Hardy spaces related to Schrödinger operators with potentials which are sums of \(L^p\)-functions
The authors consider a Schrödinger operator on \(\mathbb{R}^n\) given by \(Lf(x):=-\Delta f(x)+V(x)f(x)\), where \(x\in\mathbb{R}^n\) and \(\Delta\) denotes the Laplace operator. Let \(d\in\mathbb{N}\). The authors assume that the potential \(V\) satisfies:NEWLINENEWLINE(A1) for any \(j\in\{1,\dots,d\}\), there exists \(V_j\geq0\), \(V_j\not\equiv 0\), such that, for all \(x\in\mathbb{R}^n\), \(V(x)=\sum_{j=1}^d V_j(x)\),NEWLINENEWLINE(A2) for every \(j\in\{1,\dots,d\}\), there exists a linear subspace \(\mathbb{V}_j\) of \(\mathbb{R}^n\) of dimension \(n_j\geq 3\) such that, if \(\Pi_{\mathbb{V}_j}\) denotes the orthogonal projection on \(\mathbb{V}_j\), then, for all \(x\in\mathbb{R}^n\), \(V_j(x)=V_j(\Pi_{\mathbb{V}_j}x)\),NEWLINENEWLINE(A3) there exists \(\kappa\in(0,\infty)\) such that, for any \(j\in\{1,\dots,d\}\) and all \(r\) satisfying \(|r-n_j/2|\leq \kappa\), \(V_j\in L^r(\mathbb{V}_j)\).NEWLINENEWLINELet \(K_t:=\exp(-tL)\) and \(P_t:=\exp(t\Delta)\), with \(t\in(0,\infty)\), be the semigroups of linear operators associated with \(L\) and \(\Delta\), respectively. Let \(K_t(x,y)\) and \(P_t(x-y)\) be the integral kernels of these semigroups. Let \(M_L\) and \(M_{\Delta}\) be the associated maximal operators, i.\,e., for all \(x\in\mathbb{R}^n\), \(M_Lf(x):=\sup_{r\in(0,\infty)}|K_tf(x)|\) and \(M_{\Delta}f(x):=\sup_{t\in(0,\infty)}|P_tf(x)|\). The Hardy spaces \(H^1_L(\mathbb{R}^n)\) and \(H^1_{\Delta}(\mathbb{R}^n)\) are the subspaces of \(L^1(\mathbb{R}^n)\) defined by NEWLINE\[NEWLINEf\in H^1_L(\mathbb{R}^n)\Longleftrightarrow M_Lf\in L^1(\mathbb{R}^n),\;\;f\in H^1_{\Delta}(\mathbb{R}^n)\Longleftrightarrow M_{\Delta}f\in L^1(\mathbb{R}^n)NEWLINE\]NEWLINE with the norms \(\|f\|_{H^1_L(\mathbb{R}^n)}:=\|M_L f\|_{L^1(\mathbb{R}^n)}\) and \(\|f\|_{H^1_{\Delta}(\mathbb{R}^n)}:=\|M_{\Delta}f\|_{L^1(\mathbb{R}^n)}\). Denote by \(L^{-1}\) the operator with kernel \(\Gamma(x,y)=\int_0^{\infty}K_t(x,y)\,dt\).NEWLINENEWLINEAssume that \(f\in L^1(\mathbb{R}^n)\). The authors first prove that \(f\in H^1_L(\mathbb{R}^n)\) if and only if \((1-VL^{-1})f\in H^1_{\Delta}(\mathbb{R}^n)\). Moreover, \(\|f\|_{H^1_L(\mathbb{R}^n)}\sim\|(I-VL^{-1})f\|_{H^1_{\Delta}(\mathbb{R}^n)}\).NEWLINENEWLINEThe authors further define the weight function \(\omega\) by NEWLINE\[NEWLINE\omega(x):=\lim_{t\to\infty}\int_{\mathbb{R}^n}K_t(x,y)\,dy.NEWLINE\]NEWLINE Let \(f\in L^1(\mathbb{R}^n)\). The authors then obtain \(f\in H^1_L(\mathbb{R}^n)\) if and only if \(\omega f\in H^1_{\Delta}(\mathbb{R}^n)\). Additionally, \(\|f\|_{H^1_L(\mathbb{R}^n)}\sim\|\omega f\|_{H^1_{\Delta}(\mathbb{R}^n)}\). As a corollary of this result, the authors give an atomic characterization of the elements of \(H^1_L(\mathbb{R}^n)\).NEWLINENEWLINEFor every \(i\in\{1,\dots,n\}\), the authors introduce the Riesz transform \(\mathcal{R}_{L,\,i}\) associated with \(L\) by setting \(\mathcal{R}_{L,\,i}f:=\lim_{\varepsilon\to\infty} \int_{\varepsilon}^{\varepsilon^{-1}}\partial_iK_t f\,\frac{dt}{\sqrt{t}}\). Finally, the authors show that an \(L^1(\mathbb{R}^n)\)-function \(f\in H^1_{L}(\mathbb{R}^n)\) if and only if \(\mathcal{R}_{L,\,i}f\in L^1(\mathbb{R}^n)\) for all \(i\in\{1,\dots,n\}\). Furthermore, \(\|f\|_{H^1_L(\mathbb{R}^n)}\sim\|f\|_{L^1(\mathbb{R}^n)} +\sum_{i=1}^n\|\mathcal{R}_{L,\,i}f\|_{L^1(\mathbb{R}^n)}\).
scientific article; zbMATH DE number 6086919
scientific article