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

• Bochen Liu ^[1]
  1. [1] Southern University of Science and Technology, Shenzhen, China
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 40, Nº 3, 2024, págs. 827-858
• Idioma: inglés
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• Resumen

  • 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∥Lp(G(d,n))qdν(y), where πyμ(V):=∫y+V⊥μdHd−n=πVμ(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.