  This paper extends a stability estimate of the Sobolev Inequality established
by G. Bianchi and H. Egnell in their paper "A note on the Sobolev Inequality."
Bianchi and Egnell's Stability Estimate answers the question raised by H.
Brezis and E. H. Lieb: "Is there a natural way to bound $\| \nabla \varphi
\|_2^2 - C_N^2 \| \varphi \|_\frac{2N}{N-2}^2$ from below in terms of the
'distance' of $\varphi$ from the manifold of optimizers in the Sobolev
Inequality?" Establishing stability estimates - also known as quantitative
versions of sharp inequalities - of other forms of the Sobolev Inequality, as
well as other sharp inequalities, is an active topic. In this paper, we extend
Bianchi and Egnell's Stability Estimate to a Sobolev Inequality for "continuous
dimensions." Bakry, Gentil, and Ledoux have recently proved a sharp extension
of the Sobolev Inequality for functions on $\mathbb{R}_+ \times \mathbb{R}^n$,
which can be considered as an extension to "continuous dimensions." V. H.
Nguyen determined all cases of equality. The present paper extends the
Bianchi-Egnell stability analysis for the Sobolev Inequality to this
"continuous dimensional" generalization.
