Weighted estimates for the \(k\)-plane transform of radial functions on Euclidean spaces
The \(k\)-plane transform is an integral operator which integrates functions over lower-dimensional subspaces. For \(k\geq1\) and \(n\geq2,\) let \(G_{n,k}\) denote the Grassmannian manifold of \(k\) dimensional linear subspaces in \(\mathbb{R}^{n}\) and let \(\gamma_{n,k}\) denote the unique invariant, nonnegative Borel measure on \(G_{n,k}\) such that \(\gamma_{n,k}(G_{n,k})=1\). \ For \(\Pi\in G_{n,k}\) and \(x\in\mathbb{R}^{n}\), the \(k\)-plane transform \(T_{k}\) of a function \(f\) defined on \(\mathbb{R}^{n}\) is defined as \[ T_{k}f\left( x,\Pi\right) =\int_{\Pi}f(x-y)d\lambda_{k}y, \tag{1} \] where \(\lambda_{k}\) denotes the \(k\)-dimensional Lebesgue measure on \(\Pi\). For \(x\in\mathbb{R}^{n}\), let \(x_{\Pi}\) (respectively \(x_{\Pi^{\bot}})\) denote the projection of \(x\) on \(\Pi\) (respectively on \(\Pi^{\bot})\) and hence \(x=x_{\Pi}+x_{\Pi^{\bot}.}\) Using the translation invariance of the Lebesgue measure on \(\mathbb{R}^{k}\), it follows from (1) that \[ T_{k}f(x,\Pi)=\int_{\Pi}f(x_{\Pi^{\bot}}-y)d\lambda_{k}y. \] The paper discusses the \(L^p\rightarrow L^q\) mapping properties of the \(k\)-plane transform acting on radial functions in certain weighted \(L^p\) spaces. To this end, the authors obtain certain inequalities, the most important among them are \[ \left( \int_{G_{n,k}}\left( \int_{\Pi^{\bot}}\left| T_{k}f\left( x,\Pi\right) \right| ^{q}\left\| x_{\Pi^{\bot}}\right\| ^{\beta }d\lambda_{n-k}(x)\right) ^{\frac{r}{q}}d\gamma_{n,k}\left( \Pi\right) \right) ^{\frac{1}{r}}\leq C\left\| f\right\| _{p,\alpha,n}, \] and \[ \left( \int_{\mathbb{R}^{n}}\left( \int_{G_{n,k}}\left| T_{k}f\left( x,\Pi\right) \right| ^{r}d\gamma_{n,k}\left( \Pi\right) \right) ^{\frac{q}{r}}\left\| x\right\| ^{\beta}dx\right) ^{\frac{1}{q}}\leq C\left\| f\right\| _{p,\alpha,n}, \] where \(L_{\alpha}^{p}(\mathbb{R}^{n})\) is the space of measurable functions defined on \(\mathbb{R}^{n}\) such that \[ \left\| f\right\| _{p,\alpha,n}=\left( \int_{\mathbb{R}^{n}}\left| f(x)\right| ^{p}\left\| x\right\| ^{\alpha}dx\right) ^{\frac{1}{p} }<\infty, \] and \(\alpha\), \(\beta\), \(r\), \(p\), \(q\) are appropriate constants.
scientific article; zbMATH DE number 6105440
scientific article