x 0 x 0 δ x 0 w (x 0 ) w () K λ ( x) d w (x 0 ) f () w () f (x 0) w (x 0 ) w () K λ ( x) d w () K λ ( x) d w (x 0 ) I 1 I 2 I 3 I 4 I 5. I is sufficien o show h I i 0 (i 1, 5) s (x, λ) (x 0, λ 0 ) provided (x, λ) Z. Le us consider he inegrl I 5. By Theorem 2 in [12], we ge Therefore, we hve lim w () K λ ( x) d w (x 0 ). (x,λ) (x 0,λ 0 ) lim I 5 0. (x,λ) (x 0,λ 0 ) Now, we consider he inegrls I 1 nd I 4. From hypohesis (4) nd condiion (d) of clss A w, we hve I 1 x 0 δ f () w () f (x 0) w (x 0 ) w () K λ ( x) d sup, w () sup, K λ ( x) x 0 δ f () w () f (x 0) w (x 0 ) d Hence we obin I 1 w (x 0 δ) sup K λ (ξ) f () ξ > δ w () f (x 0) w (x 0 ) d 2 { } w (x 0 δ) sup K λ (ξ) f L1,w,b ξ > δ w (x 0 ) (b ). 2 Using similr sregy, we hve he following inequliy for he inegrl I 4 { } I 4 w (x 0 δ) sup K λ (ξ) f L1,w,b ξ > δ w (x 0 ) (b ). 2

7 M. M. Yilmz, G. Uysl, / Eur. J. Pure Appl. Mh, 10 (2) (2017), Combining hese inegrls, we ge he following inequliy: { } I 1 I 4 {w (x 0 δ) w (x 0 δ)} sup K λ (ξ) f L1,w,b ξ > δ w (x 0 ) (b ). 2 In view of condiion (b) of clss A w, I 1 I 4 0 s λ λ 0. Now, we consider he inegrl I 2. By he definiion of µ generlized Lebesgue poin, for every ε > 0 here exiss corresponding number δ > 0 such h he expression: x 0 x 0 h holds for ll 0 < h δ < δ 0. Define he funcion F () by From (6), we hve f () w () f (x 0) w (x 0 ) d < εµ (h) (5) F () x 0 f (y) w (y) f (x 0) w (x 0 ) dy. (6) df () f () w () f (x 0) w (x 0 ) d. (7) From (5) nd (6), for ll sisfying 0 < x 0 δ < δ 0 we hve By virue of (6) nd (7) we hve F () εµ (x 0 ). (8) I 2 x 0 x 0 f () w () f (x 0) w (x 0 ) K λ ( x) w () d K λ ( x) w () d [ F ()]. Using inegrion by prs nd pplying (8), we hve he following inequliy: I 2 εµ (δ) K λ (x 0 δ x) w (x 0 δ) ε I is esy o see h x 0 I 2 εµ (δ) K λ (x 0 δ x) w (x 0 δ) µ (x 0 ) d K λ ( x) w ().

8 M. M. Yilmz, G. Uysl, / Eur. J. Pure Appl. Mh, 10 (2) (2017), x 0 x ε µ (x 0 x ) d K λ (u) w (u x) x 0 x δ x 0 x δ. If we use hypohesis (4) nd inegrion by prs, hen we hve he following expression: x 0 x I 2 ε K λ (u) w (u x) {µ (x 0 x )} d. x 0 x δ x 0 x δ Using condiion (d) of clss A w, we obin I 2 ε x 0 K λ ( x) w () {µ (x 0 )} d 2 K λ (0) w (x) µ (x 0 x). Using preceding mehod, we cn esime he inegrl I 3 s I 3 ε x 0 Combining I 2 nd I 3, we obin K λ ( x) w () {µ ( x 0 )} d. I 2 I 3 ε K λ ( x) w () {µ ( x 0 )} d 2ε K λ (0) w (x) µ ( x 0 x ). Noe h he bove inequliy is obined for he cse 0 < x x 0 < δ 2. Therefore, if he poins (x, λ) Z re sufficienly ner o (x 0, λ 0 ), we hve I 2 I 3 ends o zero. Thus, he proof is compleed. In he nex heorem we suppose h, b R. Theorem 3. Suppose h w() nd K λ ( x) re lmos everywhere differenible funcions on R wih respec o vrible such h he following inequliy: d d w() d d K λ( x) > 0, for ny fixed x R. (4) holds. If x 0 R is common µ generlized Lebesgue poin of funcions f L 1,w (R) nd w L 1 (R), hen lim (x,λ) (x 0,λ 0 ) L λ(f; x) f(x 0 ) on ny plnr se Z on which he funcion K λ ( x) w() {µ( x 0 )} d 2 K λ (0) w (x) µ ( x 0 x ), 0 < δ < δ 0, where δ 0 is fixed posiive rel number, is bounded s (x, λ) ends o (x 0, λ 0 ).

