THE BRASCAMP–LIEB INEQUALITIES:FINITENESS, STRUCTURE AND EXTREMALSJONATHAN BENNETT, ANTHONY CARBERY, MICHAEL CHRIST, AND TERENCE TAOAbstract. We consider the Brascamp–Lieb inequalities concerning multilin-ear integrals of products of functions in several dimensions. We give a completetreatment of the issues of finiteness of the constant, and of the existence anduniqueness of centred gaussian extremals. For arbitrary extremals we com-pletely address the issue of existence, and partly address the issue of unique-ness. We also analyse the inequalities from a structural perspective. Our maintool is a monotonicity formula for positive solutions to heat equations in linearand multilinear settings, which was first used in thi s type of setting by Carlen,Lieb, and Loss [CLL]. In that paper, the heat flow method was used to obtainthe rank one case of Lieb’s fundamental theorem concerning exhaustion bygaussians; we extend the technique to the higher rank case, giving two newproofs of the general rank case of Lieb’s theorem.1. IntroductionImportant inequalities such as the multilinear H¨older inequality, the sharp Youngconvolution inequality and the Loomis–Whitney inequality find their natural gen-eralisation in the Brascamp–Lieb inequalities, which we now describe.Definition 1.1 (Brascamp–Lieb constant). Define a Euclidean space to be a finite-dimensional real Hilbert space1H, endowed with the usual Lebesgue measure dx;for instance, Rnis a Euclidean space for any n. If m ≥ 0 is an integer, we definean m-transformation to be a tripleB := (H, (Hj)1≤j≤m, (Bj)1≤j≤m)where H, H1, . . . , Hmare Euclidean spaces and for each j, Bj: H → Hjis a lineartransformation. We refer to H as the domain of the m-transformation B. Wesay that an m-transformation is non-degenerate if all the Bjare surjective (thusHj= BjH) and the common kernel is trivial (thusTmj=1ker(Bj) = {0}). Wedefine an m-exponent to be an m-tuple p = (pj)1≤j≤m∈ Rm+of non-negative realnumbers. We define a Brascamp–Lieb datum to be a pair (B, p), where B is anThe third author was supported in part by NSF grant DMS-040126.1It is convenient to work with arbitrary finite-dimensional Hilbert spaces instead of just copiesof Rnin order to take advantage of invari ance under Hilbert space isometries, as well as suchoperation s as restriction of a Hilbert space to a subspace, or quotienting one Hilbert space byanother. In fact one could dispense with the inner product structure altogether and work withfinite-dimensional vector spaces with a Haar measure dx, but as the notation in that setting isless familiar, especially when regarding heat equations on such domains, we shall retain the innerproduct structure for notational convenience.12 JONATHAN BENNETT, ANTHONY CARBERY, MICHAEL CHRIST, AND TERENCE TAOm-transformation and p is an m-exponent for some integer m ≥ 0. When we are ina situation which involves a Brascamp–Lieb datum (B, p), it is always understoodthat the objects H, Hj, Bj, pjdenote the relevant components of this Brascamp–Lieb datum. If (B, p) is a Brascamp–Lieb datum, we define an input for (B, p) tobe an m-tuple f := (fj)1≤j≤mof nonnegative measurable functions fj: Hj→ R+such that 0 <RHjfj< ∞, and then define the quantity 0 ≤ BL(B, p; f) ≤ +∞ bythe formulaBL(B, p; f) :=RHQmj=1(fj◦ Bj)pjQmj=1(RHjfj)pj.Note that if (B, p) is non-degenerate and the fjare bounded with compact support,then BL(B, p; f) < +∞. We then define the Brascamp–Lieb constant BL(B, p) ∈(0, +∞] to be the supremum of BL(B, p; f) over all inputs f. Equivalently, BL(B, p)is the smallest constant for which the m-linear Brascamp–Lieb inequalityZHmYj=1(fj◦ Bj)pj≤ BL(B, p)mYj=1(ZHjfj)pj(1)holds for nonnegative measurable functions fj: Hj→ R+.Remark 1.2. By testing (1) on functions which are strictly positive near the origin,one can easily verify that the Brascamp–Lieb constant must be strictly positive,though it can of course be infinite. We give this definition assuming only that theinputs fjare non-negative measurable, but it is easy to se e (using Fatou’s lemma)that one could just as easily work with s trictly positive Schwartz functions with nochange in the Brascamp–Lieb constant. One can of course define BL(B, p) whenB is degenerate but it is easily seen that this constant is infinite in that case (seealso Lemma 4.1). Thus we shall often restrict our attention to non-degenerateBrascamp–Lieb data.We now give some standard examples of Brascamp–Lieb data and their associatedBrascamp–Lieb constants.Example 1.3 (H¨older’s inequality). If B is the non-degenerate m-transformationB := (H, (H)1≤j≤m, (idH)1≤j≤m)for some Euclidean space H and some m ≥ 1, where idH: H → H denotes theidentity on H, then the multilinear H¨older inequality asserts that BL(B, p) is equalto 1 when p1+ . . . + pm= 1, and is equal to +∞ otherwise.Example 1.4 (Loomis–Whitney inequality). If B is the non-degenerate n - trans-formationB := (Rn, (e⊥j)1≤j≤n, (Pj)1≤j≤n)where e1, . . . , enis the standard basis of Rn, e⊥j⊂ Rnis the orthogonal complementof ej, and Pj: Rn→ e⊥jis the orthogonal projection onto ej, then the Loomis–Whitney inequality [LW] can be interpreted as an assertion that BL(B, p) = 1 whenp = (1n−1, . . . ,1n−1), and is infinite for any other value of p. For instance, whenn = 3 this inequality asserts thatZ Z Zf(y, z)1/2g(x, z)1/2h(x, y)1/2dxdydz ≤ kfk1/2L1(R2)kgk1/2L1(R2)khk1/2L1(R2)(2)BRASCAMP–LIEB INEQUALITIES 3whenever f, g, h are non-negative measurable functions on R2. More generally,Finner [F] established multilinear inequalities of Loomis-Whitney type involving or-thogonal projections to co-ordinate subspaces.Example 1.5 (Sharp Young inequality). The sharp Young inequality ([Be], [BL])can be viewed as an assertion that if B is the non-degenerate 3-transformationB := (Rd× Rd, (Rd)1≤j≤3, (Bj)1≤j≤3)where d ≥ 1 is an integer and the maps Bj: Rd× Rd→ Rdare defined forj = 1, 2, 3 byB1(x, y) := x; B2(x, y) := y; B3(x, y) := x − ythen we haveBL(B, (p1, p2, p3)) =3Yj=1(1 − pj)1−pjppjjd/2if p1+ p2+ p3= 2 and 0 ≤ p1, p2, p3≤ 1, with BL(B, (p1, p2, p3)) = +∞ for anyother values of (p1, p2, p3). (See Example 3.8.)Example 1.6 (Geometric Brascamp–Lieb inequality). Let B be the n on-degenerat em-transformationB := (H, (Hj)1≤j≤m, (Bj)1≤j≤m)where H is a Euclidean space, H1, . . . , Hmare subspaces of H, and Bj: H →Hjare