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The orthonormal Strichartz estimates and convergence problem of density functions related to $\partial_{x}^{3}+\partial_{x}^{-1}$

Xiangqian Yan, Yongsheng Li, Wei Yan

TL;DR

This work extends the theory of orthonormal Strichartz estimates and density-function convergence to the nonlocal Ostrovsky-type operator $\partial_x^3+\partial_x^{-1}$ on $\mathbb{R}$. By combining Schatten-space duality, dyadic analysis, and noncommutative interpolation with new ingredients addressing the singular phase at $0$, it proves maximal-in-time bounds on $[-1,1]$ against $\alpha$-dimensional measures, a Hausdorff-dimension bound on the divergence set, and small- and large-time Strichartz estimates for orthonormal systems. It also establishes probabilistic convergence of the density function via full randomization, enriching the random-data approach for dispersive operator flows. The results generalize prior theorems by Yan et al. and Zhao et al., offering a probabilistic framework and sharp analytic bounds for density convergence in this nonlocal dispersive context.

Abstract

In this article, we investigate the orthonormal Strichartz estimates and the convergence problem of the density function associated with $\partial_{x}^{3}+\partial_{x}^{-1}$. Firstly, when $γ_{0}\in\mathfrak{S}^β(\dot{H}^{s})$ with $\frac{1}{4}\leq s<\frac{1}{2},\, 0<α\leq 1$, and $1\leqβ<\fracα{1-2s}$, we prove that $\lim\limits_{t\longrightarrow0}\sum\limits_{j=1}^{+\infty}λ_{j} \left|e^{-t(\partial_{x}^{3}+\partial_{x}^{-1})}f_{j}\right|^{2}=\sum\limits_{j=1}^{+\infty}λ_{j} \left|f_{j}\right|^{2}.$ This extends the Theorem 1.1 of Yan et al. (Indiana Univ. Math. J. 71(2022), 1897-1921.). Moreover, we present the Hausdorff dimension of the divergence set of the density function related to $\partial_{x}^{3}+\partial_{x}^{-1}$, namely ${\rm dim_{H}}D(γ_{0})\leq (1-2s)β$, which extends the Theorem 1.1 of Zhao et al. (Acta Math. Sci. Ser. B (Engl. Ed.) 42(2022), 1607-1620.). % Secondly, we present the orthonormal Strichartz estimates and the Schatten bounds with space-time norms on $\mathbf{R}$. % Finally, by using full randomization, we establish the probabilistic convergence of the density function related to $\partial_{x}^{3}+\partial_{x}^{-1}$ on $\R$, which extends the Theorem 1.3 of Yan et al. (Indiana Univ. Math. J. 71(2022), 1897-1921.).

The orthonormal Strichartz estimates and convergence problem of density functions related to $\partial_{x}^{3}+\partial_{x}^{-1}$

TL;DR

This work extends the theory of orthonormal Strichartz estimates and density-function convergence to the nonlocal Ostrovsky-type operator on . By combining Schatten-space duality, dyadic analysis, and noncommutative interpolation with new ingredients addressing the singular phase at , it proves maximal-in-time bounds on against -dimensional measures, a Hausdorff-dimension bound on the divergence set, and small- and large-time Strichartz estimates for orthonormal systems. It also establishes probabilistic convergence of the density function via full randomization, enriching the random-data approach for dispersive operator flows. The results generalize prior theorems by Yan et al. and Zhao et al., offering a probabilistic framework and sharp analytic bounds for density convergence in this nonlocal dispersive context.

Abstract

In this article, we investigate the orthonormal Strichartz estimates and the convergence problem of the density function associated with . Firstly, when with , and , we prove that This extends the Theorem 1.1 of Yan et al. (Indiana Univ. Math. J. 71(2022), 1897-1921.). Moreover, we present the Hausdorff dimension of the divergence set of the density function related to , namely , which extends the Theorem 1.1 of Zhao et al. (Acta Math. Sci. Ser. B (Engl. Ed.) 42(2022), 1607-1620.). % Secondly, we present the orthonormal Strichartz estimates and the Schatten bounds with space-time norms on . % Finally, by using full randomization, we establish the probabilistic convergence of the density function related to on , which extends the Theorem 1.3 of Yan et al. (Indiana Univ. Math. J. 71(2022), 1897-1921.).

Paper Structure

This paper contains 14 sections, 24 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Background of Strichartz estimates for orthonormal functions
  3. Background of pointwise convergence problem and probabilistic convergence of single equation
  4. Background of pointwise convergence problem and probabilistic convergence of density function of operators
  5. Main contribution
  6. Some comments about new ingredients established in this paper
  7. The main results
  8. Preliminaries
  9. Probabilistic estimates of some random series
  10. Proof of Theorem 1.1: Maximal-in-time estimates for orthonormal functions on $\mathbf{B}$
  11. Proof of Theorem 1.2: The Hausdorff dimension of the divergence set
  12. Proof of Theorem 1.3: Strichartz estimates for orthonormal functions with small time
  13. Proof of Theorem 1.4: Strichartz estimates for orthonormal functions related to operator $\partial_{x}^{3}+\partial_{x}^{-1}$ with large time
  14. Proof of Theorem 1.5: Stochastic continuity at zero related to Schatten norm on $\hbox{\bf R}$

Key Result

Theorem 1.1

(Maximal-in-time estimates for orthonormal functions related to $\partial_{x}^{3} +\partial_{x}^{-1}$ on $\mathbf{B}$) Suppose that $\mathbf{B}=[-1,1]$, $\mu\in\mathcal{M}^{\alpha}(\mathbf{B})$, $\frac{1}{4}\leq s<\frac{1}{2}$, $0<\alpha\leq 1$, $1\leq\beta<\frac{\alpha}{1-2s}$ and $(f_{j})_{j} \in where $\lambda=(\lambda_{j})_{j}\in\ell^{\beta}$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 22 more