# Products of Functions in $$\mathrm {BMO}({\mathcal X})$$ BMO ( X ) and $$H^1_\mathrm{at}({\mathcal X})$$ H at 1 ( X ) via Wavelets Over Spaces of Homogeneous Type

Products of Functions in $$\mathrm {BMO}({\mathcal X})$$ BMO ( X ) and... Let $$({\mathcal X},d,\mu )$$ ( X , d , μ ) be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $$H^1_\mathrm{at}({\mathcal X})$$ H at 1 ( 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 $$f\times g$$ f × g of $$f\in H^1_\mathrm{at}({\mathcal X})$$ f ∈ H at 1 ( X ) and $$g\in \mathrm {BMO}({\mathcal X})$$ g ∈ BMO ( X ) , viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from $$H^1_\mathrm{at}({\mathcal X})\times \mathrm {BMO}({\mathcal X})$$ H at 1 ( X ) × BMO ( X ) into $$L^1({\mathcal X})$$ L 1 ( X ) and from $$H^1_\mathrm{at}({\mathcal X}) \times \mathrm {BMO}({\mathcal X})$$ H at 1 ( X ) × BMO ( X ) into $$H^{\log }({\mathcal X})$$ H log ( X ) , which affirmatively confirms the conjecture suggested by A. Bonami and F. Bernicot (This conjecture was presented by Ky in J Math Anal Appl 425:807–817, 2015). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Fourier Analysis and Applications Springer Journals

# Products of Functions in $$\mathrm {BMO}({\mathcal X})$$ BMO ( X ) and $$H^1_\mathrm{at}({\mathcal X})$$ H at 1 ( X ) via Wavelets Over Spaces of Homogeneous Type

, Volume 23 (4) – Jun 27, 2016
72 pages

/lp/springer_journal/products-of-functions-in-mathrm-bmo-mathcal-x-bmo-x-and-h-1-mathrm-at-m0KmIuRp9U
Publisher
Springer US
Subject
Mathematics; Fourier Analysis; Signal,Image and Speech Processing; Abstract Harmonic Analysis; Approximations and Expansions; Partial Differential Equations; Mathematical Methods in Physics
ISSN
1069-5869
eISSN
1531-5851
D.O.I.
10.1007/s00041-016-9483-9
Publisher site
See Article on Publisher Site

### Abstract

Let $$({\mathcal X},d,\mu )$$ ( X , d , μ ) be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $$H^1_\mathrm{at}({\mathcal X})$$ H at 1 ( 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 $$f\times g$$ f × g of $$f\in H^1_\mathrm{at}({\mathcal X})$$ f ∈ H at 1 ( X ) and $$g\in \mathrm {BMO}({\mathcal X})$$ g ∈ BMO ( X ) , viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from $$H^1_\mathrm{at}({\mathcal X})\times \mathrm {BMO}({\mathcal X})$$ H at 1 ( X ) × BMO ( X ) into $$L^1({\mathcal X})$$ L 1 ( X ) and from $$H^1_\mathrm{at}({\mathcal X}) \times \mathrm {BMO}({\mathcal X})$$ H at 1 ( X ) × BMO ( X ) into $$H^{\log }({\mathcal X})$$ H log ( X ) , which affirmatively confirms the conjecture suggested by A. Bonami and F. Bernicot (This conjecture was presented by Ky in J Math Anal Appl 425:807–817, 2015).

### Journal

Journal of Fourier Analysis and ApplicationsSpringer Journals

Published: Jun 27, 2016

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