Optimal Sobolev embeddings on Rn
- Autores: Jan Vybíral
- Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 51, Nº 1, 2007, págs. 17-44
- Idioma: inglés
- Enlaces
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Resumen
We study Sobolev-type embeddings involving rearrangement-invariant norms. In particular, we focus on the question when such embeddings are optimal. We concentrate on the case when the functions involved are defined on $\mathbb R\sp n$. This subject has been studied before, but only on bounded domains. We first establish the equivalence of the Sobolev embedding to a new type of inequality involving two integral operators. Next, we show this inequality to be equivalent to the boundedness of a certain Hardy operator on a specific new type of cone of positive functions. This Hardy operator is then used to provide optimal domain and range rearrangement-invariant norm in the embedding inequality. Finally, the limiting case of the Sobolev embedding on Rn is studied in detail.
