An inequality concerning the growth bound of an evolution family and the norm of a convolution operator

作者：Buse Constantin; O'Regan Donal; Saierli Olivia^*

来源：Semigroup Forum, 2017, 94(3): 618-631.

DOI：10.1007/s00233-016-9822-9

• Uniform exponential stability
• Growth bounds
• Exponentially bounded evolution families of operators
• Convolution operator on function spaces
• One dimensional heat equation

Let be a strongly continuous and exponentially bounded evolution family acting on a complex Banach space X and let be a certain Banach function space of X-valued functions. We prove that the growth bound of the family is less than or equal to provided that the convolution operator acts on It is well known that under the latter assumption, the convolution operator is bounded and then denotes (ad-hoc) its norm in As a consequence, we prove that if then Finally, we give an example showing that the accuracy of the estimates may be quite accurate.

• 出版日期2017-6