On some subspaces of Morrey–Sobolev spaces and boundedness of Riesz integrals
Volume 108 / 2013
Abstract
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.
