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

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

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.

References

• Gross, Leonard (1975a), "Logarithmic Sobolev inequalities", American Journal of Mathematics, 97 (4): 1061–1083, doi:10.2307/2373688, JSTOR 2373688
• Gross, Leonard (1975b), "Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form", Duke Journal of Mathematics, 42 (3): 383–396, doi:10.1215/S0012-7094-75-04237-4