2018/10/09 by Yongsheng Han, Han, Yongsheng, Ji Li +5
Mathematics · #42B35 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.1810.03788
openalex publication_date 2018/10/09 · openalex created_date 2018/10/26 · openalex updated_date 2026/07/28
We provide an atomic decomposition of the product Hardy spaces Hp(\widetildeX) which were recently developed by Han, Li, and Ward in the setting of product spaces of homogeneous type \widetildeX = X1 × X2. Here each factor (Xi,di,μi), for i = 1, 2, is a space of homogeneous type in the sense of Coifman and Weiss. These Hardy spaces make use of the orthogonal wavelet bases of Auscher and Hytönen and their underlying reference dyadic grids. However, no additional assumptions on the quasi-metric or on the doubling measure for each factor space are made. To carry out this program, we introduce product (p,q)-atoms on \widetildeX and product atomic Hardy spaces Hp,q\rm at(\widetildeX). As consequences of the atomic decomposition of Hp(\widetildeX), we show that for all q > 1 the product atomic Hardy spaces coincide with the product Hardy spaces, and we show that the product Hardy spaces are independent of the particular choices of both the wavelet bases and the reference dyadic grids. Likewise, the product Carleson measure spaces \rm CMOp(\widetildeX), the bounded mean oscillation space \rm BMO(\widetildeX), and the vanishing mean oscillation space \rm VMO(\widetildeX), as defined by Han, Li, and Ward, are also independent of the particular choices of both wavelets and reference dyadic grids.