9 M. M. Yilmz, G. Uysl, / Eur. J. Pure Appl. Mh, 10 (2) (2017), Proof. Mking similr clculions s in Theorem 2 we hve L λ (f; x) f(x 0 ) sup K λ (ξ) {w (x 0 δ) w (x 0 δ)} f L1,w (R) ξ > δ 2 {w (x 0 δ) w (x 0 δ)} w (x 0 ) K λ (ξ) dξ ξ > δ 2 ε w (x 0 ) K λ ( x) w() {µ( x 0 )} d 2ε K λ (0) w (x) µ ( x 0 x ) w () K λ ( x) d w (x 0 ). Using condiions (b) nd (c) of clss A w nd Theorem 3 in [12] we obin L λ (f; x) f(x 0 ) 0 s (x, λ) (x 0, λ 0 ). Hence he proof is compleed. Exmple 2. The following Guss-Weiersrss ype kernel nd weigh funcions sisfy he hypohesis of Theorem 3, respecively. Le Λ (0, ), λ 0 0 nd K λ : R R for ech λ Λ is given by K λ () 1 λ 2 π e λ 2 nd w : R R is given by w() { 1, if 0 1 (1 ), oherwise. For deiled nlysis of he bove funcions, uhors refer o [1]. 5. Re of Convergence In his secion, wo heorems concerning re of poinwise convergence will be given. Theorem 4. Suppose h he hypohesis of Theorem 2 is sisfied. Le (x, λ, δ) x 0 δ K λ ( x) w () {µ( x0 )} d 2 Kλ (0) w (x) µ ( x 0 x ), where 0 < δ δ 0, nd he following condiions re sisfied: i. (x, λ, δ) 0 s (x, λ) (x 0, λ 0 ) for some δ > 0.

10 M. M. Yilmz, G. Uysl, / Eur. J. Pure Appl. Mh, 10 (2) (2017), ii. For every ξ > 0 s (x, λ) (x 0, λ 0 ). K λ (ξ) o( (x, λ, δ)) iii. s (x, λ) (x 0, λ 0 ). w () K λ ( x) d w (x 0 ) o( (x, λ, δ)) Then ech common µ generlized Lebesgue poin of funcions f L 1,w, b nd w L 1, b we hve s (x, λ) (x 0, λ 0 ) L λ (f; x) o( (x, λ, δ)). Proof. Under he hypohesis of Theorem 2, we my wrie L λ (f; x) f (x) {w (x 0 δ) w (x 0 δ)} { } sup K λ (ξ) f L1,w,b ξ > δ w (x 0 ) (b ) 2 ε w (x 0 ) K λ ( x) w () {µ( x 0 )} d 2ε K λ (0) w (x) µ ( x 0 x ) w () K λ ( x) d w (x 0 ). From (i)-(iii) we hve he desired resul i.e.: L λ (f; x) o( (x, λ, δ)). Theorem 5. Suppose h he hypohesis of Theorem 3 is sisfied. Le (x, λ, δ) x 0 δ K λ ( x) w () {µ( x 0 )} d 2 K λ (0) w (x) µ ( x 0 x ) where 0 < δ δ 0, nd he following condiions re sisfied: i. (x, y, λ, δ) 0 s (x, λ) (x 0, λ 0 ) for some δ > 0.

11 M. M. Yilmz, G. Uysl, / Eur. J. Pure Appl. Mh, 10 (2) (2017), ii. For every ξ > 0 s (x, λ) (x 0, λ 0 ). iii. For every ξ > 0 s (x, λ) (x 0, λ 0 ). lim λ λ 0 K λ (ξ) o( (x, λ, δ)) >ξ K λ () d o( (x, λ, δ)) iv. s (x, λ) (x 0, λ 0 ). w () K λ ( x) d w (x 0 ) o( (x, λ, δ)) Then ech common µ generlized Lebesgue poin of funcions f L 1,w (R) nd w L 1 (R) we hve s (x, λ) (x 0, λ 0 ) L λ (f; x) o( (x, λ, δ)). Proof. Under he hypohesis of Theorem 3, we wrie L λ (f; x) sup K λ (ξ) {w (x 0 δ) w (x 0 δ)} f L1,w (R) ξ > δ 2 {w (x 0 δ) w (x 0 δ)} w (x 0 ) K λ (ξ) dξ ξ > δ 2 ε K λ ( x) w() {µ( x0 )} d 2ε Kλ (0) w (x) µ ( x 0 x ) w (x 0 ) w () K λ ( x) d w (x 0 ) nd from (i)-(iv) we hve he desired resul i.e.: L λ (f; x) o( (x, λ, δ)). Acknowledgemens The uhors hnk he referees for heir vluble commens nd suggesions.

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