Original scientific paper

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

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

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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.

Keywords

Sobolev embedding theorem, limiting case, radial-symmetric functions, regular variation

Hrčak ID:

332472

URI

https://hrcak.srce.hr/332472

Publication date:

19.5.2026.

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