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

Suppose that μ and ν are compactly supported Radon measures on Rᵈ , V∈ G(d,n) is an n -dimensional subspace, and let πV Rᵈ→ V denote the orthogonal projection. In this paper, we study the mixed-norm ∫ \|πʸμ\|L^p(G(d,n))q dν(y) , where πʸμ(V):=∫y+V^⊥μ \, dHᵈ⁻ⁿ=πV μ(πVy), assuming μ 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.