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Unique continuation for |Δu| ≤ V |∇u| and related problems.

Thomas H. Wolff — 1990

Revista Matemática Iberoamericana

Much of this paper will be concerned with the proof of the following Theorem 1. Suppose d ≥ 3, r = max {d, (3d - 4)/2}. If V ∈ Lloc r(Rd), then the differential inequality |Δu| ≤ V |∇u| has the strong unique continuation property in the following sense: If u belongs to the Sobolev space Wloc 2,p and if |Δu| ≤ V |∇u| and ...

Thin sets in nonlinear potential theory

Lars-Inge HedbergThomas H. Wolff — 1983

Annales de l'institut Fourier

Let L α q ( R D ) , α > 0 , 1 < q < , denote the space of Bessel potentials f = G α * g , g L q , with norm f α , q = g q . For α integer L α q can be identified with the Sobolev space H α , q . One can associate a potential theory to these spaces much in the same way as classical potential theory is associated to the space H 1 ; 2 , and a considerable part of the theory was carried over to this more general context around 1970. There were difficulties extending the theory of thin sets, however. By means of a new inequality, which characterizes the positive cone...

Quasi-constricted linear operators on Banach spaces

Eduard Yu. Emel'yanovManfred P. H. Wolff — 2001

Studia Mathematica

Let X be a Banach space over ℂ. The bounded linear operator T on X is called quasi-constricted if the subspace X : = x X : l i m n | | T x | | = 0 is closed and has finite codimension. We show that a power bounded linear operator T ∈ L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness χ | | · | | ( A ) < 1 for some equivalent norm ||·||₁ on X. Moreover, we characterize the essential spectral radius of an arbitrary bounded operator T by quasi-constrictedness of scalar multiples of T. Finally, we prove that every...

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