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

Suppose $μ, ν$ 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 $\int\|π^yμ\|_{L^p(G(d,n))}^q\,dν(y),\ \forall\,p,q\in[1,\infty),$ where $π_V:\mathbb{R}^d\rightarrow V$ denotes the orthogonal projection and $π^yμ(V)=\int_{y+V^\perp}μ\,d\mathcal{H}^{d-n}=π_Vμ(π_Vy),\ \text{if $μ$ 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 Dąbrowski, Orponen, Villa on $\|π_Vμ\|_{L^p(\mathcal{H}^n\times G(d,n))}$. The following consequences are also interesting. $\bullet$ We discover jump discontinuities in the range of $p$ at the critical line segment $\{(s_μ, s_ν)\in(0,d)^2: s_μ+s_ν=2n,\, 0<s_ν<n\},$ $\ \$ where $s_μ, s_ν$ are Frostman exponents of $μ, ν$ 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 $γ_{d,m}\{V\in G(d,m): V=\operatorname{Span}\{x_1-y,\dots,x_m-y\}: x_1,\dots,x_m\in E\}>0.$ $\ \$ This generalizes the visibility problem ($m=1$). In particular, when $m>\frac{d}{2}$ and $\dim_{\mathcal{H}} E$ is large enough, the exceptional set has Hausdorff dimension $0$.

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