Tomas Schonbek

One-sided compactness results for Aronszajn-Gagliardo functors

Interpolating compactness properties of operators is a long standing and important problem. In this paper, the authors consider the problem in a very general setting of Aronszajn-Gagliardo functors. In simplest terms they show that if T : A0 ! B0 is compact and T : A1 ! B1 is bounded, then T is compact on some interpolation spaces constructed in the Aronszajn-Gagliardo methods. These methods do not include the complex interpolation method, but the authors show that if the two couples are formed by Banach lattices then the theorem holds in the complex method as well. Additional results are in the context of interpolation of compactness properties in the context of N-tuples of Banach spaces. The paper includes a number of related, interesting results.

Scattering frequencies for the wave equation with a potential term

Abstract The existence of an infinite sequence of scattering frequencies for the equation □ u + qu = 0 is established, where q is a real valued potential which may assume negative values. This result generalizes some of the results obtained by Lax and Phillips in Comm. Pure Appl. Math. 22 (1969) , 737–787.

Entropy Numbers of Diagonal Operators between Vector-Valued Sequence Spaces

Upper and lower bounds are established for the entropy numbers of certain diagonal operators between Banach sequence spaces. These diagonal operators are isomorphisms between the spaces considered in the paper and weighted sequence spaces considered by Leopold so that the entropy numbers in question coincide with those considered by Leopold. The results in the paper improve the previous results in at least two ways. The estimates in the paper are ‘almost’ sharp in the sense that the upper and lower estimates differ only by logarithmic factors for a much wider range of parameters. Moreover, all the upper estimates are improvements on the previous ones, the improvement being quite significant in some cases.

Notes to a paper by C. N. Friedman

Abstract The proof of one of the main results of C. N. Friedman, Perturbations of the Schroedinger equation by potentials with small support ( J. Functional Analysis 10 (1972), 346–360) is modified basing it on a more general existence theorem. A proof of this existence theorem is provided.

Compact embeddings of Brézis-Wainger type

Let O be a bounded domain in Rn and denote by idO the restriction operator from the Besov space Bpq1+n/p(Rn) into the generalized Lipschitz space Lip(1,-a)(O). We study the sequence of entropy numbers of this operator and prove that, up to logarithmic factors, it behaves asymptotically like ek(idO) ~ k-1/p if a > max (1 + 2/p + 1/q, 1/p). Our estimates improve previous results by Edmunds and Haroske.

On Lp estimates for the wave equation

Abstract New and more elementary proofs are given of two results due to W. Littman: (1) Let n ⩾ 2, p ⩾ 2n (n − 1) . The estimate ∫∫ (¦▽u¦ p + ¦u t ¦ p ) dx dt ⩽ C ∫∫ ¦□u¦ p dx dt cannot hold for all uϵC0∞(Q), Q a cube in R n × R , some constant C. (2) Let n ⩾ 2, p ≠ 2. The estimate ∫ (¦▽(t)¦ p + ¦u t (t)¦ p ) dx ⩽ C(t) ∫ (¦▽u(0)¦ p + ¦u t (0)¦ p ) dx cannot hold for all C∞ solutions of the wave equation □u = 0 in R n x R ; all t ϵ R ; some function C: R → R .

Does Huygens' principle hold for small perturbations of the wave equation?☆

Abstract The perturbed wave equation □ u + q ( x ) u = 0 in R 3 × R with C ∞ ( R 3 ) compactly supported initial data at t = 0 is considered. It is proven that the Huygens principle does not hold for this equation if the potential is (essentially) non-negative, well-behaved at infinity and small in a suitable sense. The treatment is elementary and based on energy estimates and the positivity of the Riemann function for the wave equation in three space dimensions. The result still holds if the solution u is “small” over some space-time propagation cone. In the ease in which q has compact support, stronger results of this type for the above equation are obtained.

On a calculus for a generalized scalar operator

Abstract Let X be a Banach space; S and T bounded scalar-type operators in X . Define Δ on the space of bounded operators on X by ΔX = TX − XS if X is a bounded operator. We set up a calculus for Δ which allows us to consider f(Δ), for f a complex-valued bounded Borel measurable function on the spectrum of Δ, as an operator in the space of bounded operators whose domain is a subspace of operators which we call measure generating. This calculus is used to obtain some results on when the kernel of Δ is a complemented subspace of the space of bounded operators on X .