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Logarithmic Sobolev inequalities

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In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient . 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:

where is the -norm of , with being standard Gaussian measure on 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 on is said to satisfy the log-Sobolev inequality with constant if for any smooth function f


where is the entropy functional.

Notes

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References

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  • 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