Logarithmic Sobolev inequalities

In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient $\mathrm{\nabla }f$. These inequalities were discovered and named by Leonard Gross, who established them in dimension-independent form, in the context of constructive quantum field theory. Similar results were discovered by other mathematicians before and many variations on such inequalities are known. Gross proved the inequality:

$\underset{{\mathbb{ℝ}}^n}{\int }|f(x){|}^2\log |f(x)|d\nu (x)\le \underset{{\mathbb{ℝ}}^n}{\int }|\mathrm{\nabla }f(x){|}^{2}d\nu (x)+\Vert f{\Vert }_2^2\log \Vert f{\Vert }_2,$

where $\Vert f{\Vert }_2$ is the $L^2(\nu )$-norm of $f$, with $\nu$ being standard Gaussian measure on ${\mathbb{ℝ}}^n.$ Unlike classical Sobolev inequalities, Gross's log-Sobolev inequality does not have any dimension-dependent constant, which makes it applicable in the infinite-dimensional limit.

In particular, a probability measure $\mu$ on ${\mathbb{ℝ}}^n$ is said to satisfy the log-Sobolev inequality with constant $C>0$ if for any smooth function f

${\mathrm{Ent}}_{\mu }(f^2)\le C\underset{{\mathbb{ℝ}}^n}{\int }|\mathrm{\nabla }f(x){|}^{2}d\mu (x),$

where ${\mathrm{Ent}}_{\mu }(f^2)=\underset{{\mathbb{ℝ}}^n}{\int }{f}^2\log \frac{f^2}{\underset{{\mathbb{ℝ}}^n}{\int }{f}^{2}d\mu (x)}d\mu (x)$ is the entropy functional.

References

• Gross, Leonard (1975a), "Logarithmic Sobolev inequalities", American Journal of Mathematics 97 (4): 1061–1083, doi:10.2307/2373688 
• Gross, Leonard (1975b), "Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form", Duke Journal of Mathematics: 383–396, doi:10.1215/S0012-7094-75-04237-4, https://projecteuclid.org/euclid.dmj/1077311187