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

On Multidimensional Analogue of Marchaud Formula for Fractional Riesz-Type Derivatives in Domains in R^n

2000 Mathematics Subject Classification: 26A33, 42B20There is given a generalization of the Marchaud formula for one-dimensional fractional derivatives on an interval (a, b), −∞ < a < b ≤ ∞, to the multidimensional case of functions defined on a region in R^

Maximal operator in variable Stummel spaces

We prove that variable exponent Morrey spaces are closely embedded between variable exponent Stummel spaces. We also study the boundedness of the maximal operator in variable exponent Stummel spaces as well as in vanishing variable exponent Stummel spaces.publishe

On a Class of Fractional Type Integral Equations in Variable Exponent Spaces

2000 Mathematics Subject Classification: 45A05, 45B05, 45E05,45P05, 46E30We obtain a criterion of Fredholmness and formula for the Fredholm index of a certain class of one-dimensional integral operators M with a weak singularity in the kernel, from the variable exponent Lebesgue space L^p(·) ([a, b], ?) to the Sobolev type space L^α,p(·) ([a, b], ?) of fractional smoothness. We also give formulas of closed form solutions ϕ ∈ L^p(·) of the 1st kind integral equation M0ϕ = f, known as the generalized Abel equation, with f ∈ L^α,p(·), in dependence on the values of the variable exponent p(x) at the endpoints x = a and x = b.∗ This work was made under the project “Variable Exponent Analysis” supported by INTAS grant Nr.06-1000017-8792. * Supported by Fundação para a Ciência ea Tecnologia (FCT) (Grant No. SFRH / BD / 22977 / 2005), through Programa Operacional Ciência e Inovação 2010 (POCI2010) of the Portuguese Government, co-financed by the European Community Fund FSE

A note on Muckenhoupt weights with nonstandard growth

We provide quantitative results on the inclusion of a family of variable power weights in the Muckenhoupt A1 class, under appropriate conditions on the exponent. We also present an application related to variable exponent Muckenhoupt classes Ap(.).publishe

Local grand Lebesgue spaces on quasi-metric measure spaces and some applications

We introduce local grand Lebesgue spaces, over a quasi-metric measure space (X, d, mu), where the Lebesgue space is "aggrandized" not everywhere but only at a given closed set F of measure zero. We show that such spaces coincide for different choices of aggrandizers if their Matuszewska-Orlicz indices are positive. Within the framework of such local grand Lebesgue spaces, we study the maximal operator, singular operators with standard kernel, and potential type operators. Finally, we give an application to Dirichlet problem for the Poisson equation, taking F as the boundary of the domain.info:eu-repo/semantics/publishedVersio