Differential structure associated to axiomatic Sobolev spaces

Axiomatic Theory of Sobolev Spaces

We develop an axiomatic approach to the theory of Sobolev spaces on metric measure spaces and we show that this axiomatic construction covers the main known examples (Hajtasz Sobolev spaces, weighted Sobolev spaces, Upper-gradients, etc). We then introduce the notion of variational p-capacity and discuss its relation with the geometric properties of the metric space. The notions of p-parabolic ...

Holomorphic Sobolev Spaces Associated to Compact Symmetric Spaces

Using Gutzmer’s formula, due to Lassalle, we characterise the images of Soblolev spaces under the Segal-Bargmann transform on compact Riemannian symmetric spaces. We also obtain necessary and sufficient conditions on a holomorphic function to be in the image of smooth functions and distributions under the Segal-Bargmann transform.

Neumann Problems Associated to Nonhomogeneous Differential Operators in Orlicz–sobolev Spaces

— We study a nonlinear Neumann boundary value problem associated to a nonhomogeneous differential operator. Taking into account the competition between the nonlinearity and the bifurcation parameter, we establish sufficient conditions for the existence of nontrivial solutions in a related Orlicz–Sobolev space. Résumé. — On étudie un problème aux limites de Neumann associé à un opérateur différe...

Real interpolation of Sobolev spaces associated to a weight

We hereby study the interpolation property of Sobolev spaces of order 1 denoted by W 1 p,V , arising from Schrödinger operators with positive potential. We show that for 1 ≤ p1 < p < p2 < q0 with p > s0, W 1 p,V is a real interpolation space between W 1 p1,V and W 1 p2,V on some classes of manifolds and Lie groups. The constants s0, q0 depend on our hypotheses.

Sobolev Spaces