Logarithmic Sobolev inequalities information

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,^[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:

\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.

^ Gross 1975a
^ Gross 1975b
^ Gross 1975a

[1] Gross 1975a

[2] Gross 1975b

[3] Gross 1975a

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