A noninequality for the fractional gradient

Daniel Spector*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

In this paper we give a streamlined proof of an inequality recently obtained by the author: For every a a α∈(0,1) there exists a constant C = C(α,d)>0 such that for all u∈ Lq(Rd) for some 1 ≤ q < d/(1-α) such that Dαu :ΔI1-αu∈L1(Rd; Rd). We also give a counterexample which shows that in contrast to the case α=1, the fractional gradient does not admit an L1 trace inequality, i.e. ||Dαu||L1(Rd; Rd) cannot control the integral of u with respect to the Hausdorff content Hd-α. The main substance of this counterexample is a result of interest in its own right, that even a weak-type estimate for the Riesz transforms fails on the space L1(Hd-α),β∈[1,d).It is an open question whether this failure of a weak-type estimate for the Riesz transforms extends to β∈(0,1).

Original languageEnglish
Pages (from-to)153-168
Number of pages16
JournalPortugaliae Mathematica
Volume76
Issue number2
DOIs
Publication statusPublished - 2019
Externally publishedYes

Keywords

  • L-Sobolev inequality
  • Lorentz spaces
  • Trace inequality

ASJC Scopus subject areas

  • Mathematics(all)

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