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Discrete Besov spaces via semigroups associated to the discrete Laplacian and regularity of non-local operators

Luciano Abadias, Marta De León-Contreras, Alejandro Mahillo

Abstract

In this paper we prove characterizations of the discrete Besov spaces in terms of the heat and Poisson semigroups associated with the discrete Laplacian that will allow us to prove regularity results for the fractional powers of the discrete Laplacian and the discrete Bessel potentials. Moreover, we provide new estimates for the derivatives of the discrete heat kernel and semigroup which are of independent interest.

Discrete Besov spaces via semigroups associated to the discrete Laplacian and regularity of non-local operators

Abstract

In this paper we prove characterizations of the discrete Besov spaces in terms of the heat and Poisson semigroups associated with the discrete Laplacian that will allow us to prove regularity results for the fractional powers of the discrete Laplacian and the discrete Bessel potentials. Moreover, we provide new estimates for the derivatives of the discrete heat kernel and semigroup which are of independent interest.
Paper Structure (11 sections, 24 theorems, 138 equations)

This paper contains 11 sections, 24 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Discrete heat and Poisson semigroups
  3. Some known results
  4. Discrete heat kernel
  5. Heat and Poisson semigroups
  6. Characterization via semigroups of discrete Besov spaces
  7. Case 0 < α < 1
  8. Case 0 < α < 2
  9. General case
  10. Applications
  11. Appendix

Key Result

Theorem 1.1

Let $1 \leq p, q \leq \infty$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: A priori estimates
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • ...and 32 more