On the behaviors of rough fractional type sublinear operators on vanishing generalized weighted Morrey spaces

The aim of this paper is to get the boundedness of rough sublinear operators generated by fractional integral operators on vanishing generalized weighted Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. Also, rough fractional integral operator and a related rough fractional maximal operator which satisfy the conditions of our main result can be considered as some examples.


Introduction and useful informations
1.1. Background. The classical fractional integral (The classical fractional integral operator is also known as Riesz potential.) was introduced by Riesz in 1949 [6], defined by where Γ (·) is the standard gamma function and I α plays an important role in partial diferential equation as the inverse of power of Laplace operator. Especially, Its most significant feature is that I α maps L p (R n ) continuously into L q (R n ), with 1 q = 1 p − α n and 1 < p < n α , through the well known Hardy-Littlewood-Sobolev imbedding theorem (see pp. 119-121,Theorem 1 and its proof in [7]) for I α .

Definition 1.
Define Next, we give the definition of weighted Lebesgue spaces as follows: we shall define weighted Lebesgue spaces as Here and later, A p denotes the Muckenhoupt classes (see [2]). Now, let us consider the Muckenhoupt-Wheeden class A (p, q) in [5]. One says that w (x) ∈ A (p, q) for 1 < p < q < ∞ if and only if where the supremum is taken over all the balls B. Note that, by Hölder's inequality, for all balls B we have is valid. Now, we introduce some spaces which play important roles in PDE. Except the weighted Lebesgue space L p (w), the weighted Morrey space L p,κ (w), which is a natural generalization of L p (w) is another important function space. Then, the definition of generalized weighted Morrey spaces M p,ϕ (w) which could be viewed as extension of L p,κ (w) has been given as follows: is finite. Note that for ϕ(x, r) ≡ w(B(x, r)) κ p , 0 < κ < 1 and ϕ(x, r) ≡ 1, we have M p,ϕ (w) = L p,κ (w) and M p,ϕ (w) = L p (w), respectively.
Extending the definition of vanishing generalized Morrey spaces in [3] to the case of generalized weighted Morrey spaces defined above, we introduce the following definition.
Inherently, it is appropriate to impose on ϕ(x, t) with the following circumstances: we omit the details. Moreover, we have the following embeddings: Henceforth, we denote by ϕ ∈ B (w) if ϕ(x, r) is a positive measurable function on R n × (0, ∞) and positive for all (x, r) ∈ R n × (0, ∞) and satisfies (1.6) and (1.7). The purpose of this paper is to consider the mapping properties for the rough fractional type sublinear operators T Ω,α satisfying the following condition on vanishing generalized weighted Morrey spaces. Similar results still hold for the operators I Ω,α and M Ω,α , respectively. On the other hand, these operators have not also been studied so far on vanishing generalized weighted Morrey spaces and this paper seems to be the first in this direction. At last, here and henceforth, F ≈ G means F G F ; while F G means F ≥ CG for a constant C > 0; and p ′ and s ′ always denote the conjugate index of any p > 1 and s > 1, that is, 1 p ′ := 1 − 1 p and 1 s ′ := 1 − 1 s and also C stands for a positive constant that can change its value in each statement without explicit mention. Throughout the paper we assume that x ∈ R n and r > 0 and also let B(x, r) denotes x-centred Euclidean ball with radius r, B C (x, r) denotes its complement. For any set E, χ E denotes its characteristic function, if E is also measurable and w is a weight, w(E) := E w(x)dx.

Main Results
Our result can be stated as follows.
s ′ , q s ′ and s ′ < p, the following pointwise estimate (2.1) holds for any ball B (x 0 , r) and for all f ∈ L loc p,w (R n ). If ϕ 1 ∈ B (w p ), ϕ 2 ∈ B (w q ) and the pair (ϕ 1 , ϕ 2 ) satisfies the following conditions for every δ > 0, and Proof. Since inequality (2.1) is the heart of the proof of (2.4), we first prove (2.1).