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 R kπyμkLqp.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 language | English |
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Pages (from-to) | 827-858 |
Number of pages | 32 |
Journal | Revista Matematica Iberoamericana |
Volume | 40 |
Issue number | 3 |
DOIs | |
State | Published - 2024 |
Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2024 Real Sociedad Matemática Española.
Keywords
- analytic interpolation
- orthogonal projection
- radial projection