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

Bochen Liu

Research output: Contribution to journalArticlepeer-review

Abstract

Suppose that μ and ν are compactly supported Radon measures on Rd, V 2G.d; n/ is an n-dimensional subspace, and let πV W Rd ! V denote the orthogonal projection. In this paper, we study the mixed-norm RyμkLqp.G.d;n// dν.y/, where (FIGURE PRESENTED) assuming μ has continuous density. When n D d - 1 and p D 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.

Original languageEnglish
Pages (from-to)827-858
Number of pages32
JournalRevista Matematica Iberoamericana
Volume40
Issue number3
DOIs
StatePublished - 2024
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2024 Real Sociedad Matemática Española.

Keywords

  • analytic interpolation
  • orthogonal projection
  • radial projection

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