David E. Edmunds

Sobolev Embeddings with Variable Exponent, II

Let Ω be a bounded open subset of Rn with Lipschitz boundary, let n 0 such that for all f ∈ W1,p(x)(Ω), where ∥·∥M,Ω is the norm on a certain space of Orlicz-Musielak type and ∥·∥1,p,Ω is the norm on W1,p(x)(Ω). This inequality reduces to the usual Sobolev inequality when supΩp < n. The paper extends earlier work of the authors ([ER]) in which it was assumed that p was Lipschitz-continuous.

Hardy Operators, Function Spaces and Embeddings

1 Preliminaries.- 2 Hardy-type Operators.- 3 Banach function spaces.- 4 Poincare and Hardy inequalities.- 5 Generalised ridged domains.- 6 Approximation numbers of Sobolev embeddings.- References.- Author Index.- Notation Index.

Bounded and compact integral operators

Preface. Acknowledgments. Basic notation. 1. Hardy-type operators. 2. Fractional integrals on the line. 3. One-sided maximal functions. 4. Ball fractional integrals. 5. Potentials on RN. 6. Fractional integrals on measure spaces. 7. Singular numbers. 8. Singular integrals. 9. Multipliers of Fourier transforms. 10. Problems. References. Index.

Density of smooth functions in Wk, p(x) (Ω)

Kováčik & Rákosník investigated the spaces Lp(x) (Ω) of functions which are integrable with variable power p(x) and the corresponding counterparts of the Sobolev spaces Wk, p(x) (Ω). We continue that investigation and describe a class of functions p(x) for which the set of smooth functions on Ω is dense in Wk, p(x) (Ω). As a corollary we obtain in terms of the distance function a condition on elements of Wk, p(x) (Ω) sufficient to ensure that they belong to Wk0, p(x) (Ω).

Spectral theory for isotropic fractal drums

Abstract We study the behaviour of eigenvalues in problems which correspond to the vibrations of a drum, the whole mass of which is concentrated on a fractal subset of the drum.

ON THE BOUNDEDNESS AND COMPACTNESS OF WEIGHTED HARDY OPERATORS IN SPACES L p(x)

Abstract Necessary conditions and sufficient conditions for the boundedness/ compactness of weighted Hardy operators are established in generalized Lebesgue spaces 𝐿𝑝(𝑥).

Sharpness of embeddings in logarithmic Bessel-potential spaces

This paper is a continuation of [ 4 ], where embeddings of certain logarithmic Bessel-potential spaces (modelled upon generalised Lorentz-Zygmund spaces) in appropriate Orlicz spaces (with Young functions of single and double exponential type) were derived. The aim of this paper is to show that these embedding results are sharp in the sense of [ 8 ].

Eigenfrequencies of isotropic fractal drums

We study the behaviour of eigenvalues in problems which correspond to the vibrations of a drum, the whole mass of which is concentrated on a fractal subset of the drum.

Fractional Integrals on Measure Spaces

In this chapter we present results concerning the boundedness and compactness of integral transforms generated by various types of fractional integrals.

Potential-Type Operators in

In this paper we derive weight inequalities for one-sided and Riesz potentials in L spaces under the condition that p satisfies a weak Lipschitz condition. Compactness of these operators in L spaces is also established.