Generalized potentials on commutative hypergroups
aa r X i v : . [ m a t h . F A ] J u l Generalized potentialson commutative hypergroups by Mubariz G. Hajibayov
National Aviation AcademyandInstitute of Mathematics and Mechanics, Baku, Azerbaijan([email protected])
Abstract
By the Hardy-Littlewood-Sobolev theorem the classical Riesz potential is bounded onLebesgue spaces. E. Nakai and H. Sumitomo [16] extended that theorem to the Orliczspaces. We introduce generalized potential operators on commutative hypergroups andunder some assumptions on the kernel we showed the boundedness of these operators fromLebesgue space into certain Orlicz space. Our result is an analogue of Theorem 1.3 in [16].
Mathematics Subject Classification 2010 : 47G40, 20N20, 43A62, 26A33.
Key words and phrases : Riesz potential, hypergroup, Lebesgue space, Orlicz spase, Hardy-Littlevood maximal function.
For < α < n , the operator R α f ( x ) = Z R n | x − y | α − n f ( y ) dy is called a classical Riesz potential (fractional integral).By the classical Hardy-Littlewood-Sobolev theorem, if < p < ∞ and αp < n , then R α f is a bounded operator from L p ( R n ) into L q ( R n ) , where q = 1 p − αn (see [10], [19] ).The Hardy-Littlewood-Sobolev theorem is an important result in the potential theory. Thereare a lot of generalizations and analogues of that theorem. The boundedness of the Riesz po-tentials on spaces of homogeneous type was studied in [4] and [12]. The Hardy-Littlewood-Sobolev theorem was proved for the Riesz potentials associated to nondoubling measures in[13]. In [3] and [8], generalized potential-type integral operators were considered and (p, q)properties of these operators were proved. In [15], [16], [17], [9] the Hardy-Littlewood-Sobolevtheorem was extended to Orlicz spaces for generalized fractional integrals. In [5], [6], [7], [21],Riesz potentials on different hypergroups were defined and analogues of the Hardy-Littlewood-Sobolev theorem were given for these operators.1n this paper, we define generalized fractional integrals on commutative hypergroups andprove the analogue of Theorem 1.3 in [16] for the generalized fractional integrals on commuta-tive hypergroups. The obtained result is an extension of the Hardy-Littlewood-Sobolev theoremgiven in [5], [6], [7], [21], for Riesz potentials on different hypergroupsLet K be a set. A function ρ : K × K → [0 , ∞ ) is called quasi-metric if:1. ρ ( x, y ) = 0 ⇔ x = y ; ρ ( x, y ) = ρ ( y, x ) ;
3. there exists a constant c ≥ such that for every x, y, z ∈ Kρ ( x, y ) ≤ c ( ρ ( x, z ) + ρ ( z, y )) . Let all balls B ( x, r ) = { y ∈ K : ρ ( x, y ) < r } be λ -measurable and assume that the measure λ fulfils the doubling condition < λB ( x, r ) ≤ DλB ( x, r ) < ∞ . (1)A space ( K, ρ, λ ) which satisfies all conditions mentioned above is called a space of homoge-neous type (see [2]).In the theory of locally compact groups there arise certain spaces which, though not groups,have some of the structure of groups. Often, the structure can be expressed in terms of anabstract convolution of measures on the space.A hypergroup ( K, ∗ ) consists of a locally compact Hausdorff space K together with a bi-linear, associative, weakly continuous convolution on the Banach