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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 ...
The inequality expressing this fact has constants that do not involve the dimension of the space and, thus, the inequality holds in the setting of a Gaussian measure on an infinite-dimensional space. It is now known that logarithmic Sobolev inequalities hold for many different types of measures, not just Gaussian measures.
Gross was one of the initiators of the study of logarithmic Sobolev inequalities, which he discovered in 1967 for his work in constructive quantum field theory and published later in two foundational papers [11] [12] in which he established these inequalities for the Bosonic and Fermionic cases. The inequalities were named by Gross, who ...
In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the -norms of different weak derivatives of a function through an interpolation inequality.
where the equality on the left represents integration by parts, and the inequality to the right is a Sobolev inequality [citation needed]. In the latter, equality is attained for the function sin π x {\displaystyle \sin \,\pi x} , implying that the constant − π 2 {\displaystyle -\pi ^{2}} is the best possible.
Hardy–Littlewood inequality; Hardy–Littlewood–Sobolev inequality; Harnack's inequality; Hausdorff–Young inequality; Hermite–Hadamard inequality; Hilbert's inequality; Hölder's inequality; Jackson's inequality; Jensen's inequality; Khabibullin's conjecture on integral inequalities; Kantorovich inequality; Karamata's inequality; Korn's ...
President Donald Trump said he will provide details Thursday of what he and Elon Musk's Department of Government Efficiency (DOGE) claim are billions of dollars of wasteful and fraudulent ...
For example, the approach based on "upper gradients" leads to Newtonian-Sobolev space of functions. Thus, it makes sense to say that a space "supports a Poincare inequality". It turns out that whether a space supports any Poincare inequality and if so, the critical exponent for which it does, is tied closely to the geometry of the space.