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 ${\displaystyle \nabla f}$. 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.

Notes

1. Gross 1975a
2. Gross 1975b
3. Gross 1975a

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 Mathematical Journal, 42 (3): 383–396, doi:10.1215/S0012-7094-75-04237-4