The Jones Strong Distribution Banach Spaces

In this note, we introduce a new class of separable Banach spaces, ${SD^p}[{\mathbb{R}^n}],\;1 \leqslant p \leqslant \infty$, which contain each $L^p$-space as a dense continuous and compact embedding. They also contain the nonabsolutely integrable functions and the space of test functions ${\mathcal{D}}[{\mathbb{R}^n}]$, as dense continuous embeddings. These spaces have the remarkable property that, for any multi-index $\alpha, \; \left\| {{D^\alpha }{\mathbf{u}}} \right\|_{SD} = \left\| {\mathbf{u}} \right\|_{SD}$, where $D$ is the distributional derivative. We call them Jones strong distribution Banach spaces because of the crucial role played by two special functions introduced in his book (see \cite{J}, page 249). After constructing the spaces, we discuss their basic properties and their relationship to ${\mathcal{D}}[{\mathbb{R}^n}]$ and ${\mathcal{D'}}[{\mathbb{R}^n}]$. As an application, we obtain new a priori bounds for the Navier-Stokes equation.