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Mixed norm estimates for dilated averages over planar curves

Junfeng Li, Zengjian Lou, Haixia Yu

TL;DR

The paper analyzes mixed-norm estimates for dilated planar curve averages, establishing sharp space-time and time-space bounds for the operator along a general plane curve with curvature conditions. By reducing T to a Fourier integral operator with cinematic curvature and decomposing into dyadic pieces, the authors derive precise ranges for (1/p,1/q,1/r) through stationary-phase analysis and interpolation. They prove nontrivial, nearly sharp bounds for a broad class of curves, including the parabola, and identify borderlines where sharpness fails, supported by endpoint counterexamples and decay estimates. The results extend the understanding of L^p-improving phenomena and Strichartz-type estimates for curved averaging operators in two dimensions, with implications for related Fourier integral operators and maximal functions.

Abstract

In this paper, we investigate the mixed norm estimates for the operator $ T $associated with a dilated plane curve $(ut, uγ(t))$, defined by \[ Tf(x, u) := \int_{0}^{1} f(x_1 - ut, x_2 - uγ(t)) \, dt, \] where $ x := (x_1, x_2) $ and $γ$ is a general plane curve satisfying appropriate smoothness and curvature conditions. More precisely, we establish the $ L_x^p(\mathbb{R}^2) \rightarrow L_x^q L_u^r(\mathbb{R}^2 \times [1, 2]) $ (space-time) estimates for $ T $, whenever $(\frac{1}{p},\frac{1}{q})$ satisfy \[ \max\left\{0, \frac{1}{2p} - \frac{1}{2r}, \frac{3}{p} - \frac{r+2}{r}\right\} < \frac{1}{q} \leq \frac{1}{p} < \frac{r+1}{2r} \] and $$1 + (1 + ω)\left(\frac{1}{q} - \frac{1}{p}\right) > 0,$$ where $ r \in [1, \infty] $ and $ ω:= \limsup_{t \rightarrow 0^+} \frac{\ln|γ(t)|}{\ln t} $. These results are sharp, except for certain borderline cases. Additionally, we examine the $ L_x^p(\mathbb{R}^2) \rightarrow L_u^r L_x^q(\mathbb{R}^2 \times [1, 2]) $ (time-space) estimates for $T $, which are especially almost sharp when $p=2$ or $p\in [1, \frac{3}{2}]\cup [4, \infty]$.

Mixed norm estimates for dilated averages over planar curves

TL;DR

The paper analyzes mixed-norm estimates for dilated planar curve averages, establishing sharp space-time and time-space bounds for the operator along a general plane curve with curvature conditions. By reducing T to a Fourier integral operator with cinematic curvature and decomposing into dyadic pieces, the authors derive precise ranges for (1/p,1/q,1/r) through stationary-phase analysis and interpolation. They prove nontrivial, nearly sharp bounds for a broad class of curves, including the parabola, and identify borderlines where sharpness fails, supported by endpoint counterexamples and decay estimates. The results extend the understanding of L^p-improving phenomena and Strichartz-type estimates for curved averaging operators in two dimensions, with implications for related Fourier integral operators and maximal functions.

Abstract

In this paper, we investigate the mixed norm estimates for the operator associated with a dilated plane curve , defined by where and is a general plane curve satisfying appropriate smoothness and curvature conditions. More precisely, we establish the (space-time) estimates for , whenever satisfy and where and . These results are sharp, except for certain borderline cases. Additionally, we examine the (time-space) estimates for , which are especially almost sharp when or .

Paper Structure

This paper contains 15 sections, 11 theorems, 165 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm1']}
  4. Sufficiency of the region $(\frac{1}{p}, \frac{1}{q})$ for all $r \in [1, \infty]$
  5. The necessity of the region of $(\frac{1}{p}, \frac{1}{q})$ for all $r\in[1,\infty]$
  6. Proof of Theorem \ref{['thm3']}
  7. Sufficient Part
  8. The Case $1 \leq p \leq q \leq r \leq \infty$
  9. The Case $1 \leq p \leq r \leq q \leq \infty$
  10. The Case $1 \leq r < p \leq q \leq \infty$
  11. Necessary Part
  12. Proof of (I) in Theorem \ref{['thm3']}
  13. Proof of (II) of Theorem \ref{['thm3']}
  14. Proof of (III) of Theorem \ref{['thm3']}
  15. Proof of (IV) of Theorem \ref{['thm3']}

Key Result

Theorem 1.1

Assume $\gamma \in C^{N}(0,1]$ with $N \in \mathbb{N}$ sufficiently large, $\lim_{t \rightarrow 0^+} \gamma(t) = 0$, and $\gamma$ is monotonic on $(0,1]$. Additionally, suppose $\gamma$ satisfies the following conditions: Then, there exists a positive constant $C$ depending only on $\gamma$, $p$, $q$ and $r$, such that for all $f \in L_x^{p}(\mathbb{R}^{2})$, where $(\frac{1}{p}, \frac{1}{q})$ s

Figures (5)

  • Figure 1: The range of $(\frac{1}{p}, \frac{1}{q})$ in Theorem \ref{['thm1']} for all $r \in [1, \infty]$.
  • Figure 2: The range of $(\frac{1}{q}, \frac{1}{r})$ in Theorem \ref{['thm3']} when $p = 2$.
  • Figure 3: The range of $(\frac{1}{q}, \frac{1}{r})$ in Theorem \ref{['thm3']} for $p \in [1, \frac{3}{2}]$ and $p \in [4, \infty]$.
  • Figure 4: The range of $\left( \frac{1}{p}, \frac{1}{q} \right)$ in Theorem \ref{['thm3']} with $r = 2$.
  • Figure 5: The region of boundedness in \ref{['eq:1.7']} for the case $1 \leq r < p \leq q \leq \infty$ and $1 \leq r \leq \frac{3}{2}$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • ...and 14 more