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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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