The decomposition of weighted Herz space on \(\mathbb{R}^ n\) and its applications
Let \(0< \alpha, p< \infty\), \(1\leq q< \infty\), \(B_k= \{|x|\leq 2^k\}\), \(c_k= B_k\backslash B_{k- 1}\). Define the weighted (\(w_1\), \(w_2\) two weights) Herz spaces \(\dot K^{\alpha, p}_q(w_1, w_2)\) by \[ \dot K^{\alpha, p}_q(w_1, w_2)= \{f\in L^q_{\text{loc}}(\mathbb{R}^n- \{0\}, w_2): |f|_{\dot K^{\alpha, p}_q(w_1, w_2)}< \infty\}, \] \[ |f|_{\dot K^{\alpha, p}_q (w_1, w_2)}= \Biggl( \sum_k w_1(B_k)^{\alpha p/n} |f\chi_{c_k}|^p_{q(w_2)}\Biggr)^{1/p}. \] Define the central \((\alpha, q, w_1, w_2)\)- unit as follows. \(b(x)\) is called a \(c\)-\((\alpha, q, w_1, w_2)\)- unit, if \(\text{supp } b\subset B(0, r)\), \(|b|_{q(w_2)}\leq w_1(B(0, r))^{-\alpha/n}\). Then the main result of the paper can be formulated as: Let \(\alpha\), \(p\), \(q\) be as above, \(w_1 \in A_1\), then \(f\in \dot K^{\alpha, p}_q(w_1,w_2)\), if and only if \(f= \sum \lambda_k b_k(x)\) with \(b_k\) central \((\alpha, q, w_1, w_2)\)-units and \(\sum|\lambda_k|^p< \infty\). The same result holds for the non-homogeneous version \(K^{\alpha, p}_q(w_1, w_2)\). As applications, the paper obtains the boundedness of some sublinear operator on \(\dot K^{\alpha, p}_q\) and \(K^{\alpha, p}_q\). The last result of the paper is identifying weighted Herz spaces to weighted Hardy-Herz spaces for some indices and the special weight pair \(w_1\equiv 1\), \(w_2(x)= |x|^{- a}\), \(0\leq a< n\).
scientific article; zbMATH DE number 751241