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Mathematical Research Letters xxx, 10001{100NN (2008)
FINITE BOUNDS FOR HOLDER-BRASCAMP-LIEB
MULTILINEAR INEQUALITIES
Jonathan Bennett, Anthony Carbery, Michael Christ,
and Terence Tao
Abstract. A criterion is established for the validity of multilinear inequalities of
a class considered by Brascamp and Lieb, generalizing well-known inequalties of
Rogers and H older, Young, and Loomis-Whitney.
1. Formulation
Consider multilinear functionals
Z mY
(1.1) ( f ;f ; ;f ) = f (‘ (y))dy1 2 m j j
nR j=1
n n nj jwhere each ‘ :R !R is a surjective linear transformation, and f :R !j j
[0; +1]. Let p ; ;p 2 [1;1]. For which m-tuples of exponents and linear1 m
transformations is
( f ;f ; ;f )1 2 m
Q(1.2) sup <1?
pkfk jf ;;f j L1 m j
The supremum is taken over all m-tuples of nonnegative Lebesgue measurable
functionsf having positive, nite norms. If n =n for every indexj then (1.2)j j
1is essentially a restatement of H older’s inequality. Other well-known particu-
lar cases include Young’s inequality for convolutions and the Loomis-Whitney
2inequality [15].
In this paper we characterize niteness of the supremum (1.2) in linear alge-
braic terms, and discuss certain variants and a generalization. The problem has
a long history, including the early work of Rogers [17] and H older [12]. In this
level of generality, the question was to our knowledge rst posed by Brascamp
and Lieb [4]. A primitive version of the problem involving Cartesian product
rather than linear algebraic structure was posed and solved by Finner [10]; see
x7 below. In the case when the dimension n of each target space equals one,j
Barthe [1] characterized (1.2). Carlen, Lieb and Loss [7] gave an alternative
Received by the editors October 14, 2008.
The third author was supported in part by NSF grant DMS-040126.
1For a discussion of the history of H older’s inequality, including its discovery by Rogers
[17], see [16].
2Loomis and Whitney considered only the special case where each f is the characteristicj
function of a set.
1000110002 Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao
characterization, closely related to ours, and an alternative proof for that case.
[7] developed an inductive analysis closely related to that of Finner, whose ar-
gument in turn relied on a slicing and induction argument employed earlier by
Loomis and Whitney [15] and Calder on [6] to treat special cases. [7] also in-
troduced a version of the key concepts of critical and subcritical subspaces, a
higher-dimensional reformulation of which is essential in our work.
An alternative line of analysis exists. Although rearrangement inequalities
such as that of Brascamp, Lieb, and Luttinger [5] do not apply when the target
spaces have dimensions greater than one, Lieb [14] nonetheless showed that the
supremum in (1.2) equals the supremum over all m-tuples of Gaussian func-
3tions, meaning those of the formf = exp( Q (y;y)) for some positive de nitej j
quadratic formQ . See [7] and references cited there for more on this approach.j
In a companion paper [3] we have given other proofs of our characterization of
(1.2), by using heat ow to continuously deform arbitrary functions f to Gaus-j
sians while increasing the ratio in (1.2). That approach extends work of Carlen,
Lieb, and Loss [7] via a method which they introduced.
We are indebted to a referee, whose careful reading and comments have im-
proved the exposition.
2. Results
Denote by dim (V ) the dimension of a vector space V , and by codim (V )W
the codimension of a subspace V W in W . It is convenient to reformulate
the problem in a more invariant fashion. Let H;H ;:::;H be Hilbert spaces1 m
of nite, positive dimensions. Each is equipped with a canonical Lebesgue mea-
dim (H)sure, by choosing orthonormal bases, thus obtaining identi cations with R ,
dim (H )jR . Let ‘ :H!H be surjective linear mappings. Let f :H !R bej j j jR Qm
nonnegative. Then ( f ; ;f ) equals f ‘ (y)dy.1 m j jH j=1
Theorem 2.1. For 1jm let H;H be Hilbert spaces of nite, positive di-j
mensions. For each indexj let‘ :H!H be surjective linear transformations,j j
and let p 2 [1;1]. Then (1.2) holds if and only ifj
X
1(2.1) dim (H) = p dim (H )jj
j
and
X
1(2.2) dim (V ) p dim (‘ (V )) for every subspace V H:jj
j
This equivalence is established by other methods in [3], Theorem 1.15.
3This situation should be contrasted with that of multilinear operators of the same general
p qjform, mapping
L toL . Whenq 1, such multilinear operators are equivalent by dualityj
to multilinear forms . This is not so for q < 1, and Gaussians are then quite far from being
extremal [8].HOLDER-BRASCAMP-LIEB MULTILINEAR INEQUALITIES 10003
Given that (2.1) holds, the hypothesis (2.2) can be equivalently restated as
X
1
(2.3) codim (V ) p codim (‘ (V )) for every subspace V H;H H jj j
j
any two of these three conditions (2.1), (2.2), (2.3) imply the third. As will
be seen through the discussion of variants below, (2.2) expresses a necessary
condition governing large-scale geometry (compare Theorem 2.5), while (2.3)
expresses a necessary condition governing small-scale geometry (compare Theo-
rem 2.2). See also the discussion of necessary conditions for Theorem 2.3.