space of all bounded regularBorel measures on K with the following properties:1. For all x, y ∈ K , the convolution of the point measures δ x ∗ δ y is a probability measurewith compact support.2. The mapping: ( x, y ) supp ( δ x ∗ δ y ) of K × K into C ( K ) , is continuous where C ( K ) is the space of compact subsets of K endowed with the Michael topology, that is thetopology generated by the subbasis of all U V,W = { L ∈ C ( K ) : L ∩ V = ∅ , L ⊂ W } where V, W are open subsets of K .3. There exits an identity e ∈ K such that δ e ∗ δ x = δ x ∗ δ e = δ x for all x ∈ K .4. There exits a topological involution ∼ from K onto K such that ( x ∼ ) ∼ = x , for x ∈ K ,with ( δ x ∗ δ y ) ∼ = δ y ∼ ∗ δ x ∼ and e ∈ supp ( δ x ∗ δ y ) if and only if x = y ∼ for x, y ∈ K where for any Borel set B , µ ∼ ( B ) = µ ( { x ∼ : x ∈ B } ) (see [11], [18], [1], [14]).2f δ x ∗ δ y = δ y ∗ δ x for all x, y ∈ K , then the hypergroup K is called commutative . It is knownthat every commutative hypergroup K possesses a Haar measure which will be denoted by λ (see [18]). That is, for every Borel measurable function f on K , Z K f ( δ x ∗ δ y ) dλ ( y ) = Z K f ( y ) dλ ( y ) ( x ∈ K ) . Define the generalized translation operators T x , x ∈ K , by T x f ( y ) = Z K f d ( δ x ∗ δ y ) for all y ∈ K . If K is a commutative hypergroup, then T x f ( y ) = T y f ( x ) and the convolutionof two functions is defined by ( f ∗ g ) ( x ) = Z K T x f ( y ) g ( y ∼ ) dλ ( y ) . Let p > . By L p ( K, λ ) denote a class of all λ -measurable functions f : K → ( −∞ , + ∞ ) with k f k L p ( K,λ ) = (cid:18)R K | f ( x ) | p dλ ( x ) (cid:19) p < ∞ .A function Φ : [0 , ∞ ] → [0 , ∞ ] is called an N -function if can be represented as Φ ( r ) = r Z φ ( t ) dt, where φ : [0 , ∞ ] → [0 , ∞ ] is a left continuous nondecreasing function such that φ (0) = 0 and lim t →∞ φ ( t ) = ∞ . Let Φ is an N -function. Define the Orlicz space L Φ ( K, λ ) to be the set of all locally integrablefunctions f in K for which Z K Φ (cid:18) | f ( x ) | η (cid:19) dλ ( x ) < ∞ for some η > . Here L Φ ( K, λ ) is equipped with the norm k f k Φ = inf { η > Z K Φ (cid:18) | f ( x ) | η (cid:19) dλ ( x ) ≤ } . For
Φ ( r ) = r p , < p < ∞ , we have L Φ ( K, λ ) = L p ( K, λ ) .The notation χ A ( x ) denotes the characteristic function of set A .Define a function Λ x ( y ) = T x χ B ( e,r ) ( y ∼ ) .We will assume that there exit constants c > , c > and c > such that for every x, y ∈ K and r > supp Λ x ( · ) ⊂ B ( x, c r ) (2)3nd λB ( x, r ) T x χ B ( e,r ) ( y ∼ ) ≤ c λB ( e, r ) ≤ c r N . (3)As examples of hypergroups satisfying the conditions (2) and (3) can be taken Laguerre,Dunkland Bessel hypergroups (see [5], [6], [7]).A non-negative function a ( r ) defined on [0 , ∞ ) is called almost increasing (almost decreasing),if there exist a constant C > such that a ( t ) ≤ Ca ( t ) for all < t < t < ∞ ( < t < t < ∞ , respectively).For an increasing function a : (0 , ∞ ) → (0 , ∞ ) , define I a f ( x ) = Z K T x (cid:18) a ( ρ ( e, y )) ρ ( e, y ) N (cid:19) f ( y ∼ ) dλ ( y ) on the commutative hypergroup ( K, ∗ ) equipped with the quasi-metric ρ . If a ( r ) = r α , <α < N, then I a is the Riesz potential of order α. Now we formulate a main result of the paper.
