Endpoint properties of localized Riesz transforms and fractional integrals associated to Schrödinger operators
This is a highly elaborated paper, touching several topics from the theory of singular integrals, with sound applications to harmonic analysis, Riesz transform, Hardy function spaces and related concepts. In order to give a flavor of the material, we shall provide a sample result, considered by the authors, to constitute one of the main results of the paper. First, an admissible function \(\rho(y)\) is defined on \(\mathbb R^n\), and satisfies on this space the inequality \[ \rho(y)\leq\widetilde C[\rho(x)]^{1/(1+k_0)} [\rho(x)+|x-y|]^{k_0/ (1+k_0)}, \] with \(\widetilde C\) and \(k_0\) positive constants. Then a weighted Hardy space \(H^1_\rho(\mathbb R^n)\) is defined, starting with the classical Hardy space \(H^1(\mathbb R^n)\) and the admissible function \(\rho\). Further on, the operators (Riesz transform) \[ \widetilde R_j(f)(x)\equiv\text{p.v.}\,c_n\int_{\mathbb{R}^n}\frac{x_j-y_j}{|x-y|^{n+1}}\eta\left(\frac{|x-y|}{\rho(x)}\right)f(y)\,dy, \] are considered, as well as their adjoint operators \(R^*_j(f)(x)\). These operators are defined on \(\bigcup^\infty_{p=1}L^p(\mathbb R^n)\). The result can be stated as follows: For any admissible function \(\rho\), the following conditions are equivalent {\parindent=7mm \begin{itemize}\item[(i)] \(f \in H^1_\rho(\mathbb{R}^n)\); \item[(ii)] \(f\in L^1(\mathbb{R}^n)\) and \(\widetilde R_j(f)\in L^1(\mathbb{R}^n)\) for all \(j \in\{1,\dots,n\}\); \item[(iii)] \(f\in L^1(\mathbb{R}^n)\) and \(\widetilde R^*_j (f)\in L^1(\mathbb{R}^n)\) for all \(j\in\{1,\dots,n\}\). \end{itemize}} Moreover, for all \(f\in H^1_\rho(\mathbb{R}^n)\), \[ \|f\|_{H^1_\rho(\mathbb{R}^n)}\sim\|f\|_{L^1 (\mathbb{R}^n)}+\sum^n_{j=1}\|\widetilde R_j(f)\|_{L^1(\mathbb{R}^n)}\sim \|f\|_{L^1 (\mathbb{R}^n)}+\sum^n_{j=1}\|\widetilde R^*_j(f)\|_{L^1 (\mathbb{R}^n)}. \] The meaning of \(f\sim g\) is: \(f\sim g\) iff \(C_1f\leq g\leq C_2f\), with \(C_1,C_2\) positive constants. Several of the paper's results are expressing the continuity (boundedness) of various operators between function spaces, the admissible function being always present.
scientific article; zbMATH DE number 5566067
scientific article