A NOTE ON SOME OPERATORS ACTING ON CENTRAL MORREY SPACES. Martha Guzmán-Partida. 1. Introduction

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1 MATEMATIČKI VESNIK MATEMATIQKI VESNIK 70, 2 (208), Jue 208 research paper origiali auqi rad A NOTE ON SOME OPERATORS ACTING ON CENTRAL MORREY SPACES Martha Guzmá-Partida Abstract. We prove boudedess of maximal commutators ad covolutio operators with geeralized Poisso kerels o cetral Morrey spaces.. Itroductio Cetral Morrey-Campaato spaces have bee extesively studied durig the last years. We may highlight the cotributios made by Che ad Lau [3], García- Cuerva [4], Guliyev [7], Guliyev ad Aliyev [8], Lu ad Yag [2], ad may others. I this ote, we will be maily iterested to prove, o cetral Morrey spaces, some results related to cotiuity of maximal commutators ad certai covolutio operators with kerels that geeralize the classical Poisso kerel for the upper halfspace R + +. I order to prove the cotiuity of maximal commutators we will lea o results proved by Komori-Furuya et al. i [9]. Cocerig the boudedess of the geeralized Poisso trasform, called Weistei trasform, basically, we will employ properties of the kerel ivolved, which has bee recetly studied by J. Wittste i [5]. We fially obtai as a corollary the cotiuity of the Weistei trasform o weighted versios of local Morrey spaces. We will use stadard otatio alog this work, ad as usual, we shall deote by the letter C a costat that could be chagig lie by lie. 2. Prelimiaries The Morrey spaces were itroduced by C. Morrey i [4]. Here, we will cosider cetral versios of these spaces. They are defied as follows ( [,3,4,2]): for < p < 200 Mathematics Subject Classificatio: 42B35, 26D0, 44A35 Keywords ad phrases: Morrey space; maximal commutator; Weistei trasform. 55

2 56 A ote o some operators actig o cetral Morrey spaces ad λ R, a fuctio f L p loc (R ) belogs to the cetral Morrey space B p,λ if f B p,λ <, where ( /p f B p,λ := sup r>0 B r (0) +pλ f (x) dx) p. B r(0) It is well kow that ( B p,λ, B p,λ) are Baach spaces. We will restrict to the case /p λ, sice B p,λ reduces to zero for λ < /p. Moreover, if λ < µ the B p,λ is properly cotaied i B p,µ. Also, for λ = /p, B p, /p = L p. A alterative way to describe the spaces B p,λ is the followig ( [,3,4]): f B p,λ if ad oly if sup 2 k/p(+pλ) fχ Ck p <, () k Z where C k = { x R : 2 k < x 2 k}. The quatity i () defies a equivalet orm i B p,λ. Clearly, for /p < λ < 0, the classical Morrey spaces L p,λ o R defied by meas of the coditio ( ) /p f L p,λ := sup r>0, a R B r (a) +pλ f (x) p dx B r(a) are icluded i B p,λ, however, this iclusio is proper as the followig example shows. Let us cosider =, although appropriate modificatios will work for arbitrary. Defie ϕ (x) = k= 2 k( p(+pλ) p) χck (x). Sice for every k Z we have 2 k/p(+pλ) 2 k( p(+pλ) p) 2 k/p =, it is immediate that () is fiite, which shows that ϕ B p,λ. However, give k N, k 2, let I be a iterval whose legth is 2 k ad it is completely cotaied i C k. I this way I +pλ ϕ (x) p dx = I I +pλ 2k( p(+pλ) p)p I (2) = 2 (k )( pλ) 2 k( +pλ ) = 2 pλ 2 k( pλ+ +pλ ), ad oticig that pλ + +pλ > 0 sice /p < λ < 0, we see that the itegrals i the left-had side of (2) grow without boud as k. This shows that ϕ / L p,λ. It is also possible to idetify the preduals of the spaces B p,λ, i a similar fashio as the case of Morrey spaces. For p < r, a fuctio b : R R is called a (p, r)-cetral block, if there exists t > 0 such that supp b B t (0) ad b r t ( r p). If we defie h 0 p,r := { f = β j b j :b j is a (p, r)-cetral block ad f h 0 p,r j= where f h 0 p,r = if } <, (3) β j, (4) we obtai a Baach space ormed by (4). Moreover, the covergece of the series i j=

3 M. Guzmá-Partida 57 (3) is i L p ad absolutely a.e. With the same proof give i [] (see also [0]) we ca obtai the followig result. Propositio 2.. For p < r <, r < λ < 0, r + r = ad p = +λ we have ( h 0 p,r ) B r,λ. 3. Maximal commutators I [9], Komori-Furuya et al. studied the cotiuity of several classical operators o Morrey-Campaato type spaces. I particular, they cosidered the fractioal maximal operator M α, for 0 α <, which is defied as M α f (x) := sup x Q Q f (y) dy, α Q where Q is a cube with sides parallel to the coordiate axes (istead of cubes, we could also cosider balls). Notice that for α = 0, we recover the classical Hardy-Littlewood maximal fuctio. They proved the followig results: Theorem 3.. ( [9], Theorem 7) For < p <, 0 σ p, σ p Hardy-Littlewood maximal operator is cotiuous from B p,ν to B p,v. ν 0, the It is iterestig to observe that Theorem 3. geeralizes the case ν = 0 proved i [4] ad [3]. Theorem 3.2. ( [9, Theorem 7]) For 0 < α <, σ 0, < p σ+α, σ p ν α, < q ( ν σ+α) p, the fractioal maximal operator Mα is cotiuous from B p,ν to B q,ν+α/. Now, we shall use Theorem 3.2 i order to examie the actio of the maximal commutator with a Lipschitz fuctio i the spaces B p,λ. Maximal commutators with a give fuctio have played a importat role i the study of cotiuity properties of some classical operators. We highlight the work doe by García-Cuerva et al. i [5]. Give b L loc, the maximal commutator of the Hardy-Littlewood maximal operator M with b is defied as C b (f) (x) = sup b (x) b (y) f (y) dy. x Q Q Q If we allow a appropriate smoothess coditio o the fuctio b, let us say, that b is a Lipschitz fuctio, that is, for every x, y R b (x) b (y) C x y β (5) where C is a positive costat, ad β (0, ), we ca proceed as follows: C b (f) (x) = sup b (x) b (y) f (y) dy x Q Q β + β Q

4 58 A ote o some operators actig o cetral Morrey spaces C b Λβ sup x Q where b Λβ satisfyig (5). Now, if we choose σ 0, < p Q β Q β Q β Q f (y) dy C b Λβ M β f (x), (6) deotes the Lipschitz orm of b, i.e., the ifimum of the costats C σ+β, σ p ν β, < q ( ) ν σ ν σ+β p, by estimate (6) ad Theorem 3.2 we obtai the cotiuity of C b : B p,ν B q,ν+β/ ad C b B p,ν B C b q,ν+β/ Λ β. Thus, we have proved Propositio 3.3. For 0 < β <, b Λ β, σ 0, < p σ+β, σ p ν β, ad < q ( ) ν σ ν σ+β p, the maximal commutator Cb is cotiuous from B p,ν to B q,ν+β/ with orm C b B p,ν B C b q,ν+β/ Λ β. 4. Weistei trasform Recetly, J. Wittste [5] has studied boudary values of covolutios of weighted distributios with the kerels K α, α >, defied by Γ ((α + + ) /2) t α+ K α (x, t) := Γ ((α + ) /2) π /2 ( x 2, for (x, t) + t 2 R+ )(α++)/2 +. These kerels are related to the elliptic partial differetial equatio ( D α u := t α 2 u x u x u t 2 α ) u = 0, (7) t t with α >. Solutios to (7) are called geeralized axially symmetric potetials. Notice that whe α = 0 we recover the Laplace equatio. I the paper [5], it was prove that K α,t =, where K α,t (x) := K α (x, t), that K α is a solutio to the equatio (7) i R + +, ad K α,t δ 0 i S as t 0. I this sectio, we will examie the behavior of the family of kerels K α,t whe they act by covolutio i the cetral Morrey spaces B p, < p <. For θ >, we have the iclusio B p L p ( ( + x 2) θ/2 dx ), as it has bee proved i [, Corollary 2.5]. If we deote by Λ α the covolutio operator with the kerel K α,t, this operator preserves the space L ( ( + x 2) (α++)/2 dx ) (see [5] ad also [2, Remark 3.2]). However, we ca say more, as the followig result shows. Propositio 4.. The operator Λ α is bouded from L p ( ( + x 2) (α++)/2 dx ) to itself for p <.

