Hessian of Bellman functions and uniqueness of the Brascamp–Lieb inequality

Abstract

Abstract. Under some assumptions on the vectors a1, .., an ∈ R k and the function B : R n → R we find the sharp estimate of the expression´R k B(u1(a1 · x), . . . , un(an · x))dx in terms of´R uj (y)dy, j = 1, . . . , n. In some particular case we will show that these assumptions on B imply that there is only one Brascamp-Lieb inequality.

Generating monotone quantities for the heat equation

“…Other forms of concavity. As may be expected the Closure Properties 2.4 and 2.5 have a common generalisation involving a certain directional notion of concavity introduced in this context by Ledoux [41] and Ivanisvili-Volberg [40]. Here we adopt the terminology from [41].…”

“…Examples include the generalised geometric mean B(x) = x p1 1 · · · x pm m , where the p j ≥ 0 are such that p 1 + · · · + p m = 1, and harmonic addition B(x) = (x −1 1 + · · · + x −1 m ) −1 . In this context the function B is sometimes referred to as a Bellman function; see [40].…”

“…Similar closure properties also hold in the context of Section 3. We refer the reader to [40] and [41] for further discussion of this notion of concavity and its manifestations in geometric analysis.…”

A boundary value problem and the Ehrhard inequality

“…Without loss of generality we can assume that C is identity matrix. Indeed, we can denoteà i := C 1/2 A i for i = 1, 2, 3, andà := (à 1 ,à 2 ,à 3 ), and make change of variables x := xC −1/2 in the left hand side of (18). Thus, it is enough to show that A * A • Hess B ≤ 0 if and only if R k B(u 1 (xA 1 ), u 2 (xA 2 ), u 3 (xA 3 ))dγ k (x) ≥ (75)…”

“…In order to obtain (18) for all Borel measurable functions u j : R k j → I j we approximate pointwise almost everywhere by smooth bounded functions u n j such that Im(u n j ) ∈ I j . Finally, the Lebesgue dominated convergence theorem justifies the result (notice that all functions u 1 , u 2 , u 3 and B are uniformly bounded).…”