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Boundedness of Bilinear Bessel Potentials

Ana Čolović, Xinyu Gao

Abstract

In analogy with bilinear Riesz potentials, we introduce bilinear Bessel potentials and characterize their boundedness from $L^p\times L^q$ into Lebesgue and Lorentz spaces $L^{r,α}.$ In several cases we identify the optimal Lorentz indices by constructing explicit counterexamples.

Boundedness of Bilinear Bessel Potentials

Abstract

In analogy with bilinear Riesz potentials, we introduce bilinear Bessel potentials and characterize their boundedness from into Lebesgue and Lorentz spaces In several cases we identify the optimal Lorentz indices by constructing explicit counterexamples.
Paper Structure (19 sections, 16 theorems, 113 equations)

This paper contains 19 sections, 16 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Background
  3. Lorentz spaces and tools
  4. Properties of Bessel Kernels
  5. Necessary Conditions and Exponent Geometry
  6. Strong-Type Lebesgue Bounds
  7. $L^p\times L^q\to L^r$
  8. $L^1\times L^1\to L^{\frac{1}{2}}$
  9. Lorentz Space Estimates on the Fractional Index Surface
  10. $L^1\times L^q \longrightarrow L^{r_1,\alpha}$ for $1<q<\infty$
  11. Interior of the Fractional Index Plane
  12. Between Two Planes
  13. $p=\infty$ or $q=\infty$
  14. Triangle
  15. Fixed-input interpolation
  16. ...and 4 more sections

Key Result

Proposition 2.2

For $0<p<\infty,$$0<q\leq \infty,$ the following formula holds: with the usual modification when $q=\infty$. The following inequality represents an extension of Young's inequality in Lorentz spaces. It was proved by O'Neil and can be found in ONe.

Theorems & Definitions (27)

  • Definition 2.1: Lorentz spaces
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4: Integrability of the Bessel kernel
  • Proposition 3.1
  • proof
  • Theorem 4.1: Grafakos--Soria GraSor
  • Corollary 4.2
  • proof
  • Lemma 4.3
  • ...and 17 more