Maximal Non-compactness of Sobolev Embeddings
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Maximal Non-compactness of Sobolev Embeddings Jan Lang1
· Vít Musil2,3
· Miroslav Olšák4
· Luboš Pick5
Received: 7 April 2020 / Accepted: 17 September 2020 © Mathematica Josephina, Inc. 2020
Abstract It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity of spaces L p,∞ has been open, too. In this paper, we solve both these problems. We first show that weak Lebesgue spaces are never disjointly superadditive, so the suggested technique is ruled out. But then we show that, perhaps somewhat surprisingly, the measure of non-compactness of a sharp Sobolev embedding coincides with the embedding norm nevertheless, at least as long as p < ∞. Finally, we show that if the target space is L ∞ (which formally is also a weak Lebesgue space with p = ∞), then the things are essentially different. To give a comprehensive answer including this case, too, we develop a new method based on a rather unexpected combinatorial argument and prove thereby a general principle, whose special case implies that the measure of non-compactness, in this case, is strictly less than its norm. We develop a technique that enables us to evaluate this measure of non-compactness exactly. Keywords Ball measure of non-compactness · Maximal non-compactness · Sobolev embedding · Weak Lebesgue spaces
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Luboš Pick [email protected]
1
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174, USA
2
Dipartimento di Matematica e Informatica “Ulisse Dini”, University of Florence, Viale Morgagni 67/A, 50134 Florence, Italy
3
Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic
4
Department of Computer Science, Technikerstrasse 21a, 6020 Innsbruck, Austria
5
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
123
J. Lang et al.
Mathematics Subject Classification 46E35 · 47B06
1 Introduction Given a linear mapping acting between two (quasi)normed linear spaces, one of the most important questions is whether it is compact. Compactness is often desired or even indispensable for specific applications in different areas of mathematics. It plays an important role in theoretical parts of functional analysis such as, for instance, in the proof of the Schauder fixed point theorem, and also in the most customary applications of functional analysis such as proving existence, uniqueness, and regularity of solutions to partial differential equations via compact embeddings of Sobolev-type spaces into various other function spaces. However, more often than not, the mapping in question is not compact. For a non-compact mapping, more
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