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Bilinear spherical maximal function on the Heisenberg group

Abhishek Ghosh, Rajesh K. Singh

Abstract

We introduce the bilinear Nevo-Thangavelu spherical means on the Heisenberg group $\mathbb{H}^n,$ and derive $L^{p_1}(\mathbb{H}^n) \times L^{p_2}(\mathbb{H}^n) \to L^{p}(\mathbb{H}^n)$ estimates for the single-scale bilinear averaging operators, the (full) bilinear Nevo-Thangavelu maximal operator and finally for the bilinear lacunary maximal operator on $\mathbb{H}^n; n \geq 2$. Our result for the full maximal operator is sharp. The principal tools in our analysis include newly developed estimates for single-scale bilinear averages, Hopf's maximal ergodic theorem, and a $T^*T$ argument adapted to this setting.

Bilinear spherical maximal function on the Heisenberg group

Abstract

We introduce the bilinear Nevo-Thangavelu spherical means on the Heisenberg group and derive estimates for the single-scale bilinear averaging operators, the (full) bilinear Nevo-Thangavelu maximal operator and finally for the bilinear lacunary maximal operator on . Our result for the full maximal operator is sharp. The principal tools in our analysis include newly developed estimates for single-scale bilinear averages, Hopf's maximal ergodic theorem, and a argument adapted to this setting.
Paper Structure (8 sections, 11 theorems, 112 equations, 2 figures)

This paper contains 8 sections, 11 theorems, 112 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Single scale estimates
  4. Full maximal function
  5. Lacunary maximal function
  6. Single scale decay estimates throughout $\mathcal{D}$
  7. Proof of Theorem \ref{['Horizontal lac result']}
  8. Sharpness

Key Result

Theorem 1.1

Let $n \geq 2,$ and $d=2n+1.$ Suppose $(\frac{1}{p_1}, \frac{1}{p_2})$ is contained in the region $\mathcal{D}$, which consists of the open pentagon with the corners $O=(0,0)$, $A=(0,1)$, $E=(\frac{d-1}{d},1)$, $F=(1,\frac{d-1}{d})$, $D=(1,0)$, together with the line segments $[O,A], [A,E), (F,D], [ uniformly for all $r>0.$

Figures (2)

  • Figure 1: $\mathfrak{M}_{\mathrm{full}}$-boundedness region $\mathcal{P}$ which is the union of open pentagon with vertices $O,A,B,C,D$ and the line segments $[O,A)$, $[O,D), (D, C)$ and $(A, B)$.
  • Figure 2: $\mathfrak{M}_{\mathrm{lac}}$-boundedness region $\mathcal{R}$ which is the union of open pentagon with vertices $O,A,E,F,D$ and line segments $[O,A)$, $(A,B)$, $(C,D)$ and $(D,O]$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 2.1: HickmanJFA
  • Proposition 2.2: SS or Proposition 3.6.1, ThangBook
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 10 more