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 $$\nabla f $$. 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:



\int_{\mathbb{R}^n}\big|f(x)\big|^2 \log\big|f(x)\big| \,d\nu(x) \leq \int_{\mathbb{R}^n}\big|\nabla f(x)\big|^2 \,d\nu(x) +\|f\|_2^2\log \|f\|_2, $$

where $$ \|f\|_2$$ 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



\operatorname{Ent}_\mu(f^2) \le C \int_{\mathbb{R}^n} \big|\nabla f(x)\big|^2\,d\mu(x), $$

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