Hardy spaces associated with a pair of commuting operators (Q494974)

For \(\omega\in[0,\pi)\), let \(S_{\omega}:=\{z\in\mathbb{C}:\;|\arg z|\leq\omega\}\). A closed operator \(T\) on \(L^2(\mathbb{R}^n)\) is said to be of type \(\omega\), if the spectrum of \(T\) belongs to \(S_{\omega}\) and there exists a positive constant \(C\) such that, for any \(z\in\mathbb{C}\backslash S_{\omega}\), \[ \|(zI-T)^{-1}\|_{\mathcal{L}(L^2(\mathbb{R}^n))}\leq C|z|^{-1}, \] where \(\|\cdot\|_{\mathcal{L}(L^2(\mathbb{R}^n))}\) denotes the operator norm of \(T\) on \(L^2(\mathbb{R}^n)\). For any operator \(T_j\) of type \(\omega_j\in[0,\pi)\), with \(j\in\{1,2\}\), \(T_1\) and \(T_2\) are said to be commuting if their resolvents commute. Let \(m\in\mathbb{N}\) and \(T\) be a one-to-one operator of type \(\omega\in[0,\pi)\). It is said that the holomorphic semigroup \(\{e^{-tT}\}_{t>0}\), generated by \(T\), satisfies the \(m\)-Davies-Gaffney estimate if there exist positive constants \(C\) and \(c\) such that, for all closed sets \(E,\,F\subset\mathbb{R}^n\), \(t\in(0,\infty)\) and \(f\in L^2(\mathbb{R}^n)\) supported in \(E\), \[ \|e^{-tT}f\|_{L^2(F)}\leq C\exp\left\{-c\frac{[d(E,F)]^{2m/(2m-1)}}{t^{1/(2m-1)}}\right\}\|f\|_{L^2(E)}, \] where \(d(E,F):=\inf\{|x-y|:\;x\in E,\,y\in F\}\). Let \(L_1\) and \(L_2\) be a pair of one-to-one commuting operators of type \(\omega\in[0,\pi/2)\). Assume that for each \(j\in\{1,2\}\), \(L_j\) satisfies the \(m_j\)-Davies-Gaffney estimate with \(m_1\geq m_2>0\). Let \(p\in(0,1]\), \(H^p_{L_j}(\mathbb{R}^n)\), with \(j\in\{1,2\}\), and \(H^p_{L_1+\widetilde{L}_2}(\mathbb{R}^n)\) be the Hardy spaces associated with the operators \(L_j\) and \(L_1+\widetilde{L}_2\), respectively, where \(\widetilde{L}_2:=L^{m_1/m_2}_2\). In this interesting paper, the authors obtain the bounded joint \(H_\infty\) functional calculus in these Hardy spaces and then show that, for any \(p\in(n/(n+m_j),1]\) with \(j\in\{1,2\}\), the abstract Riesz transform \(D^{m_i}(L_1+L_2)^{-1/2}\) is bounded from the Hardy space \(H^p_{L_j}(\mathbb{R}^n)\) to the classical Hardy space \(H^p(\mathbb{R}^n)\). Furthermore, the authors prove that, for any \(p\in(0,1]\), \[ H^p_{L_1+\widetilde{L}_2}(\mathbb{R}^n)=H^p_{L_1}(\mathbb{R}^n)+H^p_{L_2}(\mathbb{R}^n). \] Moreover, a sufficient condition to guarantee \(H^p_{L_1}(\mathbb{R}^n)\subset H^p_{L_2}(\mathbb{R}^n)\) is also given in this paper.