Linear operators on weighted Herz-type Hardy spaces (Q1919018)

Let \(B_k=\{x\in \mathbb R^n; | x| \leq 2^k\}\) and \(C_k=B_k \setminus B_{k-1}\) for \(k\in \mathbb Z\). Let \(\chi_k\) denote the characteristic function of the set \(C_k\). Suppose \(0<\alpha<\infty\), \(0<p<\infty\) and \(1\leq q<\infty\). Let \(\omega_1\) and \(\omega_2\) be non-negative weight functions. The non-homogeneous Herz space \(\dot K_q^{\alpha, p}(\omega_1, \omega_2)\) is defined in terms of \(\| f\| _{\dot K_q^{\alpha, p}(\omega_1, \omega_2)}\) \(=\bigl\{\sum_{k\in \mathbb Z} [\omega_1(B_k)]^{\alpha p/n}\| f\chi_{k}\| _{L^q(\omega_2)}^p\bigr\}^{1/p}\) by letting \(\dot K_q^{\alpha, p}(\omega_1, \omega_2)=\{f\in L_{\roman{loc}}^q( \mathbb R^n\setminus \{0\}, \omega_2); \| f\| _{\dot K_q^{\alpha, p}(\omega_1, \omega_2)}<\infty\}\). The homogeneous Herz-type Hardy space \(H\dot K_q^{\alpha, p}(\omega_1, \omega_2)\) is the set \(\{f\in \mathcal S' ; Gf\in \dot K_q^{\alpha, p}(\omega_1, \omega_2) \}\), where \(Gf\) is the grand maximal function of \(f\) used in the theory of real Hardy space. The authors give the following: Suppose the linear operator \(T\) satisfies \(| Tf(x)| \leq C (\int | f(y)| ^{p_0}| x-y| ^{-n} dy)^{1/p_0}\) \((x\notin \text{supp} f)\) for some \(p_0>0\), \(\omega_1, \omega _2\in A_1\) (Muckenhoupt weight class), \(\sup_{2^{k-2}\leq | x| <2^{k+1}}\omega _2(x)\leq\) \(C \inf_{2^{k-2}\leq | x| <2^{k+1}}\omega _2(x)\). Then if \(T\) is bounded on \(L^q(\mathbb R^n)\) and \(H\dot K_q^{\alpha, p}(\mathbb R^n)\), it follows the boundedness on \(H\dot K_q^{\alpha, p}(\omega_1, \omega_2)\), provided \(0<p\leq 1\), \(1<q<\infty\), and \(\alpha \geq n(1-1/q)\). The proof is based on the characterization of Herz-type Hardy space in terms of central atoms. This result is an extension to Hardy space case of a result of Stein on weighted boundedness of singular integral operators for power weights [\textit{E. M. Stein}, Proc. Am. Math. Soc. 8, 250-254 (1957; Zbl 0077.27301)] and its extension by \textit{F. Soria} and \textit{G. Weiss} [Indiana Univ. Math. J. 43, No. 1, 187-204 (1994; Zbl 0803.42004)].