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Sobolev embeddings with variable exponent

100%

Edmunds D. E., Rákosník J.

Studia Mathematica

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2000

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tom 143

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nr 3

267-293

EN

Let Ω be a bounded open subset of $ℝ^{n}$ with Lipschitz boundary and let $p:\overline{Ω} → [1,∞)$ be Lipschitz-continuous. We consider the generalised Lebesgue space $L^{p(x)}(Ω)$ and the corresponding Sobolev space $W^{1,p(x)}(Ω)$, consisting of all $f ∈ L^{p(x)}(Ω)$ with first-order distributional derivatives in $L^{p(x)}(Ω)$. It is shown that if 1 ≤ p(x) < n for all x ∈ Ω, then there is a constant c > 0 such that for all $f∈ W^{1,p(x)}(Ω)$, $|f|_{M,Ω} ≤ c|f|_{1,p,Ω}$. Here $|·|_{M,Ω}$ is the norm on an appropriate space of Orlicz-Musielak type and $|·|_{1,p,Ω}$ is the norm on $W^{1, p(x)}(Ω)$. The inequality reduces to the usual Sobolev inequality if $sup_Ω p

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Series representation of compact linear operators in Banach spaces

100%

Edmunds D. E., Lang J.

Commentationes Mathematicae

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2016

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tom 56

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nr 1

EN

Let $p\in(1,\infty)$ and $I=(0,1)$; suppose that $T\colon L_{p}(I)\rightarrow L_{p}(I)$ is a~compact linear map with trivial kernel and range dense in $L_{p}(I)$. It is shown that if the Gelfand numbers of $T$ decay sufficiently quickly, then the action of $T$ is given by a series with calculable coefficients. The special properties of $L_{p}(I)$ enable this to be established under weaker conditions on the Gelfand numbers than in earlier work set in the context of more general spaces.

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Double exponential integrability, Bessel potentials and embedding theorems

81%

Edmunds D. E., Gurka P., Opic B.

Studia Mathematica

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1995

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tom 115

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nr 2

151-181

EN

This paper is a continuation of [5] and provides necessary and sufficient conditions for double exponential integrability of the Bessel potential of functions from suitable (generalized) Lorentz-Zygmund spaces. These results are used to establish embedding theorems for Bessel potential spaces which extend Trudinger's result.

Ograniczanie wyników

2 Studia Mathematica

1 Commentationes Mathematicae

3 Edmunds D. E.

1 Gurka P.

1 Lang J.

1 Opic B.

1 Rákosník J.

1 2016

1 2000

1 1995

Sobolev embeddings with variable exponent

Series representation of compact linear operators in Banach spaces

Double exponential integrability, Bessel potentials and embedding theorems