A. Serbetci

Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces

This paper considers the boundedness of Calderon– Zygmund singular integral operators in local and global Morrey-type spaces. We formulate sufficient conditions for the boundedness of Calderon-Zygmund singular integral operators for all admissible parameter values. In the case of local Morrey-type spaces for a certain range of parameters, these sufficient conditions are also necessary for genuine Calderon–Zygmund singular integral operators.

BOUNDEDNESS OF SUBLINEAR OPERATORS GENERATED BY CALDERÓN-ZYGMUND OPERATORS ON GENERALIZED WEIGHTED MORREY SPACES

Abstract In this paper we study the boundedness for a large class of sublinear operators T generated by Calderón-Zygmund operators on generalized weighted Morrey spaces Mp,φ(w) with the weight function w(x) belonging to Muckenhoupt’s class Ap. We find the sufficient conditions on the pair (φ1, φ2) which ensures the boundedness of the operator T from one generalized weighted Morrey space Μρ,φ1(w) to another Mp,φ2(w) for p > 1 and from M1,φ1 to the weak space WM1,φ2(w). In all cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on (φ1,φ2), which do not assume any assumption on monotonicity of φ1, φ2 in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudo- differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner- Riesz operator.

The boundedness of Hilbert transform in the local Morrey–Lorentz spaces

ABSTRACT In this paper, we investigate the boundedness of the Hilbert transform H in the local Morrey–Lorentz spaces , , . We prove that the operator H is bounded in under the condition , . In the limiting case , , we prove that the operator H is bounded from the space to the weak local Morrey–Lorentz space . Also we show that for the limiting case , , the modified Hilbert transform is bounded from the space to the bounded mean oscillation space.

Generalized Fractional Integral Operators on Generalized Local Morrey Spaces

We study the continuity properties of the generalized fractional integral operator on the generalized local Morrey spaces and generalized Morrey spaces . We find conditions on the triple which ensure the Spanne-type boundedness of from one generalized local Morrey space to another , , and from to the weak space , . We also find conditions on the pair which ensure the Adams-type boundedness of from to for and from to for . In all cases the conditions for the boundedness of are given in terms of Zygmund-type integral inequalities on and , which do not assume any assumption on monotonicity of , , and in .

Boundedness of the maximal operator in the local Morrey-Lorentz spaces

In this paper we define a new class of functions called local Morrey-Lorentz spaces Mp,q;λloc(Rn), 0<p,q≤∞ and 0≤λ≤1. These spaces generalize Lorentz spaces such that Mp,q;0loc(Rn)=Lp,q(Rn). We show that in the case λ<0 or λ>1, the space Mp,q;λloc(Rn) is trivial, and in the limiting case λ=1, the space Mp,q;1loc(Rn) is the classical Lorentz space Λ∞,t1p−1q(Rn). We show that for 0<q≤p<∞ and 0<λ≤qp, the local Morrey-Lorentz spaces Mp,q;λloc(Rn) are equal to weak Lebesgue spaces WL1p−λq(Rn). We get an embedding between local Morrey-Lorentz spaces and Lorentz-Morrey spaces. Furthermore, we obtain the boundedness of the maximal operator in the local Morrey-Lorentz spaces.MSC:42B20, 42B25, 42B35, 47G10.

Stein–Weiss inequalities for the fractional integral operators in Carnot groups and applications

In this article we consider the fractional integral operator I α on any Carnot group 𝔾 (i.e. nilpotent stratified Lie group) in the weighted Lebesgue spaces L p,ρ(x)β (𝔾). We establish Stein–Weiss inequalities for I α, and obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional integral operator I α from the spaces L p,ρ(x)β (𝔾) to L q,ρ(x)−γ (𝔾), and from the spaces L 1,ρ(x)β (𝔾) to the weak spaces WL q,ρ(x)−γ (𝔾) by using the Stein–Weiss inequalities. In the limiting case , we prove that the modified fractional integral operator is bounded from the space L p,ρ(x)β (𝔾) to the weighted bounded mean oscillation (BMO) space BMOρ(x)−γ (𝔾), where Q is the homogeneous dimension of 𝔾. As applications of the properties of the fundamental solution of sub-Laplacian ℒ on 𝔾, we prove two Sobolev–Stein embedding theorems on weighted Lebesgue and weighted Besov spaces in the Carnot group setting. As another application, we prove the boundedness of I α from the weighted Besov spaces to .

Boundedness of some sublinear operators and their commutators on generalized local Morrey spaces

ABSTRACT In this paper we prove the boundedness of certain sublinear operators , , generated by fractional integral operators with rough kernels , , from one generalized local Morrey space to another , , , and from the space to the weak space , , . In the case b belongs to the local Campanato space and is a linear operator, we find the sufficient conditions on the pair which ensures the boundedness of the commutator operators from to , , , , . In all cases the conditions for the boundedness of are given in terms of Zygmund-type integral inequalities on , which do not assume any assumption on monotonicity of in r.

Maximal operator and Calderon–Zygmund operators in local Morrey–Lorentz spaces

ABSTRACT In this paper we proved the boundedness of the Hardy–Littlewood maximal operator M, the Calderon–Zygmund operators T and the maximal Calderon–Zygmund operators on the local Morrey–Lorentz spaces . Finally, we give some applications of these results.

The boundedness of the generalized anisotropic potentials with rough kernels in the Lorentz spaces

In this paper, we study the generalized anisotropic potential integral K α, γ⊗ f and anisotropic fractional integral I Ω,α, γ f with rough kernels, associated with the Laplace–Bessel differential operator Δ B . We prove that the operator f→K α, γ⊗ f is bounded from the Lorentz spaces to for 1≤p<q≤∞, 1≤r≤s≤∞. As a result of this, we get the necessary and sufficient conditions for the boundedness of I Ω,α, γ from the Lorentz spaces to , 1<p<q<∞, 1≤r≤s≤∞ and from to , 1<q<∞, 1≤r≤∞. Furthermore, for the limiting case p=Q/α, we give an analogue of Adams’ theorem on the exponential integrability of I Ω,α, γ in .

On a certain class of spherical harmonic functions and singular pseudo-differential operators

This article gives some results for the weighted spherical harmonic functions. Some relations are found between the singular integral operators generated by generalized shift operator and singular pseudo-differential operators. Finally, the boundedness of the pseudo-differential operators studied are shown.