Let X be a Banach space and $f\in {L}_{l}^{1}oc(\mathbb{R};X)$ be absolutely regular (i.e. integrable when divided by some polynomial). If the distributional Fourier transform of f is locally integrable then f converges to 0 at infinity in some sense to be made precise. From this result we deduce some Tauberian theorems for Fourier and Laplace transforms, which can be improved if the underlying Banach space has the analytic Radon-Nikodym property.

If the second order problem $\ddot{u}+\dot{u}+Au=f$ has ${L}^{p}$ maximal regularity for some $p\in (1,\mathrm{\infty})$, then it has ${L}^{p}$ maximal regularity for every $p\in (1,\mathrm{\infty})$.

Local well-posedness of the curve shortening flow, that is, local existence, uniqueness and smooth dependence of solutions on initial data, is proved by applying the Local Inverse Function Theorem and ${L}^{2}$-maximal regularity results for linear parabolic equations. The application of the Local Inverse Function Theorem leads to a particularly short proof which gives in addition the space-time regularity of the solutions. The method may be applied to general nonlinear evolution equations, but is presented...

We consider some non-autonomous second order Cauchy problems of the form
ü + B(t)u̇ + A(t)u = f(t ∈ [0,T]), u(0) = u̇(0) = 0.
We assume that the first order problem
u̇ + B(t)u = f(t ∈ [0,T]), u(0) = 0,
has ${L}^{p}$-maximal regularity. Then we establish ${L}^{p}$-maximal regularity of the second order problem in situations when the domains of B(t₁) and A(t₂) always coincide, or when A(t) = κB(t).

Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay of operator semigroups and yields a number of new ones. Its core is a new operator-theoretical method of deriving rates of decay combining ingredients from functional calculus and complex, real and harmonic analysis. It also leads to several results of independent...

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