Sufficient Conditions for Boundedness of the Riesz Potential in Local Morrey-Type Spaces

Baku Mathematical Journal, 2022

The aim of this paper is to prove the Hardy-Littlewood-Stein-Weiss theorems for Riesz potentials in modified Morrey spaces L p,λ,|•| γ (R n).

Georgian Mathematical Journal

In this paper, we study the Riesz potential (G-Riesz potential) generated by the Gegenbauer differential operator G_{\lambda}=(x^{2}-1)^{\frac{1}{2}-\lambda}\frac{d}{dx}(x^{2}-1)^{\lambda+% \frac{1}{2}}\frac{d}{dx},\quad x\in(1,\infty),\,\lambda\in\Bigl{(}0,\frac{1}{2% }\Bigr{)}. We prove that the G-Riesz potential {I_{G}^{\alpha}} , {0

Journal of Inequalities and Applications, 2009

We consider generalized Morrey spaces M p,ω R n with a general function ω x, r defining the Morrey-type norm. We find the conditions on the pair ω 1 , ω 2 which ensures the boundedness of the maximal operator and Calderón-Zygmund singular integral operators from one generalized Morrey space M p,ω1 R n to another M p,ω2 R n , 1 < p < ∞, and from the space M 1,ω1 R n to the weak space WM 1,ω2 R n. We also prove a Sobolev-Adams type M p,ω1 R n → M q,ω2 R n-theorem for the potential operators I α. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω 1 , ω 2 , which do not assume any assumption on monotonicity of ω 1 , ω 2 in r. As applications, we establish the boundedness of some Schrödinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class. As an another application, we prove the boundedness of various operators on generalized Morrey spaces which are estimated by Riesz potentials.

2013

The problem of boundedness of the fractional maximal operator M α from complementary Morrey-type spaces to Morrey-type spaces is reduced to the problem of boundedness of the dual Hardy operator in weighted L pspaces on the cone of non-negative non-increasing functions, which allows obtaining sharp sufficient conditions for boundedness of M α .

Annales Polonici Mathematici, 2013

For 1 ≤ q ≤ α ≤ p ≤ ∞, (L q , l p) α is a complex Banach space which is continuously included in the Wiener amalgam space (L q , l p) and contains the Lebesgue space L α. We study the closure (L q , l p) α c,0 in (L q , l p) α of the space D of test functions (infinitely differentiable and with compact support in R d) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space W 1 ((L q , l p) α) (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space W 1,α) and obtain in it Sobolev inequalities and a Kondrashov-Rellich compactness theorem.

Georgian Mathematical Journal, 2013

Hacettepe journal of mathematics and statistics, 2023

In the present paper, we shall investigate a characterization for the boundedness of the B-Riesz potential and its commutators on the generalized weighted B-Morrey spaces. We also give a characterization for the generalized weighted B-Morrey spaces via the boundedness of the Riesz potential and its commutators generated by generalized translate operators associated with Laplace-Bessel differential operator.

Axioms

In this paper, we introduce grand Herz–Morrey spaces with variable exponent and prove the boundedness of Riesz potential operators in these spaces.

arXiv (Cornell University), 2013

We consider generalized Orlicz-Morrey spaces M Φ,ϕ (R n) including their weak versions W M Φ,ϕ (R n). In these spaces we prove the boundedness of the Riesz potential from M Φ,ϕ 1 (R n) to M Ψ,ϕ 2 (R n) and from M Φ,ϕ 1 (R n) to W M Ψ,ϕ 2 (R n). As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on (ϕ 1 , ϕ 2), which do not assume any assumption on monotonicity of ϕ 1 (x, r), ϕ 2 (x, r) in r.

Journal of Inequalities and Applications

In the present paper, we shall give a characterization for the Spanne and Adams type boundedness of the Riesz potential and its commutators on the generalized Orlicz-Morrey spaces, respectively. Also we give criteria for the weak versions of Spanne and Adams type boundedness of the Riesz potential on the generalized Orlicz-Morrey spaces. In all the cases the conditions for the boundedness are given in terms of Zygmund type integral inequalities involving the Young function (u) and the function ϕ(x, r) defining the space.