\(H^\infty\)-calculus for the Stokes operator on \(L_q\)-spaces
A closed operator \(A\) on a Banach space \(X\) is called sectorial if {(i)} \(A\) is densely defined, injective and has dense range, {(ii)} \((-\infty,0)\subset\rho(A)\) and there is an \(M\geq0\) such that \(\| \lambda(\lambda+A)^{-1}\| \leq M\) for all \(\lambda>0\). In this case, there is a \(\phi\in[0,\pi)\) such that the sector \[ \Sigma_{\pi-\phi}:=\{z\in\mathbb{C}\setminus\{0\}:| \arg z| <\pi-\phi\} \] is contained in \(\rho(-A)\), and \[ \sup\{\| \lambda(\lambda+A)^{-1}\| :\lambda\in\Sigma_{\pi-\phi}\}<\infty. \] The infinum of all such \(\phi\) is called the spectral angle of \(A\) and is denoted by \(\phi_A\). Thus, \(\sigma(A)\setminus\{0\}\subset\Sigma_{\phi_A}\). Given \(\phi\in(0,\pi)\), consider the space \[ \mathcal{H}^\infty(\Sigma_\phi):=\{h:\Sigma_\phi\to\mathbb{C}\mid h\roman{\; is\; holomorphic\; and\; bounded}\} \] and its subspace \[ \mathcal {H}^\infty_0(\Sigma_\phi):=\{h\in\mathcal{H}^\infty(\Sigma_\phi)\mid | h(z)| \leq C| z| ^s/(1+| z| ^{2s})\roman{\; for\; some\; }C\geq0,s>0\}. \] Let \(A\) be a sectorial operator on \(X\) with spectral angle \(\phi_A\), and let \(\phi_A<\theta<\phi <\pi\). For \(h\in\mathcal{H}^\infty_0(\Sigma_\phi)\), consider the Bochner integral \[ h(A):=\frac{1}{2\pi i}\int_\Gamma h(\lambda)(\lambda-A)^{-1}d\lambda, \] where \[ \Gamma:\mathbb{R}\to\mathbb{C},\quad\Gamma(t)=-te^{i\theta}\roman{\; if\; }t<0,\quad\Gamma(t)=te^{-i\theta}\roman{\; if\; }t\geq0. \] The operator \(A\) is said to admit a bounded \(H^\infty\)-calculus if there is a constant \(C\geq0\) such that \(\| h(A)x\| \leq C\| h\| _\infty\| x\| \) for all \(h\in\mathcal{H}^\infty_0(\Sigma_\phi)\) and all \(x\in X\). For \(q\in(1,\infty)\), let \(L_{q,\sigma}(\Omega)\) be the Banach space of all \(q\)-integrable solenoidal vector fields on a domain \(\Omega\subset\mathbb{R}^n\), \(P_\Omega=P_{\Omega,q}\) be the Helmholtz projection of \(L_q(\Omega)\) onto \(L_{q,\sigma}(\Omega)\), \(\Delta_\Omega\) the Dirichlet Laplacian in \(L_q(\Omega)\), and let \(A_\Omega=-P_\Omega\Delta_\Omega\) be the Stokes operator in \(L_{q,\sigma}(\Omega)\). The main result is as follows. Let \(n\geq3\) and let \(\Omega\subset\mathbb{R}^n\) be a \(C^3\)-domain which is either bounded, exterior, or a perturbed half-space. Then the Stokes operator \(A_\Omega\) admits a bounded \(H^\infty\)-calculus in \(L_{q,\sigma}(\Omega)\) if \(1<q<\infty\).
scientific article; zbMATH DE number 1983491