Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in hardy type spaces

The aim of the paper is twofold. First, we present a new general approach to the definition of a class of mixed norm spaces of analytic functions A(q;X)(D), 1 <= q < infinity on the unit disc D. We study a problem of boundedness of Bergman projection in this general setting. Second, we apply this general approach for the new concrete cases when X is either Orlicz space or generalized Morrey space, or generalized complementary Morrey space. In general, such introduced spaces are the spaces of functions which are in a sense the generalized Hadamard type derivatives of analytic functions having l(q) summable Taylor coefficients.Russian Fund of Basic Research [15-01-02732]; SFEDU grant [07/2017-31]info:eu-repo/semantics/publishedVersio

Fractional integrals and derivatives: mapping properties

This survey is aimed at the audience of readers interested in the information on mapping properties of various forms of fractional integration operators, including multidimensional ones, in a large scale of various known function spaces.As is well known, the fractional integrals defined in this or other forms improve in some sense the properties of the functions, at least locally, while fractional derivatives to the contrary worsen them. With the development of functional analysis this simple fact led to a number of important results on the mapping properties of fractional integrals in various function spaces.In the one-dimensional case we consider both Riemann-Liouville and Liouville forms of fractional integrals and derivatives. In the multidimensional case we consider in particular mixed Liouville fractional integrals, Riesz fractional integrals of elliptic and hyperbolic type and hypersingular integrals. Among the function spaces considered in this survey, the reader can find Holder spaces, Lebesgue spaces, Morrey spaces, Grand spaces and also weighted and/or variable exponent versions

On maximal and potential operators with rough kernels in variable exponent spaces

In the framework of variable exponent Lebesgue and Morrey spaces we prove some boundedness results for operators with rough kernels, such as the maximal operator, fractional maximal operator, sharp maximal operators and fractional operators. The approach is based on some pointwise estimates

Hardy type inequality in variable Lebesgue spaces

We prove that in variable exponent spaces $L^{p(\cdot)}(\Omega)$, where $p(\cdot)$ satisfies the log-condition and $\Omega$ is a bounded domain in $\mathbf R^n$ with the property that $\mathbf R^n \backslash \bar{\Omega}$ has the cone property, the validity of the Hardy type inequality | 1/\delta(x)^\alpha \int_\Omega \phi(y) dy/|x-y|^{n-\alpha}|_{p(\cdot)} \leqq C |\phi|_{p(\cdot)}, \quad 0<\al<\min(1,\frac{n}{p_+}), where $\delta(x)=\mathrm{dist}(x,\partial\Omega)$, is equivalent to a certain property of the domain \Om expressed in terms of \al and \chi_\Om.Comment: 16 page

Best Constant in the Weighted Hardy Inequality: The Spatial and Spherical Version

Mathematics Subject Classification: 26D10.The sharp constant is obtained for the Hardy-Stein-Weiss inequality for fractional Riesz potential operator in the space L^p(R^n, ρ) with the power weight ρ = |x|^β. As a corollary, the sharp constant is found for a similar weighted inequality for fractional powers of the Beltrami-Laplace operator on the unit sphere

Weighted Hardy and potential operators in the generalized Morrey spaces

We study the weighted p -> q-boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces L-p.phi(R-n, w) defined by an almost increasing function phi(r) and radial type weight w(vertical bar x vertical bar). We obtain sufficient conditions, in terms of some integral inequalities imposed on phi and w, for such a p -> q-boundedness. In some cases the obtained conditions are also necessary. These results are applied to derive a similar weighted p -> q-boundedness of the Riesz potential operator. (c) 2010 Elsevier Inc. All rights reserved.Lulea University of Technology; FCT, Portugal [SFRH/BPD/34258/2006]info:eu-repo/semantics/publishedVersio

Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces

We consider non-standard Holder spaces H(lambda(.))(X) of functions f on a metric measure space (X, d, mu), whose Holder exponent lambda(x) is variable, depending on x is an element of X. We establish theorems on mapping properties of potential operators of variable order alpha(x), from such a variable exponent Holder space with the exponent lambda(x) to another one with a "better" exponent lambda(x) + alpha(x), and similar mapping properties of hypersingular integrals of variable order alpha(x) from such a space into the space with the "worse" exponent lambda(x) - alpha(x) in the case alpha(x) 0. We admit variable complex valued orders alpha(x), where R alpha(x) may vanish at a set of measure zero. To cover this case, we consider the action of potential operators to weighted generalized Holder spaces with the weight alpha(x).FCT, Portugal [SFRH/BPD/34258/2006

On a 3D-hypersingular equation of a problem for a crack

We show that a certain axisymmetric hypersingular integral equation arising in problems of cracks in the elasticity theory may be explicitly solved in the case where the crack occupies a plane circle. We give three different forms of the resolving formula. Two of them involve regular kernels, while the third one involves a singular kernel, but requires less regularity assumptions on the the right-hand side of the equation.Russian Federal Targeted Programme "Scientific and Research-Educational Personnel of Innovative Russia" [02.740.11.5024

Possible Experimental Test of Continuous Medium Model for Fractal Media

We use the fractional integrals to describe fractal media. We consider the fractal media as special ("fractional") continuous media. We discuss the possible experimental testing of the continuous medium model for fractal media that is suggested in Phys. Lett. A. 336 (2005) 167-174. This test is connected with measure of period of the Maxwell pendulum with fractal medium cylinder.Comment: 9 page