NORM ESTIMATES FOR BESSEL-RIESZ OPERATORS ON GENERALIZED MORREY SPACES

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1 NORM ESTIMATES FOR ESSEL-RIESZ OPERATORS ON GENERALIZED MORREY SPACES Mochammad Idri, Hedra Guawa, ad Eridai 3 Departmet of Mathematic, Ititut Tekologi adug, adug 403, Idoeia [Permaet Addre: Departmet of Mathematic, Lambug Magkurat Uiverity, ajarbaru Campu, ajarbaru 7074, Idoeia] Departmet of Mathematic, Ititut Tekologi adug, adug 403, Idoeia 3 Departmet of Mathematic, Airlagga Uiverity, Campu C Mulyorejo, Surabaya 605, Idoeia addree: idemath@gmail.com, hguawa@math.itb.ac.id, ad 3 eridai.diadewi@gmail.com Abtract. We reviit the propertie of eel-riez operator ad preet a differet proof of the boudede of thee operator o geeralized Morrey pace. We alo obtai a etimate for the orm of thee operator o geeralized Morrey pace i term of the orm of their kerel o a aociated Morrey pace. A a coequece of our reult, we reprove the boudede of fractioal itegral operator o geeralized Morrey pace, epecially of expoet, ad obtai a ew etimate for their orm. Key word: eel-riez operator, fractioal itegral operator, geeralized Morrey pace. MSC 000: Primary 40; Secodary 6A33, 45, 6D0. Itroductio Itegral operator uch a maximal operator ad fractioal itegral operator have bee tudied exteively i the lat four decade. Here we are itereted i eel-riez operator, which are related to fractioal itegral operator. Let 0 < α < ad γ 0. The operator I α,γ which map every f L p loc (R ), p <, to I α,γ f (x) := K α,γ (x y)f(y)dy = K α,γ f(x), x R, R where K α,γ (x) := x α (+ x ), i called eel-riez operator, ad the kerel K γ α,γ i called eel-riez kerel. The boudede of thee operator o Morrey pace ad o geeralized Morrey pace wa tudied i [0, ]. Let p < ad ϕ : R + R + be of cla G p, that i ϕ i almot decreaig [ C > 0 uch that ϕ(r) Cϕ() for r ] ad ϕ p (r)r i almot icreaig [ C > 0 uch that ϕ p (r)r Cϕ p () for

2 Norm etimate for eel-riez operator o geeralized Morrey pace r ]. Clearly if ϕ i of cla G p, the ϕ atifie the doublig coditio, that i, there exit C > 0 uch that C ϕ(r) ϕ() C wheever r. We defie the geeralized Morrey pace Lp,ϕ (R ) to be the et of all fuctio f L p loc (R ) for which ( ) L p,ϕ := up f(x) p p dx <, =(a,r) ϕ(r) where deote the Lebegue meaure of. (Recall that the Lebegue meaure of = (a, r) i (a, r) = C r for every a R ad r > 0, where C > 0 deped oly o.) If p q < ad ϕ(r) := C r q (r > 0), the L p,ϕ (R ) i the claical Morrey pace L p,q (R ), which i equipped by ( ) L p,q := up q p f(x) p p dx. =(a,r) Particularly, for p = q, L p,p (R ) i the Lebegue pace L p (R ). I [], we kow that for γ > 0, K α,γ i a member of L t (R ) pace for ome value of t depedig o α ad γ. It follow from Youg iequality [4] that I α,γ f L q K α,γ L t L p, f L p (R ), wheever p < t, q = p t (where t deote the dual expoet of t) ad +γ α < t < Thi tell u that I α,γ i bouded from L p (R ) to L q (R ) with I α,γ Lp L q α. K α,γ L t. I [0], it i alo how that I α,γ i bouded o geeralized Morrey pace but without a good etimate for it orm a o Morrey pace. We hall ow refie the reult, by etimatig the orm of the operator more carefully through the memberhip of K α i Morrey pace. Note that for γ = 0, I α,0 = I α i the fractioal itegral operator with kerel K α (x) := x α. Hardy-Littlewood [7, 8] ad Sobolev [7] proved the boudede of I α o Lebegue pace. boudede of I α o Morrey pace i proved by Spae [6], ad improved by Adam [] ad Chiareza-Fraca []. Later, Nakai [3] obtaied the boudede of I α o geeralized Morrey pace, which ca be viewed a a exteio of Spae reult. I 009, Guawa-Eridai [5] proved the boudede of I α o geeralized Morrey pace which exted Adam ad Chiareza-Fraca reult. I thi paper, we give a ew proof of the boudede of I α,γ o geeralized Morrey pace. At the ame time, a upper boud for the orm of the operator i obtaied. A a coequece of our reult, we have a etimate for the orm of I α (from a geeralized Morrey pace to aother) i term of the orm of K α o the aociated Morrey pace. A lower boud for the orm of the operator i dicued i 3. The The oudede of I α,γ o Geeralized Morrey Space We begi with a lemma about the memberhip of K α i ome Morrey pace. Note that throughout thi paper, the letter C ad C k deote cotat which may chage from lie to lie. Lemma. If 0 < α <, the K α L,t (R ) where < t = α.

