Boundedness in Lebesgue spaces of Riesz potentials on commutative hypergroups

In the present paper we consider Riesz potentials on commutative hypergroups and prove the boundedness of these potentials from L (K,λ) to L (K,λ). We also prove the inequality from L (K,λ) to weak L (K,λ) for Riesz potentials on commutative hypergroups.


Introduction
For 0 < α < n, the operator is called a classical Riesz potential.
In this paper, we prove the boundedness of Riesz potentials on commutative hypergroups from L p (K, λ) to L q (K, λ) in the case of upper Ahlfors N -regular measure. We also prove the inequality from L 1 (K, λ) to weak L q (K, λ) for Riesz potentials on commutative hypergroups.
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 A space (K, ρ, λ) which satisfies all conditions mentioned above is called a space of homogeneous 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 an abstract convolution of measures on the space.
A hypergroup (K, * ) consists of a locally compact Hausdorff space K together with a bilinear, associative, weakly continuous convolution on the Banach space of all bounded regular Borel 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 measure with compact support.
2. The mapping: is continuous with respect to the Michael topology on the space C(K) of all nonvoid compact subsets of K, where this topology is generated by the sets 3. There is an identity e ∈ K with δ e * δ x = δ x * δ e = δ x for all x ∈ K.
A hypergroup K is called commutative if δ x * δ y = δ y * δ x for all x, y ∈ K. It is well known that every commutative hypergroup K possesses a Haar measure which will be denoted by λ (see [23]). That is, for every Borel measurable function f on K, Define the generalized translation operators T x , x ∈ K, by for all y ∈ K. If K is a commutative hypergroup, then T x f (y) = T y f (x) and the convolution of two functions is defined by Let (K, * ) be a commutative hypergroup, with quasi-metric ρ, Haar measure λ and N ∈ (0, ∞). We will say K is called upper Ahlfors N -regular by an identity, if there exists a constant C > 0, not depending r > 0, such that Let p > 0. By L p (K, λ) denote a class of all λ-measurable functions f : Let T be a linear operator from L p (K, λ) to L q (K, λ) , where p, q ∈ (0, ∞). T is said to be an operator of strong type (p, q) on (K, λ) , if there exists a positive constant C such that If for arbitrary β > 0 and f ∈ L p (K, λ) λ{x then T is called an operator of weak type (p, q) on (K, λ) .
The notation χ A (x) denotes the characteristic function of set A. Let (K, * ) be a commutative hypergroup, with quasi-metric ρ, Haar measure λ, upper Ahlfors N -regular by an identity. Define Hardy-Littlewood maximal function and Riesz potential on commutative hypergroup (K, * ) equipped with the quasi-metric ρ.

Main results
In this section we formulate the main results of this paper.
Theorem 3.1 Let (K, * ) be a commutative hypergroup, with quasi-metric ρ and doubling Haar measure λ, upper Ahlfors N -regular by an identity and let 0 < α < N , 1 ≤ p < N α If f ∈ L p (K), then the integral is absolutely convergent for almost every x ∈ K.
Therefore it follows that where C > 0 does not depend f , x and r. Estimate U 2 (x, r). By Hölder's inequality we have ρ(e, y) (α−N )p dλ(y) From (5) and (6), we have Minimum of the right-hand side is attained at So Hence, by the boundedness of Hardy-Littlewood maximal function (3) on L p (K, λ) we have The theorem is proved.
Theorem 3.3 Let (K, * ) be a commutative hypergroup, with quasi-metric ρ and doubling Haar measure λ, upper Ahlfors N -regular by an identity and let 0 < α < N , 1 q = 1 − α N and assume that the maximal operator M is an operator of weak type (1, 1) on (K, * ). Then the Riesz potential (4) is an operator of weak type (1, q) on (K, * ).