This talk concerns the asymptotic behavior of negative Sobolev norms as the differentiability order tends to zero. For $p\in(1,\infty)$ and $\alpha\in(0,1)$, the negative Sobolev space $W^{-\alpha,p}(\mathbb{R}^n)$ is realized through heat-kernel convolutions. Under a mild boundedness assumption, we establish that for functions belonging to the union of $W^{-\alpha,p}(\mathbb{R}^n)$,\[\lim_{\alpha\to 0^+} \alpha \|f\|_{W^{-\alpha,p}}^p = \frac{2}{p}\|f\|_{L^p}^p.\]The proof uses heat-kernel regularization, monotonicity, weak compactness in reflexive $L^p$ spaces, and an Abelian--Tauberian argument. The result is further extended to a general measure space equipped with a family of sub-additive, bounded, and continuous operators. These results complement the classical Bourgain--Brezis--Mironescu and Maz'ya--Shaposhnikova limiting formulas for positive fractional Sobolev norms.
