Vagif S. Guliyev

Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces

We consider generalized Morrey spaces with a general function defining the Morrey-type norm. We find the conditions on the pair which ensures the boundedness of the maximal operator and Calderón-Zygmund singular integral operators from one generalized Morrey space to another , , and from the space to the weak space . We also prove a Sobolev-Adams type -theorem for the potential operators . In all the cases the conditions for the boundedness are given it termsof Zygmund-type integral inequalities on , which do not assume any assumption on monotonicity of in . 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.

Boundedness of the fractional maximal operator in local Morrey-type spaces

The problem of boundedness of the fractional maximal operator M α, 0 ≤ α < n, in general local Morrey-type spaces is reduced to the problem of boundedness of the supremal operator in weighted L p -spaces on the cone of non-negative non-decreasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.

Generalized Local Morrey Spaces and Fractional Integral Operators with Rough Kernel

Let MΩ,α and IΩ,α be the fractional maximal and integral operators with rough kernels, where 0 < α < n. We study the continuity properties of MΩ,α and IΩ,α on the generalized local Morrey spaces

Boundedness of the Maximal and Singular Operators on Generalized Orlicz–Morrey Spaces

LM_{{p,\varphi}}^{{\left\{ {{x_0}} \right\}}}

On the Riesz Potential and Its Commutators on Generalized Orlicz-Morrey Spaces

. We prove that the commutators of these operators with local Campanato functions are bounded. Bibliography: 34 titles.

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

The authors study the boundedness for a large class of sublinear operator generated by Calderon-Zygmund operator on generalized Morrey spaces . As an application of this result, the boundedness of the commutator of sublinear operators on generalized Morrey spaces is obtained. In the case , and is a sublinear operator, we find the sufficient conditions on the pair () which ensures the boundedness of the operator from one generalized Morrey space to another . 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 . Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.

Φ-Admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces

We consider generalized Orlicz–Morrey spaces \(M_{\Phi},\varphi\,(\mathbb{R}_{n})\) including their weak versions. In these generalized spaces we prove the boundedness of the Hardy–Littlewood maximal operator and Calderon–Zygmund singular operators with standard kernel. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on \(\varphi(r)\) without assuming any monotonicity property of \(\varphi(r)\) , or in terms of supremal operators, related to \(\varphi(r)\).

BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS

In the paper the authors find conditions on the pair \( (\varphi_{1},\varphi_{2}) \) which ensure the Spanne type boundedness of the fractional maximal operator \( M_{\alpha} \) and the Riesz potential operator \( I_{\alpha} \) from one generalized Morrey spaces \( M_{p,{\varphi_{1}}} \) to another \( M_{q,{\varphi_{2}}}, 1 < p < q < \infty, 1/p-1/q = \alpha/n, \) and from \( M_{1,{\varphi_{1}}} \) to the weak space W \( M_{q,{\varphi_{2}}}, 1 < p < q < \infty, 1- 1/q = \alpha/n, \) We also find conditions on \( \varphi \) which ensure the Adams type boundedness of the \( M_{\alpha}\; {\rm and}\; I_{\alpha}\; {\rm from} \; M_{p,{\varphi}^{\frac {1}{p}}}\; \rm{to}\; M_{q,{\varphi}^{\frac {1}{q}}}\;\rm {for 1 < p < q < \infty \; and\; from\; M_{1,{\varphi}}\; to \;W\;M_{q,{\varphi}^{\frac{1}{p}}} \; for \; 1 < q < \infty.}\) As applications of those results, the boundeness of the commutators of operators \( I_{\alpha} and I_{\alpha} \) on generalized Morrey spaces is also obtained. In the case \( b \in BMO{\mathbb{(R)}^{n}}\; \rm and \;1 < p < q < \infty,\) we find the sufficient conditions on the pair \( (\varphi_{1},\varphi_{2}) \) which ensures the boundedness of the operators \( {M_{b,\alpha}}\; \rm {and \;[b,I_{\alpha}] \; from \; M_{p,\varphi_{1}}\; to \; M_{q,\varphi_{2}}\; with\; 1/p - 1/q = \alpha/n.} \) We also find the sufficient conditions on \( \varphi \) which ensures the boundedness of the operators \( {M_{b,\alpha}}\; \rm {and \;[b,I_{\alpha}] \; from \; M_{p,{\varphi^{\frac{1}{p}}}}\; to \; M_{q,\varphi^{\frac{1}{p}}}\; for\; 1 < p < q < \infty.} \) In all cases conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \( \rm {(\varphi_{1},\varphi_{2}) \;and \;\varphi} ,\)which do not assume any assumption on monotonicity of \( \rm {\varphi_{1},\varphi_{2} \;and \;\varphi} \;\rm{in\; r} ,\) As applications, we get some estimates for Marcinkiewicz operator and fractional powers of the some analytic semigroups on generalized Morrey spaces.