It is a fundamental problem in harmonic analysis to deduce regularity or asymptotic properties of a bounded, vector-valued function, defined on a half-line, from properties of its Laplace transform.
In the first part of this talk, we will study how the analytic extendability of the Laplace transform to certain large domains, and the boundedness therein, (almost) characterizes regularity properties like analyticity and differentiability of the original function. We will also see that it is not clear how to characterize continuity in this way; naive counterparts/generalizations of the results which hold for analyticity and differentiability admit easy counterexamples.
Characterizing continuity becomes not easier, if one considers bounded, strongly continuous semigroups: it was a longstanding open problem whether the decay to zero of the resolvent of the generator along vertical lines characterizes immediate norm-continuity of the semigroup with respect to the operator-norm. After several affirmative results in Hilbert space and for positive semigroups on $L^p$ spaces, a negative answer to this question was recently given by Tamas Matrai.
In the second part of this talk, we will give some counterexamples which are conceptually different to the one given by Matrai. In fact, we will present a new method of constructing semigroups, by considering operators and algebra homomorphisms on $L^1$ with specific properties. Our examples rule out the possibility of characterizing norm-continuity of semigroups on arbitrary Banach spaces in terms of resolvent-norm decay on vertical lines.