RNDr. Zdeněk Mihula, Ph.D.

We investigate the relationship between the compactness of embeddings of Sobolev spaces built upon rearrangement-invariant spaces into rearrangement-invariant spaces endowed with d-Ahlfors measures under certain restriction on the speed of their decay on balls. We show that the gateway to compactness of such embeddings, while formally describable by means of optimal embeddings and almost-compact embeddings, is quite elusive. It is known that such a Sobolev embedding is not compact when its target space has the optimal fundamental function. We show that, quite surprisingly, such a target space can actually be "fundamentally enlarged", and yet the resulting embedding remains noncompact. In order to do that, we develop two different approaches. One is based on enlarging the optimal target space itself, and the other is based on enlarging the Marcinkiewicz space corresponding to the optimal fundamental function.

We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in R-n with respect to upper Ahlfors regular measures nu, that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure nu and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz-Zygmund norms are also presented.

We improve the discretization technique for weighted Lorentz norms by eliminating all "non-degeneracy" restrictions on the involved weights. We use the new method to provide equivalent estimates on the optimal constant C such that the inequality

We study interpolation properties of operators (not necessarily linear) which satisfy a specific K-inequality corresponding to endpoints defined in terms of Orlicz-Karamata spaces modeled upon the example of the Gaussian-Sobolev embedding. We prove a reduction principle for a fairly wide class of such operators. (c) 2022 Elsevier Inc. All rights reserved.

We study a three-weight inequality for the superposition of the Hardy operator and the Copson operator, namely

We compute the precise value of the measure of noncompactness of Sobolev embeddings W-0(1, p) (sic) L-p(D), p is an element of(1, infinity), on strip-like domains D of the form R-k x Pi(n-k)(j=1) (r(j), q(j)). We show that such embeddings are always maximally noncompact, that is, their measure of noncompactness coincides with their norms. Furthermore, we show that not only the measure of noncompactness but also all strict s-numbers of the embeddings in question coincide with their norms. We also prove that the maximal noncompactness of Sobolev embeddings on strip-like domains remains valid even when Sobolev-type spaces built upon general rearrangement-invariant spaces are considered. As a by-product we obtain the explicit form for the first eigenfunction of the pseudo- p-Laplacian on an n-dimensional rectangle. (C) 2021 Elsevier Inc. All rights reserved.