Liu, Heping; Tang, Lin; Zhu, Hua Weighted Hardy spaces and BMO spaces associated with Schrödinger operators. (English) Zbl 1266.42060 Math. Nachr. 285, No. 17-18, 2173-2207 (2012). Let \(\mathcal{L}:=-\Delta+V\) be a Schrödinger operator on \(\mathbb{R}^d\), \(d\geq3\), where \(V\not\equiv0\) is a fixed non-negative potential and belongs to the reverse Hölder class \(RH_s(\mathbb{R}^d)\) for some \(s\geq\frac{d}{2}\); that is, there exists a positive constant \(C:=C(s,V)\) such that \((\frac{1}{|B|}\int_B[V(x)]^s\,dx)^{\frac{1}{s}}\leq \frac{C}{|B|}\int_BV(x)\,dx\) for every ball \(B\subset \mathbb{R}^d\). Moreover, the measure \(V(x)\,dx\) satisfies the doubling condition: there exists a positive constant \(C\) such that \(\int_{B(y,2r)}V(x)dx\leq C\int_{B(y,r)}V(x)dx\) for all \(y\in\mathbb{R}^d\) and \(r>0\). Let \(\{T_t\}_{t>0}\) be the semigroup of linear operators generated by \(\mathcal{L}\) and \(T_t(x,y)\) be their kernels, that is, for all \(x\in{\mathbb R}^d\), \[ T_tf(x):=e^{-t\mathcal{L}}f(x):= \int_{\mathbb{R}^d}T_t(x,y)f(y)\,dy\;\;\text{for\;all}\;t>0\;\text{and}\;f\in L^2(\mathbb{R}^d). \] A weighted Hardy-type space related to \(\mathcal{L}\) is defined by \[ H^1_{\mathcal{L}}(\omega):=\{f\in L^1(\omega):\;\mathcal{T}^*f(x)\in L^1(\omega)\} \;\;\text{and}\;\;\|f\|_{H^1_{\mathcal{L}}(\omega)}:= \|\mathcal{T}^*f\|_{L^1(\omega)}, \] where \(\mathcal{T}^*f(x):=\sup_{t>0}|T_tf(x)|\) for all \(x\in{\mathbb R}^d\) and \(\omega\) is a Muckenhoupt weight. A function \(a\) is called an atom for the weighted Hardy space \(H^1_{\mathcal{L}}(\omega)\) associated to a ball \(B(x_0,r)\) for some \(x_0\in {\mathbb R}^d\) and \(r_0\in (0,\infty)\), if \(\mathrm{supp}\,a\subset B(x_0,r)\), \(\|a\|_{L^\infty({\mathbb R}^d)}\leq \frac{1}{\omega(B(x_0,r))}\), if \(x_0\in\mathcal{B}_n\), then \(r\leq2^{1-n/2}\) and, if \(x_0\in\mathcal{B}_n\) and \(r\leq2^{-1-n/2}\), then \(\int_{{\mathbb R}^d} a(x)\,dx=0\), where \(\mathcal{B}_n:=\{x:\;2^{-(n+1)/2}< \rho(x)\leq2^{-n/2}\}\) for \(n\in \mathbb{Z}\) and \(\rho(x):=\rho(x,V):=\sup\{r>0:\;\frac{1}{r^{d-2}} \int_{B(x,r)}V(y)\,dy\leq 1\}\). The atomic norm of \(f\in H^1_{\mathcal{L}}(\omega)\) is defined by \(\|f\|_{\mathcal{L}-at(\omega)}:=\inf\{\sum|c_j|\}\), where the infimum is taken over all decompositions \(f=\sum c_ja_j\), \(\{c_j\}_j\subset \mathbb{C}\) and \(\{a_j\}_j\) are \(H^1_{\mathcal{L}}(\omega)\) atoms. The authors establish the following atomic characterization for \(H^1_{\mathcal{L}}(\omega)\): Assume that \(V\in RH_{d/2}(\mathbb{R}^d)\) is a non-negative potential and \(V\not\equiv0\), then the norms \(\|f\|_{H^1_{\mathcal{L}}(\omega)}\) and \(\|f\|_{\mathcal{L}-at(\omega)}\) are