Anisotropic Herz spaces with variable exponents (Q888130)

In Section 2, the author defines the anisotropic Herz spaces with two variables \(\dot{K}_{q(.) }^{\alpha (.) ,p}(A;\mathbb{R}^{n}) \) and \(K_{q(.) }^{\alpha (.) ,p}(A;\mathbb{R}^{n}) \) as follows: Let \(\alpha (.) :\mathbb{R}^{n}\rightarrow\mathbb{R}\) with \(\alpha (.) \in L^{\infty }(\mathbb{R}^{n}) \), \(0<p\leq \infty \), and \(q(.) \in \wp (\mathbb{R}^{n})\). We denote by \(L^{q(.) }(\mathbb{R}^{n}) \) the variable exponent Lebesgue space. The homogeneous anisotropic Herz space \(\dot{K}_{q(.) }^{\alpha (.) ,p}(A;\mathbb{R}^{n}) \) associated with the dilation \(A\) is defined by \[ \dot{K}_{q(.) }^{\alpha (.) ,p}(A;\mathbb{R}^{n}) =\left\{ f\in L_{\mathrm{loc}}^{q(.) }(\mathbb{R}^{n}) :\left\| f\right\| _{\dot{K}_{q(.) }^{\alpha (.) ,p}}<\infty \right\} , \] where \[ \left\| f\right\| _{\dot{K}_{q(.) }^{\alpha (.) ,p}}=\left\{ \sum_{k=-\infty }^{\infty }\left\| b^{k\alpha (.) }f\chi _{k}\right\| _{L^{q(.) }(\mathbb{R}^{n}) }^{p}\right\} ^{\frac{1}{p}}, \] and \(\chi _{k}=\chi _{C_{k}}\) is the characteristic function of \(C_{k}=B_{k}\backslash B_{k-1}\) for \(k\in\mathbb{Z}\). Define \(\widetilde{\chi _{k}}=\chi _{k}\) if \(k\in\mathbb{Z}_{+\text{ }}\) and \(\widetilde{\chi _{0}}=\chi _{B_{0}}.\) The non-homogeneous anisotropic Herz space \(K_{q(.) }^{\alpha (.) ,p}(A;\mathbb{R}^{n}) \) associated with the dilation \(A\) is defined by \[ K_{q(.) }^{\alpha (.) ,p}(A;\mathbb{R}^{n}) =\left\{ f\in L_{\mathrm{loc}}^{q(.) }(\mathbb{R}^{n}) :\left\| f\right\| _{K_{q(.) }^{\alpha (.) ,p}}<\infty \right\} , \] where \[ \left\| f\right\| _{K_{q(.) }^{\alpha (.) ,p}}=\left\{ \sum_{k=0}^{\infty }\left\| b^{k\alpha (.) }f\widetilde{\chi _{k}}\right\| _{L^{q(.) }(\mathbb{R}^{n}) }^{p}\right\} ^{\frac{1}{p}}. \] Here, the usual modifications are made when \(p=\infty\). In this section, the author also gives their block decomposition. In Section 3, the author proves the boundedness of some sublinear operators on \(\dot{K} _{q(.) }^{\alpha (.) ,p}(A;\mathbb{R}^{n}) \) and \(K_{q(.) }^{\alpha (.) ,p}(A;\mathbb{R}^{n}) \).