Tables of integrals

Edited by Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan, Wolfgang Arendt, Albrecht Bottcher, B. Malcolm Brown, Raul Curto, Fritz Gesztesy, Pavel Kurasov, Vern Paulsen, Mihai Putinar, Ilya M. Spitkovsky This series is devoted to the publication of current research in operator theory, with particular emphasis on applications to classical analysis and the theory of integral equations, as well as to numerical analysis, mathematical physics and mathematical methods in electrical engineering.

Operator Theory: Advances and Applications

Multiple integrals in the calculus of variations and nonlinear elliptic systems

Multiple Integrals In The Calculus Of Variations And Nonlinear Elliptic Systems

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 Calderon-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 Schrodinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Holder class. As an another application, we prove the boundedness of various operators on generalized Morrey spaces which are estimated by Riesz potentials.

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

In this paper the authors study the boundedness for a large class of sublinear operators ${T_{\alpha}, \alpha \in [0,n)}$ generated by Calderon–Zygmund operators (α = 0) and generated by Riesz potential operator (α > 0) on generalized Morrey spaces ${M_{p,\varphi}}$ . As an application of the above result, the boundeness of the commutator of sublinear operators ${T_{b,\alpha}, \alpha \in [0,n)}$ on generalized Morrey spaces is also obtained. In the case ${b \in BMO}$ and Tb,α is a sublinear operator, we find the sufficient conditions on the pair ${(\varphi_1,\varphi_2)}$ which ensures the boundedness of the operators ${T_{b,\alpha}, \alpha \in [0,n)}$ from one generalized Morrey space ${M_{p,\varphi_1}}$ to another ${M_{q,\varphi_2}}$ with 1/p − 1/q = α/n. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ${(\varphi_1,\varphi_2)}$ , which do not assume any assumption on monotonicity of ${\varphi_1, \, \varphi_2}$ in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular, Littlewood–Paley operator, Marcinkiewicz operator and Bochner–Riesz operator.

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

We consider generalized Morrey spaces ${\mathcal M}^{p(\cdot),\omega}(\Omega)$ with variable exponent $p(x)$ and a general function $\omega (x,r)$ defining the Morrey-type norm. In case of bounded sets $\Omega \subset {\mathsf R}^n$ we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev-Adams type ${\mathcal M}^{p(\cdot),\omega} (\Omega)\rightarrow {\mathcal M}^{q(\cdot),\omega} (\Omega)$-theorem for the potential operators $I^{\alpha(\cdot)}$, also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on $\omega(x,r)$, which do not assume any assumption on monotonicity of $\omega(x,r)$ in $r$.

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Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey 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.

Vagif S. Guliyev

Papers

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

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

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

Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces