English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We give an explicit construction of Haar functions associated to a system ofdyadic cubes in a geometrically doubling quasi-metric space equipped with apositive Borel measure, and show that these Haar functions form a basis for$L^p$. Next we focus on spaces $X$ of homogeneous type in the sense of Coifmanand Weiss, where we use these Haar functions to define a discrete squarefunction, and hence to define dyadic versions of the function spaces $H^1(X)$and ${\rm BMO}(X)$. In the setting of product spaces $\widetilde{X} = X_1\times \cdots \times X_n$ of homogeneous type, we show that the space ${\rmBMO}(\widetilde{X})$ of functions of bounded mean oscillation on$\widetilde{X}$ can be written as the intersection of finitely many dyadic${\rm BMO}$ spaces on $\widetilde{X}$, and similarly for $A_p(\widetilde{X})$,reverse-H\"older weights on $\widetilde{X}$, and doubling weights on$\widetilde{X}$. We also establish that the Hardy space $H^1(\widetilde{X})$ isa sum of finitely many dyadic Hardy spaces on $\widetilde{X}$, and that thestrong maximal function on $\widetilde{X}$ is pointwise comparable to the sumof finitely many dyadic strong maximal functions. These dyadic structuretheorems generalize, to product spaces of homogeneous type, the earlierEuclidean analogues for ${\rm BMO}$ and $H^1$ due to Mei and to Li, Pipher andWard.

Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type