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

Suppose $\mu, \nu$ are compactly supported Radon measures on $\mathbb{R}^d$ and $V\in G(d,n)$ is an $n$-dimensional subspace. In this paper we systematically study the mixed-norm $$\int\|\pi^y\mu\|_{L^p(G(d,n))}^q\,d\nu(y),\ \forall\,p,q\in[1,\infty),$$ where $\pi_V:\mathbb{R}^d\rightarrow V$ denotes the orthogonal projection and $$\pi^y\mu(V)=\int_{y+V^\perp}\mu\,d\mathcal{H}^{d-n}=\pi_V\mu(\pi_Vy),\ \text{if $\mu$ has continuous density}.$$ When $n=d-1$ and $p=q$, our result significantly improves a previous result of Orponen. In the proof we consider integer exponents first, then interpolate analytically, not only on $p,q$, but also on dimensions of measures. We also introduce a new quantity called $s$-amplitude, to present our results and illustrate our ideas. This mechanism provides new perspectives on operators with measures, thus has its own interest. We also give an alternative proof of a recent result of D\k{a}browski, Orponen, Villa on $\|\pi_V\mu\|_{L^p(\mathcal{H}^n\times G(d,n))}$. The following consequences are also interesting. $\bullet$ We discover jump discontinuities in the range of $p$ at the critical line segment $$\{(s_\mu, s_\nu)\in(0,d)^2: s_\mu+s_\nu=2n,\, 0<s_\nu<n\},$$ $\ \$ where $s_\mu, s_\nu$ are Frostman exponents of $\mu, \nu$ respectiely. This is unexpected and surprising. $\bullet$ Given $1\leq m\leq d-1$ and $E, F\subset\mathbb{R}^d$, we obtain dimensional threshold on whether there exists $y\in F$ such that $$\gamma_{d,m}\{V\in G(d,m): V=\operatorname{Span}\{x_1-y,\dots,x_m-y\}: x_1,\dots,x_m\in E\}>0.$$ $\ \$ This generalizes the visibility problem ($m=1$). In particular, when $m>\frac{d}{2}$ and $\dim_{\mathcal{H}} E$ is large enough, the exceptional set has Hausdorff dimension $0$.