3 Mochammad Idri, Hedra Guawa, ad Eridai Proof. Let 0 < α <. Take a arbitrary = (a, R) where a R ad R > 0. For < t = we oberve that t K α(x)dx (0, R) t (0,R) x (α ) dx C R ( t ) R ( t ) = C. y takig the upremum over = (a, R), we obtai K α L,t C. Hece K α L,t (R ). Remark. For 0 < α < ad γ > 0, we kow that K α,γ L t (R ) for +γ α < t < α α, []. y the icluio property of Morrey pace (ee [6]), we have K α,γ L t (R ) = L t,t (R ) L,t (R ), for t ad +γ α < t < cotaied i L,t (R ) for < t = α. Moreover, becaue K α,γ(x) K α (x) for every x R, K α,γ i alo α. A a couterpart of the reult i [0, ], we have the followig theorem o the boudede of I α,γ o Morrey pace. Note particularly that the etimate hold for p =. Theorem. If 0 < α < ad γ 0, the I α,γ i bouded from L p,q (R ) to L p,q (R ) with I α,γ f L p,q C K α,γ L,t L p,q, f L p,q (R ), wheever p q < α, p = p, ad q = q t, with < t = α t ad +γ α < t < α (for γ > 0). (for γ 0) or Theorem. i i fact a pecial cae of the boudede of I α,γ o geeralized Morrey pace, which i tated i the followig theorem. Theorem.3 Let 0 < α < ad γ 0. If ϕ : R + R + i of cla G p uch that R ϕ(r)r t dr C ϕ(r)r t for every R > 0, the I α,γ i bouded from L p,ϕ (R ) to L p,ψ (R ) where ψ(r) := ϕ(r)r t, with I α,γ f L p,ψ C K α,γ L,t L p,ϕ, f L p,ϕ (R ), wheever p < α ad p = p, with < t = α +γ α < t < α (for γ > 0). (for γ 0) or t ad Proof. Suppoe that γ > 0 ad all the hypothee hold. For f L p,ϕ (R ) ad = (a, R) where a R ad R > 0, write f := f + f := f χ + f χ c, where = (a, R) ad c deote it complemet. To etimate I α,γ f, we oberve that for every x, Hölder iequality give I α,γ f (x) K α,γ (x y) f(y) dy p = Kα,γ(x y) f(y) p ( p p p K ) ( Kα,γ(x y) f(y) p p dy α,γ (x y) f(y) p p p dy p p K α,γ (x y) f(y) p p p ) p dy. 3

4 Norm etimate for eel-riez operator o geeralized Morrey pace Meawhile, we have ( α,γ (x y) f(y) p p p dy p p K Therefore we obtai ( I α,γ f (x) ( ) p Kα,γ(x ( p ) ( ) p y)dy f(y) p dy. ) ( ) Kα,γ(x p y) f(y) p dy Kα,γ(x ( ) p y)dy f(y) p dy ) Kα,γ(x p y) f(y) p dy C R ( t )( p )+ ϕ p We take the p -th power ad itegrate both ide over to get I α,γ f (x) p dx Kα,γ(x y) f(y) p dy dx y Fubii theorem, we have I α,γ f (x) p dx whece (C R ( t )( p )+ ϕ p (R) K α,γ f(y) p ( ) Kα,γ(x y)dx dy (R) K α,γ p p L,t (C R ( t )( p )+ ϕ p (R) K α,γ C R ( t ) K α,γ L,t f(y) p dy p L,t (R ( t )( p )+ ϕ p (R) K α,γ p L,t C R ( t )+ ϕ p (R) K α,γ L,tp L p,ϕ (R ( t )( p )+ ϕ p (R) K α,γ p C ψ p (R) K α,γ p L,t p L p,ϕ, ( ) p I α,γ f (x) p dx ψ(r) L,t C Kα,γ L,t L p,ϕ. Next, we etimate I α,γ f. For every x = (a, R), we oberve that I α,γ f (x) K α,γ (x y) f(y) dy c K α,γ (x y) f(y) dy = x y R C k R x y < k+ R K α,γ ( k R) K α,γ (x y) f(y) dy k R x y < k+ R ( K α,γ ( k R)( k R) p C L p,ϕ f(y) dy K α,γ ( k R)( k R) ϕ( k R). L,t p ) p L p.,ϕ p ) p L p,ϕ p ) p L p,ϕ p ) p L p,ϕ f(y) p dy k R x y < k+ R ) p p. L p,ϕ 4

