Advances in Differential Equations

Functional calculi for linear operators in vector-valued $L^p$-spaces via the transference principle

Matthias Hieber and Jan Prüss

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $-A$ be the generator of a bounded $C_0$-group or of a positive contraction semigroup, respectively, on $L^p(\Omega,\mu,Y)$, where $(\Omega,\mu)$ is measure space, $Y$ is a Banach space of class $\cal H \cal T$ and $1<p<\infty$. If $Y=\mathbb{C}$, it is shown by means of the transference principle due to Coifman and Weiss that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda\pm\pi/2|<\theta\}$, where $\theta>0$ and on each sector $\Sigma_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda|<\theta\}$ with $\theta<\pi/2$, respectively. Several extensions of these results to the vector-valued case $L^p(\Omega,\mu,Y)$ are presented. In particular, let $-A$ be the generator of a bounded group on a Banach spaces of class $\cal H\cal T$. Then it is shown that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta$, $\theta > 0$, and that $-A^2$ admits an $H^\infty$-calculus on each sector $\Sigma_\theta$, where $\theta > 0$. Applications of these results deal with elliptic boundary value problems on cylindrical domains and on domains with non smooth boundary.

Article information

Adv. Differential Equations, Volume 3, Number 6 (1998), 847-876.

First available in Project Euclid: 18 April 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A60
Secondary: 35J25: Boundary value problems for second-order elliptic equations 35J40: Boundary value problems for higher-order elliptic equations 42B15: Multipliers 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 47N20: Applications to differential and integral equations


Hieber, Matthias; Prüss, Jan. Functional calculi for linear operators in vector-valued $L^p$-spaces via the transference principle. Adv. Differential Equations 3 (1998), no. 6, 847--876.

Export citation