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 [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:

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

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.

Notes

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

References

• Gross, Leonard (1975a), "Logarithmic Sobolev inequalities" (PDF), American Journal of Mathematics: 1061–1083
• Gross, Leonard (1975b), "Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form", Duke Journal of Mathematics: 383–396