2004/04/02 by Grigor Karagulyan, Michael T. Lacey, Karagulyan, Grigor +2
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations #math.CA
paper · pdf · doi:10.48550/arxiv.math/0404028
12 pages
arxiv created 2004/04/02 · openalex publication_date 2004/04/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In Rd, define a maximal function in the directions v∈ \directions⊂\x | \abs x=1\ by M^\directions f(x)=supv∈\directions sup\zve ∫-\ze^\ze \absf(x-vy) dy. For a function f on \ZRd, let S_\zw f denote the Fourier restriction of f to a region \zw. We are especially interested taking \zw to be a sector of Rd with base points at the origin. A sector is a product of the interval (0,∞) with respect to a choice of (non orthogonal) basis. What is most important is that the basis is a subset of \directions. Consider a collection \zW of pairwise disjoint sectors \zw as above. Assume that M^\directions maps Lp into Lp, for some 1