5 Mochammad Idri, Hedra Guawa, ad Eridai For every k Z, we have K α,γ ( k R) C ( k R) ( Sice R ϕ(r)r t dr C ϕ(r)r t, we get Kα,γ(x y)dy k R x y < k+ R I α,γ f (x) C K α,γ L,t L p,ϕ C K α,γ L,t L p,ϕ ) ( k R) t ϕ( k R) C K α,γ L,t L p,ϕϕ(r)r t = C K α,γ L,t L p,ϕψ(r). R ϕ(r)r t dr C ( k R) t Kα,γ L,t. Raiig to the p -th power ad itegratig over, we obtai I α,γ f (x) p dx C ( pψ p K α,γ L,t L p,ϕ) (R), whece ( ) I α,γ f (x) p p dx ψ(r) C Kα,γ L,t L p,ϕ. Combiig the two etimate for I α,γ f ad I α,γ f, we obtai ( ) I α,γ f(x) p p dx ψ(r) C Kα,γ L,t L p,ϕ. Sice thi iequality hold for every a R ad R > 0, it follow that I α,γ f L p,ψ C K α,γ L,t L p,ϕ, a deired. We may repeat the ame argumet ad ue Lemma. to obtai the ame iequality for the cae where γ = 0 ad < t = α. Remark. Theorem. ad.3 give u upper etimate for the orm of the eel-riez operator (from oe Morrey pace to aother). I particular, for γ = 0, we have a etimate for the orm of the fractioal itegral operator I α i term of the orm of it kerel (o the aociated Morrey pace), which follow from the iequality I α f L p,ψ C K α L,t L p,ϕ, for p < α ad p = p, with < t = α. I the followig ectio, we dicu lower etimate for the orm of the operator i term of the orm of the eel-riez kerel (o ome Morrey pace). 5

6 Norm etimate for eel-riez operator o geeralized Morrey pace 3 A Etimate for the Norm of the Operator Recall that if (X, X ) ad (Y, Y ) are ormed pace ad that T : (X, X ) (Y, Y ) i a liear operator, the the orm of T (from X to Y ) i defied by T f Y T X Y := up. f 0 X Kowig that the eel-riez operator I α,γ i a liear operator o Morrey pace, we would like to etimate the orm of I α,γ from a (geeralized) Morrey pace to aother. We obtai the followig reult. Theorem 3. Let 0 < α <, γ 0, ad ϕ i of cla G p where p < α. If ϕ(r)r i almot icreaig ad for every R > 0 we have (i) R ϕ(r)r t dr C ϕ(r)r t, (ii) R (r)r dr 0 ϕp C ϕ p (R)R, ad (iii) R r 0 dr C 3R, where p ϕ (r)r ϕ (R)R < t ad < < t = α (for γ 0) or p t, < t, ad +γ α < t < α (for γ > 0), the we have C 4 K α,γ L p,t I α,γ L p,ϕ L p,ψ C 5 K α,γ L,t, wheever p = p ad ψ(r) := ϕ(r)r t. I particular, for γ = 0, p < t, ad < < t = α, we have wheever p = p ad ψ(r) := ϕ(r)r t. C 4 K α L p,t I α L p,ϕ L p,ψ C 5 K α L,t, Proof. Suppoe that γ > 0 ad all the hypothee hold. y Theorem.3, we already have I α,γ L p,ϕ L C K p α,γ,ψ L,t. To prove the lower etimate, put ρ(r) := ϕ(r)r. Let = (a, R) where a R ad R > 0. y our aumptio o ϕ, we have ( ) ( t ψ(r) ρ ) R ( x ) dx C ϕ(r)r r dr C. 0 ϕ (r)r Now take f 0 (x) := ϕ( x ). Here f 0 L p,ϕ. Moreover, oe may compute that I α,γ f 0 (x) (x, x ) for every x R. It follow that K α,γ (x y)f 0 (y)dy C K α,γ (x)ϕ( x ) x = C ρ( x )K α,γ (x), ρ( )K α,γ L p,ψ C I α,γ f 0 L p,ψ C I α,γ L p,ϕ L p,ψ. Next, by Hölder iequality, we have ( ) ( ) ( K p p α,γ(x)dx ρ [ ( x ) dx ρ( x )Kα,γ (x) ] ) p dx p, 6

