Tepper L Gill
The Jones Strong Distribution Banach Spaces
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ABSTRACT.  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 $lpha, \; \left\| {{D^lpha }{\mathbf{u}}} 
ight\|_{SD} = \left\| {\mathbf{u}} 
ight\|_{SD}$, 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