Sharp spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates (Q2866746)

Let \((X,d,\mu)\) be a metric measure space satisfying the volume doubling condition: \(\mu(B(x,2r)<C\mu(B(x,r)\) \((x\in X,r>0)\). From the doubling condition it follows that there exist \(C,n>0\) such that \(\mu(B(x,\lambda r)<C\lambda^n\mu(B(x,r)\) \((x\in X,r>0, \lambda\geq1)\). Consider \(n\) as small as possible. Let \(L\) be a non-negative self-adjoint operator on \(L^2(X)\) satisfying the following two conditions.NEWLINENEWLINE(H1) (\textit{Davies-Gaffney estimate}) There exist two positive constants \(C,c\) such that for all open sets \(U_1,U_2\subset X\) and \(t>0\) \( |\langle e^{-tL}f_1,f_2\rangle| \leq C\exp\bigl({-\frac {d(U_1,U_2)^2}{ct}}\bigr)\|f_1\|_{L^2(X)}\|f_2\|_{L^2(X)} \) for every \(f_j\in L^2(X)\) with \(\text{supp}f_j\subset U_j\) \((j=1,2)\).NEWLINENEWLINE(H2) (\textit{restriction-type estimate}) For each \(R>0\) and all Borel functions \(F:\mathbb R\to\mathbb C\) with \(\text{supp}F\subset [0,R]\), there exist some \(1\leq p_0<2\) and \(1<q<\infty\) such that \( \|F(\sqrt L)P_{B(x,r)}\|_{L^{p_0}\to L^2} \leq C\mu(B(x,r))^{1/2-1/p_0}(Rr)^{n(1/p_0-1/2)}\|\delta_RF\|_{L^q} \) for all \(x\in X\) and all \(r\geq 1/R\), where \(\delta_R F(s):=F(Rs)\) and \(P_Bf(x):=\chi_B(x)f(x)\).NEWLINENEWLINEUnder the above setting, the author gives the following: Let \(\phi\) be a non-trivial smooth function with copmact support on \((0,\infty)\). Suppose that \(0<p\leq1\). Let \(F\) be a bounded Borel function for which there exists some constant \(s>n(1/p-1/2)\) such that \(\sup_{t>0}\|\phi\delta_t F\|_{W^{s,q}(\mathbb R)}<\infty\). Then the operator \(F(\sqrt L)\) is bounded on the Hardy space \(H^p_L(X)\) associated to the operator \(L\).NEWLINENEWLINEAs an application, a sharp result for the boundedness of Bochner-Riesz means on \(H^p_L(X)\) is given. Examples which satisfies (H1) and (H2) are discussed.