Maximal regularity of discrete and continuous time evolution equations

Abstract

Abstract. We consider the maximal regularity problem for the discrete time evolution equation u n+1 −T u n = f n for all n ∈ N 0 , u 0 = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup (T n ) n∈N 0 and of the resolvent R(λ, T ), secondly by the maximal regularity of the continuous time evolution equation u (t) − Au(t) = f (t) for all t > 0, u(0) = 0, where A := T − I. By recent results of Weis, this continuous maximal regularity is characterized by R-boundedness properties of the continuous time semigroup (e t(T −I) ) t≥0 and again of the resolvent R(λ, T ). As an important tool we prove an operator-valued Mikhlin theorem for the torus T providing conditions on a symbol M ∈ L ∞ (T; L(X)) such that the associated Fourier multiplier T M is bounded on l p (X).

Well posedness for semidiscrete fractional Cauchy problems with finite delay

“…The following Fourier multiplier theorem for operator valued symbols is due to S. Blunck [9]. This theorem is stated for the UMD class of Banach spaces.…”

“…By hypothesis (H) α , the left hand side is invertible for |z| = 1, z = 1 and therefore, in view of (4.6), the discrete Fourier transform of K α f (n − 1) coincides with R α f (n) for all n ∈ N, by uniqueness. Moreover, we define 9) and considering the second equation in (4.4) we have that the discrete Fourier transform of P α f (n − 1) coincides with U α f (n) for all n ∈ N, by uniqueness. It proves (i).…”

Square Functions for Ritt Operators on Noncommutative $L^p$-Spaces

“…Details and complements can be found in [ALM], [Bl1], [Bl2], [LM2], [Lyu], [NaZ], [Nev] and [Vit]. Let X be a Banach space.…”

Operator-valued multiplier theorems characterizing Hilbert spaces

“…(X)) be such that both {M(t) : t e {-it, 0)U(0,7r)}and{(e"-l)(e" + l)Af'(r) : / € (-w, 0) U (0, n)} are ^-bounded. Then M is an L p -Fourier multiplier on l"(Z;X) [5]. …”