Smoothness of extremizers for certain inequalities of the Radon transform
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909716707278848 |
|---|---|
| author | Flock, Taryn C. |
| author_facet | Flock, Taryn C. |
| contents | The Radon transform is a bounded operator from $L^p$ of Euclidean space to $L^q$ of the manifold of all affine hyperplanes in $\mathbb{R}^n$ for certain exponents depending dimension. Extremizers have been determined for certain values of $q$ and $p$, but most remain open.
We show that extremizers are infinitely differentiable whenever the exponents in the associated Euler-Lagrange equation, $q-1$ and $\frac1{p-1}$, are integers. The proof adapts the method of Christ and Xue, to the case where the underlying space is a manifold.
The proof is carried out in the setting of the $k$-plane transform, which takes functions on $\mathbb{R}^n$ to functions on the manifold of all affine $k$-planes in $\mathbb{R}^n$ by integrating the function over the $k$-dimensional plane. We show that when $q-1$ and $\frac1{p-1}$ are intergers, all nonnegative critical points of the functional
\[ \|T_{n,k}f\|_{L^q(M)}/\|f\|_{L^p(\mathbb{R}^n)}\]
are infinitely differentiable, all derivatives are in $L^p$ and exhibit some additional decay measured in a weighted $L^p$-space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00783 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smoothness of extremizers for certain inequalities of the Radon transform Flock, Taryn C. Classical Analysis and ODEs 42B20 The Radon transform is a bounded operator from $L^p$ of Euclidean space to $L^q$ of the manifold of all affine hyperplanes in $\mathbb{R}^n$ for certain exponents depending dimension. Extremizers have been determined for certain values of $q$ and $p$, but most remain open. We show that extremizers are infinitely differentiable whenever the exponents in the associated Euler-Lagrange equation, $q-1$ and $\frac1{p-1}$, are integers. The proof adapts the method of Christ and Xue, to the case where the underlying space is a manifold. The proof is carried out in the setting of the $k$-plane transform, which takes functions on $\mathbb{R}^n$ to functions on the manifold of all affine $k$-planes in $\mathbb{R}^n$ by integrating the function over the $k$-dimensional plane. We show that when $q-1$ and $\frac1{p-1}$ are intergers, all nonnegative critical points of the functional \[ \|T_{n,k}f\|_{L^q(M)}/\|f\|_{L^p(\mathbb{R}^n)}\] are infinitely differentiable, all derivatives are in $L^p$ and exhibit some additional decay measured in a weighted $L^p$-space. |
| title | Smoothness of extremizers for certain inequalities of the Radon transform |
| topic | Classical Analysis and ODEs 42B20 |
| url | https://arxiv.org/abs/2508.00783 |