Glasnik Matematicki, Vol. 60, No. 1 (2025), 127-145. \( \)

THE LIMITING CASE IN THE SOBOLEV EMBEDDING THEOREM AND RADIAL-SYMMETRIC FUNCTIONS

Peter Grandits

Institute for Mathematical Methods in Economics, TU Wien, Wiedner HauptstraĂźe 8-10, 1040 Wien, Austria
e-mail:pgrand@fam.tuwien.ac.at

Abstract.   Denoting by \(B_{r_0}\) the open ball with radius \(r_0\), centered at the origin, we consider the so called “limiting case” in the Sobolev embedding theorem, \( W^{j+m,p}(B_{r_0})\to W^{j,q}(B_{r_0}), \) namely the case \(mp=n\), \(1\lt p\leq q\), where the embedding for \(q=\infty\) does not hold. We show that in the case \(j=1\), contrary to the case \(j=0\), radial-symmetric counterexamples, that is radial-symmetric functions in \(W^{m+1,p}(B_{r_0}) \setminus W^{1,\infty}(B_{r_0})\) do not exist, if one assumes \(C^2\)-regularity away from the origin. Moreover, we characterize in dimension \(n=2\) the set \(W^{m+1,p}(B_{r_0}) \setminus W^{1,\infty}(B_{r_0})\), i.e. \(W^{2,2}(B_{r_0}) \setminus W^{1,\infty}(B_{r_0})\) within a reasonable large class of functions.

2020 Mathematics Subject Classification.   46E35

Key words and phrases.   Sobolev embedding theorem, limiting case, radial-symmetric functions, regular variation

https://doi.org/10.3336/gm.60.1.08

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