In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient ${\displaystyle \nabla f}$. These inequalities were discovered and named by Leonard Gross, who established them in dimension-independent form,^[1][2] 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^[3] proved the inequality:

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

where ${\displaystyle \|f\|_{2))$ is the ${\displaystyle L^{2}(\nu )}$-norm of ${\displaystyle f}$, with ${\displaystyle \nu }$ being standard Gaussian measure on ${\displaystyle \mathbb {R} ^{n}.}$ 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 ${\displaystyle \mu }$ on ${\displaystyle \mathbb {R} ^{n))$ is said to satisfy the log-Sobolev inequality with constant ${\displaystyle C>0}$ if for any smooth function f

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

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