Products of Functions in \mathopBMO(\mathcal X) and H¹_{\rm at}(\mathcal X) via Wavelets over Spaces of Homogeneous Type

2015/06/19 by Fu, Xing, Yang, Dachun, Liang, Yiyu
#30L99 #42C40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 42B30 #Secondary 47A07

paper · doi:10.48550/arxiv.1506.05910

Abstract

Let (\mathcal X,d,μ) be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and H¹_{\rm at}(\mathcal X) be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytönen, the authors prove that the product f× g of f∈ H¹_{\rm at}(\mathcal X) and g∈\mathopBMO(\mathcal X), viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from H¹_{\rm at}(\mathcal X)×\mathopBMO(\mathcal X) into L¹(\mathcal X) and from H¹_{\rm at}(\mathcal X)×\mathopBMO(\mathcal X) into H^log(\mathcal X), which affirmatively confirms the conjecture suggested by A. Bonami and F. Bernicot (This conjecture was presented by L. D. Ky in [J. Math. Anal. Appl. 425 (2015), 807-817]).

Related

• Paraproducts and Products of functions in BMO(\mathbb Rⁿ) and H¹(\mathbb Rⁿ) through wavelets
• Wavelet Characterizations of the Atomic Hardy Space H¹ on Spaces of Homogeneous Type
• Atomic decomposition of product Hardy spaces via wavelet bases on spaces of homogeneous type
• Product of functions in \BMO and \H¹ in non-homogeneous spaces
• Bilinear Decompositions of Products of Hardy and Lipschitz Spaces Through Wavelets