equivalent. The authors also characterize \(H^1_{\mathcal{L}}(\omega)\) by the Riesz transforms \(R_j:=\frac{\partial}{\partial x_j} \mathcal{L}^{-1/2}\) for \(j\in\{1,\ldots,d\}\): if \(V\in RH_{d}(\mathbb{R}^d)\) is a non-negative potential, then there exists a positive constant \(C\) such that \[ C^{-1}\|f\|_{H^1_{\mathcal{L}}(\omega)}\leq \|f\|_{L^1(\omega)}+\sum_{j=1}^d\|R_jf\|_{L^1(\omega)} \leq C\|f\|_{H^1_{\mathcal{L}}(\omega)}\;\text{for all} \;f\in H^1_{\mathcal{L}}(\omega). \] The authors also prove that the dual space of \(H^1_{\mathcal{L}}(\omega)\) is \(\text{BMO}_{\mathcal{L}}(\omega)\), where \(f\in \text{BMO}_{\mathcal{L}}(\omega)\) means that \(f\in \text{BMO}(\omega)\) and there exists a positive constant \(C\) such that \(\frac{1}{\omega(B)}\int_B|f(y)|\,dy\leq C\) for all \(B:=B(x,R)\) with \(x\in{\mathbb R}^d\) and \(R>\rho(x)\). A positive measure \(\mu\) on \(\mathbb{R}^{d+1}_+:=\mathbb{R}^{d}\times(0,\infty)\) is called an \(\omega-\)Carleson measure if \[ \|\mu\|_{\mathcal{C}}:=\sup_{x\in \mathbb{R}^{d},r>0} \frac{\mu(B(x,r)\times(0,r))}{\omega((B(x,r))}<\infty. \] The authors prove that, if \(f\in \text{BMO}_{\mathcal{L}}(\omega)\), \(V\not\equiv0\) is a non-negative potential in \(RH_s(\mathbb{R}^d)\) for some \(s\geq\frac{d}{2}\) and \(\omega\) a weight in \(A_1\), then \(d\mu_f(x,t):=|Q_tf(x)|^2\frac{|B(x,r)|}{\omega(B(x,r))} \,\frac{dxdt}{t}\) is an \(\omega-\)Carleson measure, where \[ (Q_tf)(x):=t^2\left(\left.\frac{dT_s}{ds}\right|_{s=t^2}f\right)(x),\;(x,t)\in \mathbb{R}^{d+1}_+, \] and, conversely, if \(f\in L^1((1+|x|)^{-(d+1)}dx)\) and \(d\mu_f(x,t)\) is a \(\omega-\)Carleson measure, then \(f\in \text{BMO}_{\mathcal{L}}(\omega)\). Finally, the authors establish the boundedness of the Hardy-Littlewood maximal operator and the semigroup maximal operator on \(\text{BMO}_{\mathcal{L}}(\omega)\). Reviewer: Yang Dachun (Beijing) Cited in 11 Documents MSC: 42B35 Function spaces arising in harmonic analysis 42B30 \(H^p\)-spaces 42B25 Maximal functions, Littlewood-Paley theory 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B37 Harmonic analysis and PDEs Keywords:weighted Hardy space; weighted BMO space; Schrödinger operator; atom; semigroup; Riesz transform PDFBibTeX XMLCite \textit{H. Liu} et al., Math. Nachr. 285, No. 17--18, 2173--2207 (2012; Zbl 1266.42060) Full Text: DOI References: [1] Bennet, Weak-L and BMO, Ann. Math. 113 pp 601– (1981) · Zbl 0465.42015 [2] Bongioanni, Riesz transforms related to Schrödinger operators acting on BMO type spaces, J. Math. Anal. Appl. 357 pp 115– (2009) · Zbl 1180.42013 [3] Bui, Weighted hardy spaces, Math. 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