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Hardy inequalities and nonlocal capacity

Tomasz Grzywny, Julia Lenczewska

Abstract

In this article, we introduce and study capacities related to nonlocal Sobolev spaces, with focus on spaces corresponding to zero-order nonlocal operators. In particular, we prove Hardy-type inequalities to obtain Sobolev embeddings and use them to estimate the nonlocal capacities of a ball.

Hardy inequalities and nonlocal capacity

Abstract

In this article, we introduce and study capacities related to nonlocal Sobolev spaces, with focus on spaces corresponding to zero-order nonlocal operators. In particular, we prove Hardy-type inequalities to obtain Sobolev embeddings and use them to estimate the nonlocal capacities of a ball.

Paper Structure

This paper contains 11 sections, 18 theorems, 104 equations.

Table of Contents

  1. Preliminaries
  2. Lower Matuszewska index and regular variation
  3. Concentration functions
  4. Dense subspace and min-max property of $W^{\nu}_p$
  5. Properties of nonlocal capacity
  6. Equivalent definitions of $\mathrm{Cap}_{\nu,p}$
  7. Capacitary inequality
  8. Connection between $\mathrm{Cap}_{\nu,p}$ and $\nu$-perimeter
  9. Hardy inequality, Sobolev embedding and ball's capacity
  10. Hardy inequalities and nonlocal Sobolev embedding
  11. Ball's capacity

Key Result

Lemma 1.1

If $L$ has lower Matuszewska index at zero strictly bigger than $-p$, then there exists $r_0>0$ such that If, additionally, $L$ has lower Matuszewska index at infinity strictly bigger than $-p$, then eq:hp-L holds for all $r>0$.

Theorems & Definitions (29)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • Lemma 1.3
  • Proposition 2.1
  • Corollary 2.2
  • Definition 2.1
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • ...and 19 more