2015/09/12 by Kairema, Anna, Li, Ji, Pereyra, M. Cristina +1 · 2 citations
#42 #43 #46 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1509.03761
We give an explicit construction of Haar functions associated to a system of dyadic cubes in a geometrically doubling quasi-metric space equipped with a positive Borel measure, and show that these Haar functions form a basis for Lp. Next we focus on spaces X of homogeneous type in the sense of Coifman and Weiss, where we use these Haar functions to define a discrete square function, and hence to define dyadic versions of the function spaces H1(X) and \rm BMO(X). In the setting of product spaces \widetildeX = X1 × ⋯ × Xn of homogeneous type, we show that the space \rm BMO(\widetildeX) of functions of bounded mean oscillation on \widetildeX can be written as the intersection of finitely many dyadic \rm BMO spaces on \widetildeX, and similarly for Ap(\widetildeX), reverse-Hölder weights on \widetildeX, and doubling weights on \widetildeX. We also establish that the Hardy space H1(\widetildeX) is a sum of finitely many dyadic Hardy spaces on \widetildeX, and that the strong maximal function on \widetildeX is pointwise comparable to the sum of finitely many dyadic strong maximal functions. These dyadic structure theorems generalize, to product spaces of homogeneous type, the earlier Euclidean analogues for \rm BMO and H1 due to Mei and to Li, Pipher and Ward.