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).
We consider the problem of L p -boundedness of higher order Riesz transforms ∇ m L −1/2 associated to elliptic operators L of order 2m on R D . As an application of the recently solved Kato conjecture, we show ∇ m L −1/2 ∈ L(L p (R D )) for all p ∈ ( 2D 2m+D ∨1 , 2]. This generalizes the result of Auscher and Tchamitchian restricted to the case D ≤ 2m.
bounded and analytic on L r for all r strictly between p and q. This is a discrete analogue of the well-known corresponding result for analytic semigroups (e tA ). As recently shown by the author, the analyticity of T is a necessary condition for the maximal regularity of the discrete time evolution equation u n+1 &Tu n = f n for all n # Z + , u 0 =0. In the second part of this paper we establish the following two sufficient conditions for its maximal regularity: T is a subpositive analytic contraction, or T is an integral operator satisfying certain Poisson bounds. These results are discrete analogues of the corresponding results for the maximal regularity of the evolution equation u$(t)&Au(t)= f (t) for all t # R + , u(0)=0, due to Lamberton, Weis, Coulhon and Duong and Hieber and Pru ss. For the Poisson bound result of Coulhon and Duong and Hieber and Pru ss we give a slight improvement and a short proof.
Academic Press
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