Mixed-norm of orthogonal projections and analytic interpolation on dimensions of measures
Bochen Liu
Southern University of Science and Technology, Shenzhen, China
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Abstract
Suppose that and are compactly supported Radon measures on , is an -dimensional subspace, and let denote the orthogonal projection. In this paper, we study the mixed-norm , where
assuming has continuous density. When and , our result significantly improves a previous result of Orponen on radial projections. We also discuss about consequences including jump discontinuities in the range of , and -planes determined by a set of given Hausdorff dimension. In the proof, we run analytic interpolation not only on and , 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 -amplitude, that is crucial for our interpolation and gives an alternative definition of Hausdorff dimension.
Cite this article
Bochen Liu, Mixed-norm of orthogonal projections and analytic interpolation on dimensions of measures. Rev. Mat. Iberoam. 40 (2024), no. 3, pp. 827–858
DOI 10.4171/RMI/1472