A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces

Abstract: In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator (B-maximal operator) on ( ) (⋅) +  Lp γ k n , , variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the B-maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the B-maximal operator is not bounded on ( ) (⋅) +  Lp γ k n , , variable exponent Lebesgue spaces in the case of = − p 1. We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional B-maximal function) on ( ) (⋅) +  Lp γ k n , , variable exponent Lebesgue spaces.

Abstract: In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator (B-maximal operator) on ( ) (⋅) + L p γ k n , , variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the B-maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the B-maximal operator is not bounded on ( ) variable exponent Lebesgue spaces in the case of = − p 1. We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional B-maximal function) on ( )
In recent years, there have been important developments in the theory of variable exponent Lebesgue spaces. On variable exponent Lebesgue spaces, the boundedness of some operators in harmonic analysis, such as maximal operator, and singular integral operator, has an important role. On variable exponent Lebesgue spaces, the boundedness of the classical maximal operator and singular integral operators has been studied by Diening [15,16], Cruz-Uribe et al. [17], Cruz-Uribe et al. [18], and Adamowicz et al. [19].
The purpose of this study is to extend the theory of variable exponent Lebesgue spaces. The boundedness of the B-maximal operator plays an important role in obtaining the boundedness of the convolutiontype singular integral operators related to the Laplace-Bessel differential operator. We will obtain that a necessary condition for the boundedness of the B-maximal operator on ( ) variable exponent Lebesgue spaces, using + translation according to n-variables. Moreover, we will obtain that the B-maximal operator is not bounded on , variable exponent Lebesgue spaces in the case of = − p 1. This article is organized as follows. In Section 2, we give some results that are useful for us. In Section 3, we obtain a necessary condition for the boundedness of the B-maximal operator on variable exponent Lebesgue spaces. Moreover, we prove that the B-maximal operator is not bounded on variable exponent Lebesgue spaces for = − p 1. In Section 4, we prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional B-maximal function) on variable exponent Lebesgue spaces.

Notations and preliminaries
The generalized translation operator is defined by [8,20]. It is well known that the generalized translation operator is closely connected with Δ B -Laplace-Bessel differential operator.
The B-convolution operator associated with the generalized translation operator is defined by f g x f y T g x y y d .
is finite. Now, we will introduce the space ( ) and state some fundamental properties of this space. For a measurable function (⋅) → [ ∞) Then, we denote the set of variable exponent functions by ( ) We say that (⋅) p is log-Hölder continuous at infinity and denote this by , to consist of measurable functions f such that the modular f on , is a Banach space with respect to the norm , we denote the conjugate exponent.
be Lebesgue measurable and denote the set of simple functions on up to the equivalence of the norms 3 The B-maximal operator on ( ) In this section, we obtain the boundedness of B-maximal operators, which play an important role in harmonic analysis, on variable exponent Lebesgue spaces. We also establish a necessary condition for the B-maximal operator to be bounded on where the supremum is taken over all balls (or cubes) ∈ B n which contain x. In this paper, we consider the Hardy-Littlewood maximal operator related to the Laplace-Bessel differential operator (B-maximal operator) (see [7]): where C is independent of x, r, t, and f .
where C is independent of f .
The above theorem gives that the B-maximal operator is bounded on ( ) Lebesgue spaces. We will now show that the B-maximal operator is bounded on variable exponent Lebesgue spaces. In order for the boundedness of the B-maximal operator on variable Lebesgue spaces, ( ) p x must provide the log-Hölder continuous condition.  . Then, the following conditions are equivalent: We have a constant > C 0 so that for any ∈ + B , where C is independent of f .

Proof.
To prove the boundedness of the B-maximal operator on ( ) , , we will use the maximal function on a space of homogeneous type. Therefore, we will first introduce the fractional maximal function on a space of homogeneous type. Given a pseudo-metric space ( ) X ρ , and a positive measure μ. Then, we say that ( ) X ρ μ , , is a homogeneous-type space, if the measure μ satisfies the doubling condition where C is independent of x and > r 0, and ( ) = { ∈ ( ) < } B x r y X ρ x y r , : , It is clear that this measure satisfies the doubling condition (1) (see [14]). Also in [3,7,24], it was proved that From (5) and the boundedness of the maximal operator M μ on ( ) , we obtain the boundedness of the B-maximal operator on ( )  Now, we will prove ∈ ( )   for = α 0 (see [6]). . Then, where C is a constant independent of f .
Proof. We use the fractional maximal function on a space of homogeneous type. Therefore, we will firstly introduce the fractional maximal function on a space of homogeneous type. Let ( ) X ρ , be a pseudo-metric space, ( ) X ρ μ , , be a homogeneous type space, and ( ) B x r , be as above. Then, the fractional maximal function is defined as: (see [25,26]), i.e., To complete the proof of Theorem 5, we will use the following statements. In the case = (see [7]). Now we will show that