On some subspaces of Morrey–Sobolev spaces and boundedness of Riesz integrals

Mouhamadou Dosso¹ ; Ibrahim Fofana¹ ; Moumine Sanogo²

¹ UFR de Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan, Côte d'Ivoire
² D.E.R. de Mathématiques et Informatique Université de Bamako Bamako, Mali

Annales Polonici Mathematici, Tome 108 (2013) no. 2, pp. 133-153.

Résumé

For $1\leq q\leq \alpha \leq p\leq \infty$, $(L^q,l^p)^{\alpha}$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^q,l^p)$ and contains the Lebesgue space $L^{\alpha}$. We study the closure $(L^q,l^p)^{\alpha}_{c,0}$ in $(L^q,l^p)^{\alpha}$ of the space $\mathcal{D}$ of test functions (infinitely differentiable and with compact support in $\mathbb R^d$) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W^1((L^q,l^p)^{\alpha})$ (a subspace of a Morrey–Sobolev space, but a superspace of the classical Sobolev space $W^{1,\alpha}$) and obtain in it Sobolev inequalities and a Kondrashov–Rellich compactness theorem.

Détail

DOI : 10.4064/ap108-2-2

Keywords: leq leq alpha leq leq infty p alpha complex banach space which continuously included wiener amalgam space p contains lebesgue space alpha study closure p alpha p alpha space mathcal test functions infinitely differentiable compact support mathbb obtain norm inequalities riesz potential operators riesz transforms these spaces introduce sobolev type space p alpha subspace morrey sobolev space superspace classical sobolev space alpha obtain sobolev inequalities kondrashov rellich compactness theorem