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}\colon \mathbb{R}^{d}\rightarrow V denote the orthogonal projection. In this paper, we study the mixed-norm \int \|\pi^{y}\mu\|_{L^p(G(d,n))}^{q} d\nu(y) , where \pi^{y}\mu(V):=\int_{y+V^\perp}\mu \, d\mathcal{H}^{d-n}=\pi_{V} \mu(\pi_{V}y), 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.