Theorem 1.1
Let ( K, ∗ ) be a commutative hypergroup, with the quasi-metric ρ and doublingHaar measure λ satisfying the conditions (2) and (3) . Assume that < p < ∞ and a = a ( r ) is non-negative almost increasing function on [0 , ∞ ) , a ( r ) r λ is almost decreasing for some <λ < Np and Z a ( t ) t dt < ∞ . Then the operator I a is bounded from L p ( K, λ ) into the Orlicz space L Φ ( K, λ ) , where the N -function is defined by its inverse Φ − ( r ) = r Z A (cid:16) t − N (cid:17) t − p ′ dt, where A ( r ) = r R a ( t ) t dt . If we take a ( r ) = r α , < α < N, then we have Hardy-Littlewood-Sobolev theorem for theRiesz potential I α f ( x ) = Z K T x ρ ( e, y ) α − N f ( y ∼ ) dλ ( y ) on the commutative hypergroup ( K, ∗ ) . Corollary 1.2
Let ( K, ∗ ) be a commutative hypergroup, with the quasi-metric ρ and doublingHaar measure λ satisfying the conditions (2) and (3) . If < α < N, < p < Nα and p − q = αN , then I α is a bounded operator from L p ( K, λ ) into L q ( K, λ ) . Preliminaries
Define Hardy-Littlewood maximal function
M f ( x ) = sup r> λB ( e, r ) (cid:0) | f | ∗ χ B ( e,r ) (cid:1) ( x ) on commutative hypergroup ( K, ∗ ) equipped with the pseudo-metric ρ . Lemma 2.1
Let ( K, ∗ ) be a commutative hypergroup, with quasi-metric ρ and doubling Haarmeasure λ . Assume that there exist constants c > and c > such that for every x, y ∈ K and r > supp Λ x ( · ) ⊂ B ( x, c r ) and λB ( x, r ) T x χ B ( e,r ) ( y ∼ ) ≤ c λB ( e, r ) . Then1) The maximal operator M satisfies a weak type (1 , inequality, that is, there exists aconstant C > such that for every f ∈ L ( K, λ ) and α > λ { x : M f ( x ) > α } ≤ Cα Z K | f ( x ) | dλ ( x ) .
2) The maximal operator M is of strong type ( p, p ) , for < p ≤ ∞ , that is, k M f k L p ( K,λ ) ≤ C p k f k L p ( K,λ ) , (4) for some constant C p and every f ∈ L p ( K, λ ) .Proof. It is clear that there exists nonnegative integer m such that c ≤ m and λB ( x, c r ) ≤ D m λB ( x, r ) , where D is a constant on doubling condition (1). Then we have M f ( x ) = sup r> λB ( e, r ) Z K T x | f ( y ) | χ B ( e,r ) ( y ∼ ) dλ ( y )= sup r> λB ( e, r ) Z K | f ( y ) | T x χ B ( e,r ) ( y ∼ ) dλ ( y ) ≤ sup r> λB ( e, r ) Z B ( x,c r ) | f ( y ) | T x χ B ( e,r ) ( y ∼ ) dλ ( y )= sup r> λB ( x, r ) Z B ( x,c r ) | f ( y ) | T x χ B ( e,r ) ( y ∼ ) λB ( x, r ) λB ( e, r ) dλ ( y ) ≤ c sup r> λB ( x, r ) Z B ( x,c r ) | f ( y ) | dλ ( y ) ≤ c D m M ρ f ( x ) , M ρ f ( x ) = sup r> λB ( x, r ) Z B ( x,r ) | f ( y ) | dλ ( y ) is a maximal operator on ( K, ρ, λ ) . It