The role of the Hardy type inequalities in the theory of function spaces

Petru Mironescu (1, 2)

1 ICJ - Institut Camille Jordan
2 Simion Stoilow Institute of Mathematics of the Romanian Academy

We illustrate the crucial importance of the Hardy type inequalities in the study of function spaces, especially of fractional regularity. Immediate applications include Sobolev and Morrey type embeddings, and properties of the superposition operator $f\mapsto \Phi\circ f$. Another fundamental consequence is the trace theory of weighted Sobolev spaces. In turn, weighted Sobolev spaces are useful in the regularity theory of the superposition operators. More involved applications, that we present in the final section, are related to Sobolev spaces of maps with values into manifolds.

Mots clés: Hardy inequalities; Sobolev spaces; fractional Sobolev spaces; Sobolev and Morrey embeddings; Nemitzkii operators; maps with valued into manifolds; differential forms; pullback

Domaines: Analyse classique [math.CA]

Petru Mironescu. The role of the Hardy type inequalities in the theory of function spaces. 2018. ⟨hal-01819040v1⟩