7 Mochammad Idri, Hedra Guawa, ad Eridai whece t p ( ) ( K p p α,γ(x)dx t ψ(r) ( ψ(r) C I α,γ L p,ϕ L p,ψ. ) ρ ( x ) dx [ ρ( x )Kα,γ (x) ] p dx ) p y takig the upremum over = (a, R), we coclude that a deired. C K α,γ L p,t I α,γ L p,ϕ L,ψ, p The ame argumet applie for the cae where γ = 0, with p < t ad < < t = α. Remark. Oe may oberve that the cotat C 4 ad C 5 i Theorem 3. deped o ϕ,, p,, ad t, but ot o α ad γ. Although the lower ad the upper boud are ot comparable, we may till get ueful iformatio from thee etimate, epecially for the orm of the operator I α from L p,ϕ (R ) to L p,ψ (R ). Oberve that for p < t = α, we have K α p C L = p,t (α )p + C α. Hece, if all the hypothee i Theorem 3. hold for the cae where γ = 0, the we obtai I α L p,ϕ L C p,ψ α, which blow up whe α 0 +. For ϕ(r) := r q with p < q < mi{, α }, our reult reduce to the etimate I α L p,q L p,q C α where p = p ad q = q α. A imilar behavior of the orm of I α from L p (R ) to L p (R ) for p = p α whe α 0+ i oberved i [, Chapter 4]. Ackowledgemet. The firt ad ecod author are upported by IT Reearch & Iovatio Program 06. All author would like to thak the aoymou referee for hi/her ueful commet o the earlier verio of thi paper. Referece [] D. R. Adam, A ote o Riez potetial, Duke Math. J. 4 (975), [] F. Chiareza ad M. Fraca, Morrey pace ad Hardy-Littlewood maximal fuctio, Red. Mat. 7 (987), [3] Eridai, H. Guawa, ad E. Nakai, O geeralized fractioal itegral operator, Sci. Math. Jp. 60 (004), [4] L. Grafako, Claical Fourier Aalyi, Graduate Text i Matemathic, Vol. 49, Spriger, New York, 008. [5] H. Guawa ad Eridai, Fractioal itegral ad geeralized Ole iequalitie, Kyugpook Math. J. 49 (009), [6] H. Guawa, D. I. Hakim, K. M. Limata, ad A. A. Mata, Icluio propertie of geeralized Morrey pace, Math. Nachr. 90 (07), [DOI: 0.00/maa ]. 7

8 Norm etimate for eel-riez operator o geeralized Morrey pace [7] G. H. Hardy ad J. E. Littlewood, Some propertie of fractioal itegral. I, Math. Zeit. 7 (97), [8] G. H. Hardy ad J. E. Littlewood, Some propertie of fractioal itegral. II, Math. Zeit. 34 (93), [9] M. Idri ad H. Guawa, The boudede of geeralized eel-riez operator o geeralized Morrey pace, preeted at The Aia Mathematical Coferece 06, ali, 06. [0] M. Idri, H. Guawa, ad Eridai, The boudede of eel-riez operator o geeralized Morrey pace, Autral. J. Math. Aal. Appl. 3 (06), Iue, Article 9, 0. [] M. Idri, H. Guawa, J. Lidiari, ad Eridai, The boudede of eel- Riez operator o Morrey pace, AIP Coferece Proceedig 79, 0000 (06) [DOI: 0.063/ ]. [] E. H. Lieb ad M. Lo, Aalyi, d ed., America Mathematical Society, Providece, Rhode Ilad, 00. [3] E. Nakai, Hardy-Littlewood maximal operator, igular itegral operator ad Riez potetial o geeralized Morrey pace, Math. Nachr. 66 (994), [4] E. Nakai, O geeralized fractioal itegral o the weak Orlicz pace, MO ϕ, the Morrey pace ad the Campaato pace, i M. Cwikel et al. (Ed.), Fuctio Space, Iterpolatio Theory ad Related Topic, De Gruyter, erli, 00, [5] E. Nakai, O Orlicz-Morrey pace, Reearch Report [ acceed o Augut 7, 05]. [6] J. Peetre, O the theory of L p,λ pace, J. Fuct. Aal. 4 (969), [7] S. L. Sobolev, O a theorem i fuctioal aalyi (Ruia), Math. Sob. 46 (938), [Eglih tralatio i Amer. Math. Soc. Tral. er.. 34 (963), 39 68]. 8

The Australian Journal of Mathematical Analysis and Applications

The Australian Journal of Mathematical Analysis and Applications The Australia Joural of Mathematical Aalysis ad Applicatios Volume 13, Issue 1, Article 9, pp 1-10, 2016 THE BOUNDEDNESS OF BESSEL-RIESZ OPERATORS ON GENERALIZED MORREY SPACES MOCHAMMAD IDRIS, HENDRA GUNAWAN