5 M. Guzmá-Partida 59 Proof. Let us deote w α (x) = ( + x 2 ) (α++)/2. Usig Jese s iequality, Toelli theorem ad radiality of K α,t we obtai p Λ α (f) p L p (wα dx) R = K α,t (x y) f (y) dy wα (x) dx R f (y) p ( K α,t w ) α (y) dy. R R The ext step is to estimate K α,t wα (y). Accordig to [5, pp ], K α,t wα (y) C α, wα (y), where C α, is a costat oly depedig o ad α. Therefore, Λ α (f) p L p (wα dx) C α, f (y) p wα (y) dy = C α, f p R L p (wα dx). This cocludes the proof. Remark 4.2. It is also true that for f L ( p wα dx ), p <, we have K α,t f f i L ( p wα dx ) as t 0. The proof of this assertio is basically the same as that give i [5, Theorem 4.3], (see also [2, Theorem 3.6]). Now, we will prove the desired cotiuity. Theorem 4.3. The Weistei trasform Λ α is bouded from B p ito itself, < p <. Proof. As i the case of the Poisso kerel, we ca obtai the followig estimate (see [6, pp. 54 ad 77]) for each t > 0 { t α+ f (y) K α,t f(x) C,α (t 2 + y x 2) dy (α++)/2 + k=0 y x t 2 k t< y x 2 k+ t { C,α t y x t f (y) dy + t α+ } f (y) (t 2 + y x 2) dy (α++)/2 (2 k t) k=0 y x 2 k+ t } f (y) dy C,α Mf(x). (8) Usig the fact that M : B p B p is a cotiuous operator (Theorem 3.), we obtai the boudedess of the operator Λ α from B p to B p. We ca improve the previous result usig weighted versios of the cetral Morrey spaces B p. These are defied as follows. Let w be a weight o R, that is, w L loc ad 0 < w < a.e. } o R. We will deote by B p (w) the space B p (w) := {f L ploc,w : f B p(w) <, where L p loc,w is the space of fuctios locally i L p (w) ad f Bp (w) = sup k Z w (C k ) /p fχ Ck Lp (w). Matsuoka has proved i [3]: Propositio 4.4. ( [3]) Let < p < ad w A p. The, the Hardy-Littlewood maximal operator M is bouded from B p (w) ito B p (w).

6 60 A ote o some operators actig o cetral Morrey spaces Here, the set A p deotes the classical family of weights w satisfyig the coditio ( ) ( p w (x) dx w (x) dx) /(p ) C B B B B for every ball B ad some positive costat C. I view of Propositio 4.4 ad estimate (8) we ca obtai Corollary 4.5. If w A p the the Weistei trasform Λ α is bouded from B p (w) ito B p (w). Refereces [] J. Alvarez, M. Guzmá-Partida, J. Lakey, Spaces of bouded λ-cetral mea oscillatio, Morrey spaces, ad λ-cetral Carleso measures, Collect. Math. 5 (2000), 47. [2] J. Alvarez, M. Guzmá-Partida, S. Pérez-Esteva, Harmoic extesios of distributios, Math. Nachr. 280 (2007), [3] Y. Che, K. Lau, Some ew classes of Hardy spaces, J. Fuct. Aal. 84 (989), [4] J. García-Cuerva, Hardy spaces ad Beurlig algebras, J. Lodo Math. Soc. 39 (989), [5] J. García-Cuerva, E. Harboure, C. Segovia, J. L. Torrea, Weighted orm iequalities for commutators of strogly sigular itegrals, Idiaa Uiv. Math. J. 40 (99), [6] J. García-Cuerva, J. L. Rubio de Fracia, Weighted orm iequalities ad related topics, North-Hollad, Amsterdam, 985. [7] V. S. Guliyev, Geeralized weighted Morrey spaces ad higher order commutators of subliear operators, Eurasia Math. J. 3 (202), [8] V. S. Guliyev, S. A. Aliyev, Boudedess of the parametric Marcikiewicz itegral operator ad its commutators o geeralized Morrey spaces, Georgia Math. J. 9 (202), [9] Y. Komori-Furuya, K. Matsuoka, E. Nakai ad Y. Sawao, Itegral operators o B σ-morrey- Campaato spaces, Rev. Mat. Complut. 26 (203), 32. [0] Y. Komori, T. Mizuhara, Notes o commutators ad Morrey spaces, Hokkaido Math. J. 32 (2003), [] R. Log, The spaces geerated by blocks, Scietia Siica Ser. A 27 (984), [2] S. Lu, D. Yag, The cetral BMO spaces ad Littlewood-Paley operators, Approx. Theory Appl. (995), [3] K. Matsuoka, O some weighted Herz spaces ad the Hardy-Littlewood maximal operator, Proc. It. Symp. Baach ad Fuctio Spaces II, Kitakyushu, Japa (2006), [4] C. Morrey, O the solutios of quasi-liear elliptic partial differetial equatios, Tras. Amer. Math. Soc. 43 (938), [5] J. Wittste, Geeralized axially symmetric potetials with distributioal boudary values, Bull. Sci. Math. 39 (205), (received ; i revised form ; available olie 207) Departameto de Matemáticas, Uiversidad de Soora, Hermosillo, Soora 83000, México martha@mat.uso.mx