Let $$(\mathcal {X},d,\mu )$$ ( X , d , μ ) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let T be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and $$b\in \widetilde{\mathrm{RBMO}}(\mu )$$ b ∈ RBMO ~ ( μ ) (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator $$T_b:=bT-Tb$$ T b : = b T - T b generated by T and b from the atomic Hardy space $$\widetilde{H}^1(\mu )$$ H ~ 1 ( μ ) with the discrete coefficient into the weak Lebesgue space $$L^{1,\,\infty }(\mu )$$ L 1 , ∞ ( μ ) . From this and an interpolation theorem for sublinear operators which is also proved in this paper, the authors further show that the commutator $$T_b$$ T b is bounded on $$L^p(\mu )$$ L p ( μ ) for all $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) . Moreover, the boundedness of the commutator generated by the generalized fractional integral $$T_\alpha \,(\alpha \in (0,1))$$ T α ( α ∈ ( 0 , 1 ) ) and the $$\widetilde{\mathrm{RBMO}}(\mu )$$ RBMO ~ ( μ ) function from $$\widetilde{H}^1(\mu )$$ H ~ 1 ( μ ) into $$L^{1/{(1-\alpha )},\,\infty }(\mu )$$ L 1 / ( 1 - α ) , ∞ ( μ ) is also presented.
Analysis and Mathematical Physics – Springer Journals
Published: Jun 1, 2016
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.