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A note on bilinear multipliers with convex singularities

Valentina Ciccone

Abstract

We study bounds in the local $L^2$ range of exponents for bilinear multipliers whose symbol is the characteristic function of the epigraph of certain convex curves. We realize these bounds as a consequence of estimates that we establish, via simple arguments, for the associated exotic paraproducts. As a further application, we observe bounds beyond the local $L^2$ range for bilinear multipliers whose symbol is the characteristic function of the epigraph of convex polygonal curves associated with these paraproducts.

A note on bilinear multipliers with convex singularities

Abstract

We study bounds in the local range of exponents for bilinear multipliers whose symbol is the characteristic function of the epigraph of certain convex curves. We realize these bounds as a consequence of estimates that we establish, via simple arguments, for the associated exotic paraproducts. As a further application, we observe bounds beyond the local range for bilinear multipliers whose symbol is the characteristic function of the epigraph of convex polygonal curves associated with these paraproducts.
Paper Structure (5 sections, 6 theorems, 71 equations)

This paper contains 5 sections, 6 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Boundedness of the staircase paraproducts
  3. Proof of Theorem \ref{['thm_1_over_x_power']} and of Theorem \ref{['thm_power_positive']}
  4. Further examples and generalizations
  5. Proof of Theorem \ref{['thm_polygonal']}

Key Result

Theorem 1

Let $\lbrace a_j\rbrace_{j\in\mathbb{N}_0}, \;\lbrace b_j \rbrace_{j\in\mathbb{N}_0}$ be two strictly decreasing sequences of real numbers satisfying hypothesis $\mathrm{(Hyp \; 1)}$ -- respectively, $\mathrm{(Hyp \; 2)}$. Let $B_{m_{\mathtt{a,b}}}$ be the associated multiplier operator defined as i holds for all Schwartz functions $f,g\in\mathscr{S}(\mathbb{R})$ whenever and, in such range of ex

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6