TY - JOUR
T1 - A Note on $H^p_w$-Boundedness of Riesz Transforms and $\theta$-Calderόn-Zygmund Operators Through Molecular Characterization
AU - Luong Dang Ky
JO - Analysis in Theory and Applications
VL - 3
SP - 251
EP - 264
PY - 2011
DA - 2011/08
SN - 27
DO - http://doi.org/10.1007/s10496-011-0251-z
UR - https://global-sci.org/intro/article_detail/ata/4598.html
KW - Muckenhoupt weight, Riesz transform, Calderón-Zygmund operator.
AB - <p style="text-align: justify;">Let $0 &lt;p \leq 1$ and $w$ in the Muckenhoupt class $A_1$. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and Yang<sup>[11]</sup> established that the Riesz transforms $R_j$, $j = 1,2, \cdots ,n$, are bounded on $H^p_w(\mathbf{R}^n)$. 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 [11], consists in passing from atoms to all functions in $H^p_w(\mathbf{R}^n)$. Furthermore, the $H^p_w$-boundedness of $\theta$-Calderón-Zygmund operators are also given through molecular characterization and atomic decomposition.</p>