Let U={U(t,s)}t≥s≥0\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {U}=\{U(t,s)\}_{t\ge s\ge 0}$$\end{document} be a strongly continuous and exponentially bounded evolution family acting on a complex Banach space X and let X\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {X}$$\end{document} be a certain Banach function space of X-valued functions. We prove that the growth bound of the family U\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {U}$$\end{document} is less than or equal to -1c(U,X)\documentclass[12pt]{minimal}
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\begin{document}$$-\frac{1}{c(\mathcal {U}, \mathcal {X})}$$\end{document} provided that the convolution operator f↦U∗f\documentclass[12pt]{minimal}
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\begin{document}$$f\mapsto \mathcal {U}*f$$\end{document} acts on X.\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {X}.$$\end{document} It is well known that under the latter assumption, the convolution operator is bounded and then c(U,X)\documentclass[12pt]{minimal}
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\begin{document}$$c(\mathcal {U}, \mathcal {X})$$\end{document} denotes (ad-hoc) its norm in L(X).\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}(\mathcal {X}).$$\end{document} As a consequence, we prove that if sups≥0∫s∞‖U(t,s)‖dt=u1(U)<∞,\documentclass[12pt]{minimal}
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\begin{document}$$\sup \nolimits _{s\ge 0}\int \nolimits _{s}^\infty \Vert U(t,s)\Vert dt=u_1(\mathcal {U})<\infty ,$$\end{document} then ω0(U)u1(U)≤-1.\documentclass[12pt]{minimal}
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\begin{document}$$\omega _0(\mathcal {U})u_1(\mathcal {U})\le -1.$$\end{document} Finally, we give an example showing that the accuracy of the estimates may be quite accurate.
