Modulation type spaces for generators of polynomially bounded groups and Schrödinger equations

Abstract

We introduce modulation type spaces associated with the generators of polynomially bounded groups. Besides strongly continuous groups we study in detail the case of bi-continuous groups, e.g. weak *-continuous groups in dual spaces. It turns out that this gives new insight in situations where generators are not densely defined. Classical modulation spaces are covered as a special case but the abstract formulation gives more flexibility. We illustrate this with an application to a nonlinear Schrödinger equation.

The Global Cauchy Problem for the NLS With Higher Order Anisotropic Dispersion

“…Putting everything together, we arrive at the condition 12) which is satisfied for sufficiently small R. Similarly, we obtain…”

“…For q < ∞, the solution from Theorems 1 and 2 is a continuous function with values in the corresponding modulation space, that is, indeed a mild solution. For the more delicate situation q = ∞, see [12].…”

On continuity properties of semigroups in real interpolation spaces

“…The integral here is the limit lim a→∞ in operator norm of the integrals a 0 e −λt T (t)x dt which in turn have to be understood as τ X -Riemann integrals (we refer also to [14,Prop. A.2] and to the arguments in the proof of Proposition 2.9).…”

“…The paper owes much to our previous work [14] on abstract modulation type spaces where we studied polynomially bounded groups. Nevertheless, there are substantial differences; the functional calculus in [14] is based on the Fourier transform, and we could work with compactly supported C ∞ -functions on the real line; here, it is based on the Laplace transform, and we have to work with holomorphic functions. Moreover, abstract modulation type spaces are instrinsically inhomogeneous, which means that certain difficulties simply cannot arise, whereas here we have both homogeneous and inhomogeneous abstract Besov spaces.…”

“…Finally, it turned out that we had to relax the assumptions on the topology in [13] slightly (see Remarks 2.2 and 2.4). This problem had been overlooked in [14].…”