is well known that the maximal operator M ρ is of weaktype (1 , and is bounded on L p ( K, λ ) (see [2], [20]). This fact and the inequality M f ( x ) ≤ c D m M ρ f ( x ) completes the proof. (cid:3) We may suppose that f ( x ) ≥ and by the linearity of the operator I a , it suffices to prove that k I a f k Φ ≤ C < ∞ for k f k L p ( K,λ ) ≤ . accordance with Hedbergs trick,we split I a f ( x ) in thestandard way I a f ( x ) = Z B ( e,r ) a ( ρ ( e, y )) ρ ( e, y ) N T x f ( y ∼ ) dλ ( y )+ Z X \ B ( e,r ) a ( ρ ( e, y )) ρ ( e, y ) N T x f ( y ∼ ) dλ ( y ) = A r ( x ) + B r ( x ) . Estimate A r ( x ) . Since a ( t ) t N is almost decreasing, we have A r ( x ) = ∞ X k =0 Z − k − r ≤ ρ ( e,y ) < − k r a ( ρ ( e, y )) ρ ( e, y ) N T x f ( y ∼ ) dλ ( y ) ≤ C ∞ X k =0 a (cid:0) − k − r (cid:1) (2 − k − r ) N Z − k − r ≤ ρ ( e,y ) < − k r T x f ( y ∼ ) dλ ( y ) ≤ CM f ( x ) ∞ X k =0 a (cid:0) − k − r (cid:1) ≤ CM f ( x ) ∞ X k =0 2 − k r Z − k − r a ( t ) t dt. Therefore, A r ( x ) ≤ CA ( r ) M f ( x ) , A ( r ) = r Z a ( t ) t dt. (5)Now estimate B r ( x ) . By the H¨older inequality and the condition k f k L p ( K,λ ) ≤ , we obtain B r ( x ) ≤ Z K \ B ( e,r ) ( T x f ( y ∼ )) p dλ ( y ) p Z K \ B ( e,r ) (cid:18) a ( ρ ( e, y )) ρ ( e, y ) N (cid:19) p ′ dλ ( y ) p ′ ≤ Z K \ B ( e,r ) (cid:18) a ( ρ ( e, y )) ρ ( e, y ) N (cid:19) p ′ dλ ( y ) p ′ ∞ X k =0 Z k r ≤ ρ ( e,y ) < k +1 r (cid:18) a ( ρ ( e, y )) ρ ( e, y ) N (cid:19) p ′ dλ ( y ) p ′ ≤ C ∞ X k =0 a (cid:0) k r (cid:1) (2 k r ) N ! p ′ Z ρ ( e,y ) < k +1 r dλ ( y ) p ′ ≤ C ∞ X k =0 a (cid:0) k r (cid:1) (2 k r ) N ! p ′ (cid:0) k +1 r (cid:1) N p ′ ≤ C ∞ X k =0 a (cid:0) k r (cid:1) (2 k r ) Np ! p ′ p ′ ≤ C ∞ X k =0 (cid:0) a (cid:0) k r (cid:1)(cid:1) p ′ k +1 r Z k r (cid:18) t Np (cid:19) p ′ t dt p ′ ≤ C ∞ X k =0 2 k +1 r Z k r (cid:18) a ( t ) t Np (cid:19) p ′ t dt p ′ = C ∞ Z r (cid:18) a ( t ) t Np (cid:19) p ′ t dt p ′ ≤ C a ( r ) r β ∞ Z r (cid:16) t β − Np (cid:17) p ′ t − dt p ′ ≤ C a ( r ) r Np Therefore B r ( x ) ≤ CA ( r ) r − Np (6)From (5) and (6), we have I a f ( x ) ≤ C (cid:16) M f ( x ) + r − Np (cid:17) A ( r ) . Then I a f ( x ) ≤ C h M f ( x ) r Np + 1 i Φ − (cid:18) r N (cid:19) (7)by Theorem 4.9 in [9]. If we choose r = [ M f ( x )] − pN , then the inequality (7) turns into I a f ( x ) ≤ C Φ − ([ M f ( x )] p ) Z K Φ (cid:18) I a f ( x ) C (cid:19) dλ ( x ) ≤ Z K [ M f ( x )] p dλ ( x ) ≤ , where we have used (4) and the fact that k f k L p ( K,λ ) ≤ . Hence k I a f k Φ ≤ C, which completes the proof. Acknowledgement.
