Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus
Let \(X\) be a metric space, equipped with a distance function \(d\) and a non-negative, doubling Borel measure \(\mu\) on \(X\) with \(\mu(X)=\infty\). Let \(0\leq \omega<\nu<\pi\) and let \(S_\omega\) be the closed sector in the complex plane \({\mathbb C}\) defined by \[ S_\omega:=\{z\in{\mathbb C}:\,\, |\arg z|\leq \omega\}\cup\{0\} \] and \(S^0_\omega\) its interior. Let \(H(S^0_\nu)\) be the space of all holomorphic function on \(S^0_\nu\) and \[ H_\infty(S^0_\nu):=\{b\in H(S^0_\nu):\,\,\|b\|_\infty<\infty\}, \] where \(\|b\|_\infty:=\sup\{|b(z)|:\,\,z\in S^0_\nu\}\), and \[ \Psi(S^0_\nu):=\left\{\psi\in H_\infty(S^0_\nu):\,\, \exists s>0, |\psi(z)|\leq c\frac{|z|^s}{(1+|z|^{2s})}\right\}. \] Throughout the paper, \(L\) is assumed to be a linear operator of type \(\omega\) (that is, \(\sigma(L)\subset S_\omega\)) on \(L^2(X)\) with \(\omega<\pi/2\) and satisfy the following two conditions: {(H1)} The operator \(L\) has a bounded \(H_\infty\)-calculus on \(L^2(X)\). That is, there exists \(c_{\nu,\,2}>0\) such that for \(b\in H_\infty(S^0_\nu)\), \(b(L)\in {\mathcal L}(L^2, L^2)\) and for any \(g\in L^2(X)\), \[ \|b(L)g\|_{L^2(X)}\leq c_{\nu,\,2}\|b\|_\infty\|g\|_{L^2(X)}; \] {(H2)} The analytic semigroup \(\{e^{-tL}\}_{t>0}\) satisfies the Davies-Gaffney estimate. That is, there exist constants \(C\), \(c>0\) such that for any open subsets \(U_1\), \(U_2\subset X\), \(f_i\in L^2(X)\) with \({\text{\,supp\,}}(f_i)\subset U_i\), \(i=1,\,2\), and \(t>0\), \[ |\langle e^{-tL}f_1, f_2\rangle|\leq C\exp\left(-\frac{{\mathrm{\,dist\,}}(U_1, U_2)^2}{ct}\right)\|f_1\|_{L^2(X)}\|f_2\|_{L^2(X)}, \] where \({\text{\,dist\,}}(U_1, U_2):=\inf_{x\in U_1,\,y\in U_2}d(x, y)\). Fix \(\alpha\geq 0\), \(0<p\leq1\), and integers \(s\geq [\frac{n\alpha}{2}]\) and \(M>[\frac{n(2-p)}{4p}]\). The authors first introduce Lipschitz spaces \(\Lambda^{\alpha,\,s}_L(X)\), the molecular Hardy space \(H^p_{L,\,\text{mol},\,M}(X)\), the Hardy space \(H^p_{L,\,S_h}(X)\) defined in terms of the square function \(S_h\) associated with the heat semigroup generated by \(L\), and the atomic Hardy space \(H^p_{L,\,\text{at},\,M}(X)\) under the following additional assumption: {(H3)} \(L\) generates the family of bounded operators \(\cos t\sqrt L\), \(t\geq0\), on \(L^2(X)\). Under {(H1)} and {(H2)}, the authors first show that the spaces \(H^p_{L,\,\text{mol},\,M}(X)\) and \(H^p_{L,\,S_h}(X)\) coincide with equivalent norms, and that the spaces \(H^p_{L,\,\text{at},\,M}(X)\) and \(H^p_{L,\,S_h}(X)\) coincide with equivalent norms under the additional assumption {(H3)}. Write \[ H^p_L(X):=H^p_{L,\,\text{mol},\,M}(X)=H^p_{L,\,S_h}(X), \] and \[ H^p_L(X):=H^p_{L,\,\text{at},\,M}(X) \] under {(H3)}. Under {(H1)} and {(H2)}, the authors also show that the dual space of \(H^p_{L}(X)\) is \(\Lambda^{1/p-1,\,M}_{L^\ast}(X)\), and that if \(T\) is a sublinear operator from \(H^{p_1}_L(X)+H^{p_2}_L(X)\), \(0<p_1<p_2<\infty\), into measurable functions on \(X\), which is of weak-type \((H^{p_1}, p_1)\) and \((H^{p_2}, p_2)\), then \(T\) is bounded from \(H^p_L(X)\) into \(L^p(X)\) for \(p_1<p<p_2\). As applications, for \(0<p\leq 1\), the authors show that the operator \(L\) has a bounded holomorphic functional calculus on \(H_L^p(X)\) and the Riesz transform \(DL^{-\frac12}\) associated with \(L\) is bounded from \(H^p_L(X)\) into \(L^p(X)\), where \(D\) is a densely defined linear operator on \(L^2(X)\) satisfying some extra conditions.
scientific article; zbMATH DE number 6159679