Hypersingular integrals and potential type operators in variable Lebesgue spaces

Rafeiro, Humberto Gil Silva

http://hdl.handle.net/10400.1/737

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Authors

Rafeiro, Humberto Gil Silva

Advisor(s)

Samko, S.

Abstract(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.