This work was supported by the Science Development Foundationunder the President of the Republic of Azerbaijan Grant EIF-2012-2(6)-39/10/1. The authorwould like to express his thanks to Academician Akif Gadjiev for valuable remarks.
References [1] W. R. Bloom and H. Heyer, Harmonic analysis of probability measures on hypergroups, de Gruyter Stud. Math., vol. 20, Walter de Gruyter & Co., Berlin, 1995.[2] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaceshomog`enes.(French)
Lecture Notes in Math., , Springer-Verlag, Berlin-New York,1971[3] A. D. Gadjiev, On generalized potential-type integral operators,
Functiones et Approxima-tio, UAM, (1997), 37-44.[4] A. E. Gatto and S. Vagi, Fractional integrals on spaces of homogeneous type, Analysis andPartial Differential Equations, (1990), 171-216.[5] V. S. Guliyev, On maximal function and fractional integral, associated with the Besseldifferential operator,
Math. Inequal. Appl., (2003), 317-330.[6] V. S. Guliyev, Y. Y. Mammadov, On fractional maximal function and fractional integralsassociated with the Dunkl operator on the real line,
J. Math. Anal. Appl., (2009),449-459.[7] V. S. Guliyev, M.N. Omarova, On fractional maximal function and fractional integral onthe Laguerre hypergroup,
J. Math. Anal. Appl., (2008), 1058-1068.[8] M.G.Hajibayov ( L p ; L q ) properties of the potential-type integrals associated to non-doubling measures, Sarajevo J. Math. (2006), 173-180.[9] M. G. Hajibayov and S. G. Samko, Generalized potentials in variable exponent Lebesguespaces on homogeneous spaces,
Math. Nachr., (2011), 53-66.[10] L. Hedberg, On certain convolution inequalities,
Proc. Amer. Math. Soc., (1972), 505-510. 811] R. L. Jewett, Spaces with an abstract convolution of measures. Adv. in Math., (1975),1-101.[12] V. M. Kokilashvili and A. Kufner, Fractional inteqrals on spaces of homogeneous type,
Comment. Math. Univ. Carolinae, (1989), 511-523.[13] V. Kokilashvili and A. Meskhi, Fractional integrals on measure spaces,
Frac. Calc. Appl.Anal., (2001), 1-24.[14] M. Lashkarizadeh Bami, The semisimplicity of L ( K, w ) of a weighted commutative hy-pergroup K , Acta Math. Sin. (Engl.ser), (2008), 607-610.[15] E. Nakai, On generalized fractional integrals,
Taiwanese J. Math. (2001), 587-602.[16] E. Nakai and H. Sumitomo, On generalized Riesz potentials and spaces of some smoothfunctions,
Sci. Math. Japonicae,
54 (3) (2001), 463-472.[17] E. Nakai, On generalized fractional integrals in the Orlicz spaces on spaces of homoge-neous type,
Sci. Math. Japonicae, (2001), 473 -487.[18] R. Spector, Measures invariantes sur les hypergroupes(French),
Trans. Amer. Math. Soc., (1978), 147-165.[19] E. Stein, Singular integrals and diferentiability properties of functions,
Princeton Mathe-matical Series,
No. 30 Princeton University Press, Princeton, N.J. 1970.[20] J. O. Str¨omberg and A. Torchinsky, Weighted Hardy spaces. Lecture Notes in Math., ,Springer-Verlag, Berlin, 1989.[21] S. Thangavelu and Y. Xu, Riesz transform and Riesz potentials for Dunkl transform,
J. ofComput. and Appl. Math.,199(1)