Mixed-norm of orthogonal projections and analytic interpolation on dimensions of measures

Abstract

Suppose that $\mu$ and $\nu$ are compactly supported Radon measures on ${\mathbb{R}}^d$, $V\in G(d,n)$ is an $n$-dimensional subspace, and let ${\pi }_V:{\mathbb{R}}^d\to V$ denote the orthogonal projection. In this paper, we study the mixed-norm $\int \Vert {\pi }^y\mu {\Vert }_{L^p(G(d,n))}^{q}d\nu (y)$, where

assuming $\mu$ has continuous density. When $n=d-1$ and $p=q$, our result significantly improves a previous result of Orponen on radial projections. We also discuss about consequences including jump discontinuities in the range of $p$, and $m$-planes determined by a set of given Hausdorff dimension. In the proof, we run analytic interpolation not only on $p$ and $q$, but also on dimensions of measures. This is partially inspired by previous work of Greenleaf and Iosevich on Falconer-type problems. We also introduce a new quantity called $s$-amplitude, that is crucial for our interpolation and gives an alternative definition of Hausdorff dimension.