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On square functions associated to sectorial operators. (English) Zbl 1066.47013
The author presents new results on square functions associated to sectorial operators. For a given sectorial operator \(A\) of type \(\omega\in (0,\pi)\) on a Banach space \(X\), a number \(\theta\in(\omega,\pi)\), and a function \(f \in H_0^\infty(\Sigma_{\theta})\), one can define an operator \(f(A)\in B(X)\) as follows. Let \(\gamma\in(\omega,\pi)\) be an intermediate angle and consider the oriented contour \(\Gamma_{\gamma}\) defined by \[ \Gamma_{\gamma}(t)=\begin{cases} -te^{i\gamma} &t\in \mathbb R_-,\\ te^{-i\gamma} &t\in \mathbb R_+.\end{cases} \] Then let \(f(A)=\frac{1}{2\pi i}\int_{\Gamma_{\gamma}} f(z)R(z,A)dz\). \(A\) admits a bounded \(H^{\infty}(\Sigma_{\theta})\) functional calculus if \(f(A)\) is bounded for any \(f \in H^{\infty}(\Sigma_{\theta})\). In that case, the mapping \(f\mapsto f(A)\) is a bounded homomorphism from \(H^{\infty}(\Sigma_{\theta})\) into \(B(X)\), provided that \(H^{\infty}(\Sigma_{\theta})\) is equipped with the norm \(\| f\|_{\infty,\theta}= \sup \{|f(z)|: z\in\Sigma_{\theta}\}\).
The main result of the article is as follows. Theorem 1.1. Let \(A\) be an \(R\)-sectorial operator of \(R\)-type \(\omega\in (0,\pi)\) on a space \(L^p(\Omega)\), with \(1\leq p < \infty\). Let \(\theta\in(\omega,\pi)\) and let \(F\) and \(G\) be two nonzero functions belonging to \(H_0^{\infty}(\Sigma_{\theta})\).
(1) There exists a constant \(K>0\) such that for any \(f\in H^{\infty}(\Sigma_{\theta})\) and any \(x \in L^p(\Omega)\), \[ \left\|\left(\int_0^{\infty} |f(A)F(tA)x|^2dt/t\right)^{1/2}\right\|_{L^p(\Omega)} \leq K\| f\|_{\infty,\theta} \left\|\left(\int_0^{\infty} |G(tA)x|^2 dt/t\right)^{1/2}\right\|_{L^p(\Omega)}. \] (2) There exists a constant \(K>0\) such that \[ K^{-1}\| x\|_G\leq \| x\|_F\leq K\| x\|_G,\quad x\in L^p(\Omega). \] The proof of Theorem 1.1 is based on results of Ausher-Duong-Mcintosh, Weis and others.

MSC:
47A60 Functional calculus for linear operators
47D06 One-parameter semigroups and linear evolution equations
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