Advances in Differential Equations

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

Matthias Hieber and Jan Prüss

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Abstract

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

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

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366292551

Mathematical Reviews number (MathSciNet)
MR1659281

Zentralblatt MATH identifier
0956.47008

Subjects
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

Citation

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. https://projecteuclid.org/euclid.ade/1366292551.


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