A note on $H^p_w$-boundedness of Riesz transforms and $θ$-Calderón-Zygmund operators through molecular characterization

Details A note on $H^p_w$-boundedness of Riesz transforms and $θ$-Calderón-Zygmund operators through molecular characterization A note on $H^p_w$-boundedness of Riesz transforms and $θ$-Calderón-Zygmund operators through molecular characterization

2011-06-23

arxiv.org/abs/1106.4724v3

Mathematics

Classical Analysis and ODEs

to appear in Anal. Theory. Appl. 27 (2011), no. 3, 251-264

Scientific paper

Let $0 < p \leq 1$ and $w$ in the Muckenhoupt class $A_1$. Recently, by using the weighted atomic decomposition and molecular characterization; Lee, Lin and Yang \cite{LLY} (J. Math. Anal. Appl. 301 (2005), 394--400) established that the Riesz transforms $R_j, j=1, 2,...,n$, are bounded on $H^p_w(\mathbb R^n)$. In this note we extend this to the general case of weight $w$ in the Muckenhoupt class $A_\infty$ through molecular characterization. One difficulty, which has not been taken care in \cite{LLY}, consists in passing from atoms to all functions in $H^p_w(\mathbb R^n)$. Furthermore, the $H^p_w$-boundedness of $\theta$-Calder\'on-Zygmund operators are also given through molecular characterization and atomic decomposition.

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