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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.13561 |
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Table of Contents:
- 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.).