Some sharp inequalities of Mizohata--Takeuchi-type
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911991628562432 |
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| author | Carbery, Anthony Iliopoulou, Marina Wang, Hong |
| author_facet | Carbery, Anthony Iliopoulou, Marina Wang, Hong |
| contents | Let $Σ$ be a strictly convex, compact patch of a $C^2$ hypersurface in $\mathbb{R}^n$, with non-vanishing Gaussian curvature and surface measure $dσ$ induced by the Lebesgue measure in $\mathbb{R}^n$. The Mizohata--Takeuchi conjecture states that
\begin{equation*}
\int |\widehat{gdσ}|^2w \leq C \|Xw\|_\infty \int |g|^2
\end{equation*}
for all $g\in L^2(Σ)$ and all weights $w:\mathbb{R}^n\rightarrow [0,+\infty)$, where $X$ denotes the $X$-ray transform. As partial progress towards the conjecture, we show, as a straightforward consequence of recently-established decoupling inequalities, that for every $ε>0$, there exists a positive constant $C_ε$, which depends only on $Σ$ and $ε$, such that for all $R \geq 1$ and all weights $w:\mathbb{R}^n\rightarrow [0,+\infty)$ we have \begin{equation*}
\int_{B_R} |\widehat{gdσ}|^2w \leq C_εR^ε\sup_T \left(\int _T w^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}}\int |g|^2,
\end{equation*}
where $T$ ranges over the family of all tubes in $\mathbb{R}^n$ of dimensions $R^{1/2} \times \dots \times R^{1/2} \times R$. From this we deduce the Mizohata--Takeuchi conjecture with an $R^{\frac{n-1}{n+1}}$-loss; i.e., that \begin{equation*}
\int_{B_R} |\widehat{gdσ}|^2w \leq C_εR^{\frac{n-1}{n+1}+ ε}\|Xw\|_\infty\int |g|^2
\end{equation*}
for any ball $B_R$ of radius $R$ and any $ε>0$. The power $(n-1)/(n+1)$ here cannot be replaced by anything smaller unless properties of $\widehat{gdσ}$ beyond 'decoupling axioms' are exploited. We also provide estimates which improve this inequality under various conditions on the weight, and discuss some new cases where the conjecture holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_11877 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some sharp inequalities of Mizohata--Takeuchi-type Carbery, Anthony Iliopoulou, Marina Wang, Hong Classical Analysis and ODEs Analysis of PDEs Let $Σ$ be a strictly convex, compact patch of a $C^2$ hypersurface in $\mathbb{R}^n$, with non-vanishing Gaussian curvature and surface measure $dσ$ induced by the Lebesgue measure in $\mathbb{R}^n$. The Mizohata--Takeuchi conjecture states that \begin{equation*} \int |\widehat{gdσ}|^2w \leq C \|Xw\|_\infty \int |g|^2 \end{equation*} for all $g\in L^2(Σ)$ and all weights $w:\mathbb{R}^n\rightarrow [0,+\infty)$, where $X$ denotes the $X$-ray transform. As partial progress towards the conjecture, we show, as a straightforward consequence of recently-established decoupling inequalities, that for every $ε>0$, there exists a positive constant $C_ε$, which depends only on $Σ$ and $ε$, such that for all $R \geq 1$ and all weights $w:\mathbb{R}^n\rightarrow [0,+\infty)$ we have \begin{equation*} \int_{B_R} |\widehat{gdσ}|^2w \leq C_εR^ε\sup_T \left(\int _T w^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}}\int |g|^2, \end{equation*} where $T$ ranges over the family of all tubes in $\mathbb{R}^n$ of dimensions $R^{1/2} \times \dots \times R^{1/2} \times R$. From this we deduce the Mizohata--Takeuchi conjecture with an $R^{\frac{n-1}{n+1}}$-loss; i.e., that \begin{equation*} \int_{B_R} |\widehat{gdσ}|^2w \leq C_εR^{\frac{n-1}{n+1}+ ε}\|Xw\|_\infty\int |g|^2 \end{equation*} for any ball $B_R$ of radius $R$ and any $ε>0$. The power $(n-1)/(n+1)$ here cannot be replaced by anything smaller unless properties of $\widehat{gdσ}$ beyond 'decoupling axioms' are exploited. We also provide estimates which improve this inequality under various conditions on the weight, and discuss some new cases where the conjecture holds. |
| title | Some sharp inequalities of Mizohata--Takeuchi-type |
| topic | Classical Analysis and ODEs Analysis of PDEs |
| url | https://arxiv.org/abs/2302.11877 |