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On sharp heat kernel estimates in the context of Fourier-Dini expansions

Bartosz Langowski, Adam Nowak

Abstract

We prove sharp estimates of the heat kernel associated with Fourier-Dini expansions on $(0,1)$ equipped with Lebesgue measure and the Neumann condition imposed on the right endpoint. Then we give several applications of this result including sharp bounds for the corresponding Poisson and potential kernels, sharp mapping properties of the maximal heat semigroup and potential operators and boundary convergence of the Fourier-Dini semigroup.

On sharp heat kernel estimates in the context of Fourier-Dini expansions

Abstract

We prove sharp estimates of the heat kernel associated with Fourier-Dini expansions on equipped with Lebesgue measure and the Neumann condition imposed on the right endpoint. Then we give several applications of this result including sharp bounds for the corresponding Poisson and potential kernels, sharp mapping properties of the maximal heat semigroup and potential operators and boundary convergence of the Fourier-Dini semigroup.

Paper Structure

This paper contains 21 sections, 20 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Facts and formulas concerning the Bessel functions $J_{\nu}$ and $I_{\nu}$
  5. Fourier-Dini system
  6. Main result
  7. Jacobi trigonometric Lebesgue measure setting scaled to $(0,1)$
  8. Preparatory results
  9. Proof of Theorem \ref{['thm:main']}
  10. Comments on the proof method
  11. Applications
  12. Maximal operators and boundary convergence of the Fourier-Dini semigroup
  13. Sharp bounds for the Fourier-Dini-Poisson kernel
  14. Sharp bounds for Fourier-Dini potential kernels
  15. Sharp $L^p-L^q$ mapping properties of Fourier-Dini potential operators
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $\nu > -1$ and $H \in \mathbb{R}$ be fixed. Then the Fourier-Dini system is an orthonormal basis in $L^2((0,1),dx)$.

Theorems & Definitions (38)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3: NSS1NSS2NSS3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • ...and 28 more