Remarks on the nonexistence of doubling measures.
New examples of weakly compact approximation in Banach spaces.
Hamilton-Jacobi flows and characterization of solutions of Aronsson equations
In this note, we verify the conjecture of Barron, Evans and Jensen [3] regarding the characterization of viscosity solutions of general Aronsson equations in terms of the properties of associated forward and backwards Hamilton-Jacobi flows. A special case of this result is analogous to the characterization of infinity harmonic functions in terms of convexity and concavity of the functions and , respectively.
Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces
We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the first order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces defined intrinsically on F. Our method employs the definitions of the function...
Weak Compactness of Multiplication Operators on Spaces of Bounded Linear Operators.
Superharmonicity of nonlinear ground states.
The objective of our note is to prove that, at least for a convex domain, the ground state of the p-Laplacian operator Δpu = div (|∇u|p-2 ∇u) is a superharmonic function, provided that 2 ≤ p ≤ ∞. The ground state of Δp is the positive solution with boundary values zero of the equation div(|∇u|p-2 ∇u) + λ |u|p-2 u = 0 in the bounded domain Ω in the n-dimensional...
Isomorphisms of Royden Type Algebras Over
Let and be the unit circle and the unit disc in the plane and let us denote by the algebra of the complex-valued continuous functions on which are traces of functions in the Sobolev class . On we define the following norm where is the harmonic extension of to . We prove that every isomorphism of the functional algebra is a quasitsymmetric change of variables on .
Representing non-weakly compact operators
For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E**/E) is defined by R(S)(x** + E) = S**x** + E(x** ∈ E**). We study mapping properties of the correspondence S → R(S), which provides a representation R of the weak Calkin algebra L(E)/W(E) (here W(E) denotes the weakly compact operators on E). Our results display strongly varying behaviour of R. For instance, there are no non-zero compact operators in Im(R) in the case of and C(0,1), but R(L(E)/W(E)) identifies isometrically with...
Weak essential spectra of multiplication operators on spaces of bounded linear operators.
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