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 spaces via hyperbolic fillings of the underlying metric space.

Study of the Singular Yamabe Problem in Some Bounded Domain of ℝn

In this paper, we extend the result of R. Mazzeo and F. Pacard in the following direction: Given Ω any bounded open regular subset of ℝto have a positive weak solution in Ω with 0 boundary data, which is singular at each x

A MULTIPLICITY RESULT FOR ELLIPTIC EQUATIONS AT CRITICAL GROWTH IN LOW DIMENSION

We consider the problem -Δu = |u|2*-2u + λu in Ω, u = 0 on ∂Ω, where Ω is an open regular subset of ℝN (N ≥ 3), [Formula: see text] is the critical Sobolev exponent and λ is a constant in ]0, λ1[ where λ1 is the first eigenvalue of -Δ. In this paper we show that, when N ≥ 4, the problem has at least [Formula: see text] (pairs of) solutions, improving a result obtained in [4] for N ≥ 6.

The Reflection Principle for Banach Space-Valued Analytic Functions

We give sufficient conditions for the continuation of an analytic function with values in a Branch space. For analytic functions taking complex numbers as values, the principle is known as the Schwarz Reflection Principle.A function defined on a domain of the complex plane with values in a Banach space X is analytic if it possesses at each point Z0 of the domain a convergent power series in z, with coefficients in X.THEOREM. Let D be a domain in the upper half-plane, and E a regular subset of the boundary of D. Suppose that E is an interval of the real axis (a,b). Let f be an analytic function defined on D, continuous up to E, taking values in a Banach space X. Let the image of D under f be Ω, and let Γ be the part of the boundary of Ω which is the image of E under f.