Stability theory for semigroups using $(L^p,L^q)$ Fourier multipliers

Abstract

We study polynomial and exponential stability for $C_0$-semigroups using the recently developed theory of operator-valued $(L^p,L^q)$ Fourier multipliers. We characterize polynomial decay of orbits of a $C_0$-semigroup in terms of the $(L^p,L^q)$ Fourier multiplier properties of its resolvent. Using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space. We do not assume that the semigroup is uniformly bounded, our results depend only on spectral properties of the generator. As a corollary of our work on polynomial stability we reprove and unify various existing results on exponential stability, and we also obtain a new theorem on exponential stability for positive semigroups.

40 pages. To appear in Journal of Functional Analysis

References in corpus (6)

• Optimal rates of decay for operator semigroups on Hilbert spaces
• Optimal energy decay in a one-dimensional coupled wave-heat system
• Quantified versions of Ingham's theorem
• Fourier multiplier theorems on Besov spaces under type and cotype conditions
• A Katznelson-Tzafriri theorem for measures
• Asymptotics for infinite systems of differential equations

Cited by in corpus (6)

• Optimal rates of decay for operator semigroups on Hilbert spaces
• Semi-uniform stability of operator semigroups and energy decay of damped waves
• Fourier multiplier theorems on Besov spaces under type and cotype conditions
• Operator-valued $(L^p,L^q)$ Fourier multipliers and stability theory for evolution equations
• Stability of (eventually) positive semigroups on spaces of continuous functions
• Improved polynomial decay for unbounded semigroups