On continuity properties of semigroups in real interpolation spaces

Starting from a bi-continuous semigroup in a Banach space X (which might actually be strongly continuous), we investigate continuity properties of the semigroup that is induced in real interpolation spaces between X and the domain D(A) of the generator. Of particular interest is the case (X,D(A))$_{θ}$,$_{∞}$. We obtain topologies with respect to which the induced semigroup is bi-continuous, among them topologies induced by a variety of norms. We illustrate our results with applications to a nonlinear Schrödinger equation and to the Navier–Stokes equations on $\mathbb{R}$$^{d}$

New criteria for the H∞H^\infty-calculus and the Stokes operator on bounded Lipschitz domains

We show that the Stokes operator A on the Helmholtz space Lp (Ω) for a bounded Lipschitz domain Ω ⊂ Rd, d ≥ 3, has a bounded H ∞- calculus if |1p − 1/2| ≤ 1/2d . Our proof uses a new comparison theorem A and the Dirichlet Laplace −∆ on Lp(Ω)d, which is based on “off-diagonal” estimates of the Littlewood-Paley decompositions of A and −∆. This comparison theorem can be formulated for rather general sectorial operators and is well suited to extrapolate the H ∞-calculus from L2(U ) to the Lp(U )-scale or part of it. It also gives some information on coincidence of domains of fractional powers

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

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 exibility. We illustrate this with an application to a nonlinear Schrödinger equation. Mathematics Subject Classification (2010). 47D06, 47A60, 47D08, 35Q55

Seismic imaging with generalized Radon transforms: stability of the Bolker condition

Generalized Radon transforms are Fourier integral operators which are used, for instance, as imaging models in geophysical exploration. They appear naturally when linearizing about a known background compression wave speed. In this work we first consider a linearly increasing background velocity in two spatial dimensions. We verify the Bolker condition for the zero-offset scanning geometry and provide meaningful arguments for it to hold even if the common offset is positive. Based on this result we suggest an imaging operator for which we calculate the top order symbol in the zero-offset case to study how it maps singularities. Second, to support the usage of background models obtained from linear regression we present a stability result for the Bolker condition under perturbations of the background velocity and of the offset

Imaging with the elliptic Radon transform in 3D from an analytical and numerical perspective

The three-dimensional elliptic Radon transform (eRT) averages distributions over ellipsoids of revolution. It thus serves as a linear model in seismic imaging where one wants to recover the earth’s interior from reflected wave fields. As there is no inversion formula known for the eRT, approximate formulas have to be used. In this paper we suggest several of those, microlocally analyze their properties, provide and implement an adapted algorithm whose performance we test by diverse numerical experiments. Our previous results of [Inverse Problems, 34 (2018), 014002 & 114001] are thus generalized to three space dimensions