Stochastic evolution equations in UMD Banach spaces

We discuss existence, uniqueness, and space-time Hölder regularity for solutions of the parabolic stochastic evolution equation

     

Separable anisotropic differential operators in weighted abstract spaces and applications

In this paper operator-valued multiplier theorems in Banach-valued weighted Lp spaces are studied. Also weighted Sobolev-Lions type spaces are discussed when E₀, E are two Banach spaces and E₀ is continuously and densely embedded on E. Several conditions are found that ensure the continuity of the embedding operators that are optimally regular in these spaces in terms of interpolations of E₀. These results permit us to show the separability of the anisotropic differential operators in an E-valued weighted Lp space. By using these results the maximal regularity of infinite systems of quasi elliptic partial differential equations are established.     

Resolution, in L-spaces, of transmission problems set in an unbounded domains

The goal of this work is to give a complete study of some abstract transmission problems (P^δ), for every δ > 0, set in unbounded domain composed of a half-line ]−∞, 0[ and a thin layer ]0, δ[. Existence and uniqueness results are obtained for strict solutions in UMD Banach spaces, by using essentially the semigroup theory and the Dore-Venni’s Theorem given in [8].     

Maximal regularity of discrete second order Cauchy problems in Banach spaces

We characterize the discrete maximal regularity for second order difference equations by means of spectral and R-boundedness properties of the resolvent set.     

An operator-valued Tb theorem

We establish conditions similar to the Tb theorem of David, Journé and Semmes which guarantee the boundedness of an integral transformation T with L(X)-valued kernel on , where 1<p<∞ and X is a Banach space with the unconditionality property of martingale differences (UMD).     

‐maximal regularity for fractional difference equations on UMD spaces

Let T be a bounded linear operator defined on a Banach space X. We introduce an operator theoretical method for linear fractional difference equations based on the notion of α‐resolvent sequence of bounded and linear operators. Then, we define and characterize the ‐ maximal regularity of solutions for the problem solely in terms of the R‐boundedness of the set      

A characterization of periodic solutions for time‐fractional differential equations in UMD spaces and applications

We study the fractional differential equation (*) D^αu(t) + BD^βu(t) + Au(t) = f(t), 0 ? t ? 2π (0 ? β < α ? 2) in periodic Lebesgue spaces L^p(0, 2π; X) where X is a Banach space. Using functional calculus and operator valued Fourier multiplier theorems, we characterize, in UMD spaces, the well posedness of (*) in terms of R‐boundedness of the sets {(ik)^α((ik)^α + (ik)^βB + A)^?1}_{k∈ Z} and {(ik)^βB((ik)^α + (ik)^βB + A)^?1}_{k∈ Z} . Applications to the fractional problems with periodic boundary condition, which includes the time diffusion and fractional wave equations, as well as an abstract version of the Basset‐Boussinesq‐Oseen equation are treated. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

The group reduction for bounded cosine functions on UMD spaces

It is shown that if A generates a bounded cosine operator function on a UMD space X, then i(−A)^1/2 generates a bounded C ₀-group. The proof uses a transference principle for cosine functions.