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 [math]\displaystyle{ \nabla f }[/math]. These inequalities were discovered and named by Leonard Gross, who established them [1][2] 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[3] proved the inequality:
- [math]\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, }[/math]
where [math]\displaystyle{ \|f\|_2 }[/math] is the [math]\displaystyle{ L^2(\nu) }[/math]-norm of [math]\displaystyle{ f }[/math], with [math]\displaystyle{ \nu }[/math] being standard Gaussian measure on [math]\displaystyle{ \mathbb{R}^n. }[/math] 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 [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ \mathbb{R}^n }[/math] is said to satisfy the log-Sobolev inequality with constant [math]\displaystyle{ C\gt 0 }[/math] if for any smooth function f
- [math]\displaystyle{ \operatorname{Ent}_\mu(f^2) \le C \int_{\mathbb{R}^n} \big|\nabla f(x)\big|^2\,d\mu(x), }[/math]
where [math]\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) }[/math] is the entropy functional.
Notes
References
- Gross, Leonard (1975a), "Logarithmic Sobolev inequalities", American Journal of Mathematics 97 (4): 1061–1083, doi:10.2307/2373688
- Gross, Leonard (1975b), "Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form", Duke Journal of Mathematics: 383–396, doi:10.1215/S0012-7094-75-04237-4, https://projecteuclid.org/euclid.dmj/1077311187
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