Hardy-Littlewood-Fefferman-Stein type inequalities, 3

Abstract

(You can skip this section if you are not interested in Section 10.) Theorem 8.1.

$$ Let{\rm{ }}f{\rm{ }} \in {\rm{ }}\bigcup\limits_{p \in \left[ {1, + \infty } \right]} {L^p } , $$

$$ 0 < b,{\rm{ 0 < }}\varepsilon {\rm{ < }}m, $$

(8.1)

$$ q \in \left( {0,1} \right], $$

(8.2)

$$ \left\{ {\varphi _1 , \cdots ,\varphi _J } \right\} \subset \Lambda _{b + \varepsilon } ,\sum\limits_{i = 1}^J {\left\| {\varphi _i } \right\|} _{B,b + \varepsilon ,m} < \infty $$

(8.3)

$$ F\varphi _i \in C^{\left[ {n + b + m + 2} \right]} \left( {{\rm{R}}^n \backslash \left\{ 0 \right\}} \right),{\rm{ }}\left( {i = 1, \cdots ,J} \right), $$

(8.4)

$$ \sup \left\{ {\sum\limits_{i = 1}^J {\left| {F\varphi _i \left( {t\xi } \right)} \right|:t > 0} } \right\} > 1{\rm{ }}for{\rm{ }}any{\rm{ }}\xi \in {\rm{R}}^n \backslash \left\{ 0 \right\}, $$

(8.5)

$$ \psi \in \Lambda _b ,\left\| \psi \right\|_{B,b,m} < + \infty , $$

(8.6)

$$ \int {\psi \left( x \right)} x^\alpha dx = 0{\rm{ }}if{\rm{ }}\left| \alpha \right| < m. $$

(8.7)