A compact embedding theorem for generalized Sobolev spaces

On the characterization of the compact embedding of Sobolev spaces

For every positive regular Borel measure, possibly infinite valued, vanishing on all sets of p-capacity zero, we characterize the compactness of the embedding W (R ) ∩ L(R , μ) ↪→ L(R ) in terms of the qualitative behavior of some characteristic PDE. This question is related to the well posedness of a class of geometric inequalities involving the torsional rigidity and the spectrum of the Diric...

A Fixed Point Theorem in Generalized D-metric Spaces

A Haar-rado Type Theorem for Minimizers in Sobolev Spaces

Let ω : [0,+∞[→ [0,+∞[ be a modulus of continuity. Consider the integral functional

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.

A proof of Sobolev’s Embedding Theorem for Compact Riemannian Manifolds

Observe that H 0 (M) = L p(M). Also, Hk := H2 k is a Hilbert space under the L2-inner product. F k contains only smooth functions. In general, a sequence in F k will not converge in the H k norm to a function in F k , so we need to complete the space to have anything useful. An alternate approach would have been to start with functions in Lp rather than completing the space of smooth functions ...