  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.
