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This paper extends a stability estimate of the Sobolev Inequality establishedby 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 SobolevInequality?" Establishing stability estimates - also known as quantitativeversions of sharp inequalities - of other forms of the Sobolev Inequality, aswell as other sharp inequalities, is an active topic. In this paper, we extendBianchi and Egnell's Stability Estimate to a Sobolev Inequality for "continuousdimensions." Bakry, Gentil, and Ledoux have recently proved a sharp extensionof 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 theBianchi-Egnell stability analysis for the Sobolev Inequality to this"continuous dimensional" generalization.

An Extension of the Bianchi-Egnell Stability Estimate to Bakry, Gentil,  and Ledoux's Generalization of the Sobolev Inequality to Continuous  Dimensions