Orthonormal Strichartz inequalities and their applications on abstract measure spaces
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909321988669440 |
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| author | Feng, Guoxia Mondal, Shyam Swarup Song, Manli Wu, Huoxiong |
| author_facet | Feng, Guoxia Mondal, Shyam Swarup Song, Manli Wu, Huoxiong |
| contents | The main objective of this paper is to extend certain fundamental inequalities from a single function to a family of orthonormal systems. In the first part of the paper, we consider a non-negative, self-adjoint operator $L$ on $L^2(X,μ)$, where $(X,μ)$ is a measure space. Under the assumption that the kernel $K_{it}(x,y)$ of the Schrödinger propagator $e^{itL}$ satisfies a uniform $L^\infty$-decay estimate of the form
\begin{equation*}
\sup_{x,y\in X}|K_{it}(x,y)|\lesssim |t|^{-\frac{n}{2}},\,|t|<T_0, \text{ for some }n\geq1,
\end{equation*} where $T_0\in(0,+\infty]$, we establish Strichartz estimates for the Schrödinger propagator $e^{itL}$ and using a duality principle argument by Frank-Sabin \cite{FS}, we extend it for a system of infinitely many fermions on $L^2(X)$. We also obtain orthonormal Strichartz estimates for a class of dispersive semigroup $U(t)=e^{itϕ(L)}ψ(\sqrt{L}),$ where $ϕ: \mathbb{R}^+\rightarrow \mathbb{R}$ is a smooth function and $ψ\in C_c^\infty([\frac{1}{2},2])$. As an application of these orthonormal versions of Strichartz estimates, we prove the well-posedness for the Hartree equation in the Schatten spaces.
In the next part of the paper, we obtain some new orthonormal Strichartz estimates, which extend prior work of Kenig-Ponce-Vega \cite{Kenig-Ponce-Vega} for single functions. Using those orthonormal versions of Kenig-Ponce-Vega result, we prove the orthonormal restriction theorem for the Fourier transform on some particular noncompact hypersurface of the form $S=\{(ξ, ϕ(ξ): ξ\in \mathbb{R})\}$, where $ϕ$ satisfies certain growth condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14044 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orthonormal Strichartz inequalities and their applications on abstract measure spaces Feng, Guoxia Mondal, Shyam Swarup Song, Manli Wu, Huoxiong Functional Analysis Primary 42B37, 47D08, 35B65 The main objective of this paper is to extend certain fundamental inequalities from a single function to a family of orthonormal systems. In the first part of the paper, we consider a non-negative, self-adjoint operator $L$ on $L^2(X,μ)$, where $(X,μ)$ is a measure space. Under the assumption that the kernel $K_{it}(x,y)$ of the Schrödinger propagator $e^{itL}$ satisfies a uniform $L^\infty$-decay estimate of the form \begin{equation*} \sup_{x,y\in X}|K_{it}(x,y)|\lesssim |t|^{-\frac{n}{2}},\,|t|<T_0, \text{ for some }n\geq1, \end{equation*} where $T_0\in(0,+\infty]$, we establish Strichartz estimates for the Schrödinger propagator $e^{itL}$ and using a duality principle argument by Frank-Sabin \cite{FS}, we extend it for a system of infinitely many fermions on $L^2(X)$. We also obtain orthonormal Strichartz estimates for a class of dispersive semigroup $U(t)=e^{itϕ(L)}ψ(\sqrt{L}),$ where $ϕ: \mathbb{R}^+\rightarrow \mathbb{R}$ is a smooth function and $ψ\in C_c^\infty([\frac{1}{2},2])$. As an application of these orthonormal versions of Strichartz estimates, we prove the well-posedness for the Hartree equation in the Schatten spaces. In the next part of the paper, we obtain some new orthonormal Strichartz estimates, which extend prior work of Kenig-Ponce-Vega \cite{Kenig-Ponce-Vega} for single functions. Using those orthonormal versions of Kenig-Ponce-Vega result, we prove the orthonormal restriction theorem for the Fourier transform on some particular noncompact hypersurface of the form $S=\{(ξ, ϕ(ξ): ξ\in \mathbb{R})\}$, where $ϕ$ satisfies certain growth condition. |
| title | Orthonormal Strichartz inequalities and their applications on abstract measure spaces |
| topic | Functional Analysis Primary 42B37, 47D08, 35B65 |
| url | https://arxiv.org/abs/2409.14044 |