Logarithmic Sobolev inequalities explained

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 _{R ^{n}} | f(x) | ^{2} log | f(x) | d\nu(x) \leq \int _{R ^{n}} | \nabla f(x) | ^{2} d\nu(x) + \|f\| ^{2log} _{2} \|f\| _{2,}

where

\|f\| _{2} is the L ^{2(\nu)} -norm of f , with \nu being standard Gaussian measure on 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 R ^{n} is said to satisfy the log-Sobolev inequality with constant C> 0 if for any smooth function f\operatorname{Ent} ^{2 )} _{\mu(f} \le C \int _{R ^{n}} | \nabla f(x) | ^{2d\mu(x),}

where

\operatorname{Ent} ^{2 )} _{\mu(f} = \int _{R ^{n}} f ^{2log (f ^{2})/(\int _{R ^{n}} f ^{2d\mu(x)})} d\mu(x) is the entropy functional