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Research Project

## OPERADORES HIPERSINGULARES E OPERADORES DE TIPO DE POTENCIAL NOS ESPAÇOS DE LEBESGUE COM EXPOENTE VARIÁVEL

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## Authors

## Publications

Hardy type inequalityin variable lebesgue spacesExpand Expand

Publication . Rafeiro, Humberto; Samko, Stefan

We prove that in variable exponent spaces where L-p(.)(Omega), where p(.) satisfies the log-condition and Omega is a bounded domain in R-n with the property that R-n\(Omega) over bar has the cone property, the validity of the Hardy type inequality parallel to 1/delta(x)(alpha)integral(Omega)phi(y)/vertical bar x-y vertical bar(n-alpha)dy parallel to(p(.)) <= C parallel to phi parallel to(p(.)), 0 < alpha < min (1, n/p(+)), where delta(x) is approximately equal to dist(x, partial derivative Omega), is equivalent to a certain property of the domain Omega expressed in terms of alpha and chi(Omega).

Hypersingular integrals and potential type operators in variable Lebesgue spacesExpand Expand

Publication . Rafeiro, Humberto Gil Silva; Samko, S.

The study in the thesis is made within the frameworks of weighted variable Lebesgue
spaces and operator theory in such spaces, and is related to the following main results.
We give a characterization of the range of the one-dimensional Riemann-Liouville
fractional integral operator over weighted variable exponent Lebesgue spaces which is
given in terms of convergence of the corresponding hypersingular integrals. These range
is shown to coincide with a weighted Sobolev-type space.
We obtain a criterion of Fredholmness and formula for the Fredholm index of a certain
class of one-dimensional integral operators with a weak singularity in the kernel, from
the variable exponent Lebesgue space to a Sobolev-type space of fractional smoothness.
Within these spaces, formulas of closed form solutions for the so-called generalized Abel
equation are given.
We study the inversion problem for the Riesz and Bessel potential operators in variable
exponent Lebesgue spaces. The inverse operator is constructed by using approximative
inverse operators. This generalizes some classical results to the variable exponent setting.
The Grünwald-Letnikov approach to fractional differentiation, in the case of fractional
powers of the minus Laplace operator, is shown to coincide with the Riesz fractional
derivative in the variable setting, allowing us to give a characterization of the variable
exponent Bessel potential space in terms of the rate of convergence of the Poisson semigroup.
The validity of a multidimensional Hardy type inequality in variable Lebesgue spaces
is shown to be equivalent to a certain property of the domain.
A famous theorem of Krasnoselskii states that compactness of a regular linear integral
operator in Lebesgue spaces follows from that of a majorant operator. This theorem is
extended to the case of arbitrary Banach function spaces, from which—as a particular
case—we obtain such a dominated compactness theorem for weighted variable Lebesgue
spaces. We also extend Kolmogorov’s compactness criterion to the case of variable
Lebesgue spaces, requiring only the “uniformness” condition and allowing also the case
where p¡ Æ 1.

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## Funders

## Funding agency

Fundação para a Ciência e a Tecnologia

## Funding programme

## Funding Award Number

SFRH/BD/22977/2005