Function Spaces, Entropy Numbers, Differential Operators

Abstract

The distribution of the eigenvalues of differential operators has long fascinated mathematicians. Advances have shed light upon classical problems in this area, and this book presents a fresh approach, largely based upon the results of the authors. The emphasis here is on a topic of central importance in analysis, namely the relationship between i) function spaces on Euclidean n-space and on domains; ii) entropy numbers in quasi-Banach spaces; and iii) the distribution of the eigenvalues of degenerate elliptic (pseudo) differential operators. The treatment is largely self-contained and accessible to non-specialists. Both experts and newcomers alike will welcome this unique exposition.

Function spaces with dominating mixed smoothness

“…Next we define the notion of entropy numbers and recall their basic properties. We refer to [10] and references given there for details.…”

“…The crucial property of entropy numbers was observed by Carl [6], who proved that the entropy numbers of a compact operator T ∈ L(A, A) dominate in some sense its eigenvalues. In general, we use the method of [10] in this part.…”

!Entropy Numbers of Embeddings of Weighted Besov Spaces. Ii

“…The decay of the numbers e n (T : X → Y ) describes the compactness of T in a qualitative way. In particular, T is compact if and only if this problem has a certain history, for which we refer to [8]. The interesting phenomenon observed in this situation is the following.…”

“…. , of bounded linear operators T ∈ L(X, Y ) between quasi-Banach spaces are a well-established field of research (see, for example, the monographs [6,8,21,33]). The decay of the numbers e n (T : X → Y ) describes the compactness of T in a qualitative way.…”

Connect the dots: how many random points can a regular curve pass through?

“…The modern trend in studying functional classes views the traditional regularity classes based on Hölder, Lipschitz and Sobolev regularity simply as special cases of the more general Besov and Triebel scales [22,8,12 We extend the notion of growth exponent slightly:…”