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 of Fourier Analysis and Applications – Springer Journals
Published: Jun 27, 2016
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