In the rank one case, when each target space H is one-dimensional, a nec-j
essary and su cient condition for inequality (1.2) was rst obtained by Barthe
[1]. Carlen, Lieb, and Loss [7] gave a di erent proof of the inequality for the
rank one case, and a di erent characterization which is closely related to ours.
Write‘ (x) =hx;vi. It was shown in [7] that (1.2) is equivalent, in the rank onej jP P
1 1case, to having p = dim (H) and p dim (span (fv : j2 Sg))jj j j2S j
for every subset S off1; 2; ;mg; a set of indices S was said to be subcritical
if this last inequality holds, and to be critical if it holds with equality. In the
higher-rank case, we have formulated these concepts as properties of subspaces
of H, rather than of subsets off1; 2; ;mg.
To elucidate the connection between the two formulations in the rank one
case, dene W = spanfv :j2Sg, and say that a set of indices S is maximalS j
~if there is no larger set S of indices satisfying W =W . All sets of indices are~ SS
subcritical, if and only if all maximal sets of indices are subcritical. Ifj2S then
? ?codim (‘ (W )) = 1; if j2= S and S is maximal then codim (‘ (W )) = 0;H j H jj jS S
?and codim(W ) = dim (span (fv : j2 Sg)). Thus if S is maximal, then thejS Pn 1 ? ?subcriticality of S is equivalent to p codim (‘ (W )) codim(W ).H jj=1 j j S SPn 1As noted above, under the condition p = dim (H), this is equivalentj=1 jP 1
to our subcriticality condition dim (V ) p dim (‘ (V )) for the subspacejjj
?V =W .S
The necessity of (2.1) follows from scaling: if f (x ) =g ( x ) for each 2j j jj Q
+ dim (H) R then ( ff g) is proportional to , while kf k is proportionalpj j jjQ
dim (H )=pj jto . That (2.2) is also necessary will be shown inx5 in thej
course of the proof of the more general Theorem 2.3.
Remark 2.1. can be alternatively expressed as a constant multiple of theR Q
integral f d , where is a linear subspace of H and is Lebesguej j j j
measure on . More exactly, is the range of the map H 3 x7! ‘ (x).j j
Denote by the restriction to of the natural projection : H ! H .j j i i j
Then condition (2.2) can be restated as
X
1~ ~ ~(2.4) dim ( ) p dim ( ( )) for every linear subspace .jj
j10004 Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao
A local variant is also natural. Consider
Z Y
(2.5) (f ; ;f ) = f ‘ (y)dy:loc 1 m j j
fy2H:jyj 1g j
Theorem 2.2. Let H;H ;‘ , and f :H ! [0;1) be as in Theorem 2.1. Letj j j j
p 2 [1;1] for 1 j m. A necessary and su cient condition for there toj
exist C <1 such that
Y
p(2.6) (f ; ;f )C kfk jloc 1 m L
j
for all nonnegative measurable functionsf is that every subspaceV ofH satis esjP 1(2.3): codim (V ) p codim (‘ (V )).H H jj j j
This is equivalent to Theorem 8.17 of [3], proved there by a di erent method.
Certain cases of 2.2 follow from Theorem 2.1; if there exist exponents
r satisfying the hypotheses (2.1) and (2.2) of Theorem 2.1, such that r j j
p for all j, then the conclusion of Theorem 2.2 follows directly from that ofj
0Theorem 2.1 by H older’s inequality, since kfk r C kfk p . But not allj j j jL L
cases of Theorem 2.2 are subsumed in Theorem 2.1 in this way. See Remark 7.1
for examples.
The next theorem, in which some but not necessarily all coordinates of y are
constrained to a bounded set, uni es Theorems 2.1 and 2.2.
Theorem 2.3. Let H;H ; ;H be nite-dimensional Hilbert spaces and as-0 m
sume that dim (H )> 0 for allj 1. Let‘ :H!H be linear transformationsj j j
for 0jm, which are surjective for all j 1. Let p 2 [1;1] for 1jm.j
Then there exists C <1 such that
Z m mY Y
p(2.7) f ‘ (y)dyC kfk jj j j L
fy2H:j‘ (y)j 1g0 j=1 j=1
for all nonnegative Lebesgue measurable functions f if and only ifj
mX
1(2.8) dim (V ) p dim (‘ (V )) for all subspaces V kernel (‘ )j 0j
j=1
and
mX
1
(2.9) codim (V ) p codim (‘ (V )) for all subspaces V H.H H jj j
j=1
This subsumes Theorem 2.2, by taking H = H and ‘ : H ! H to be0 0
the identity; (2.8) then only applies tof0g, for which it holds automatically,
so that the only hypothesis is then (2.9). On the other hand, Theorem 2.1
is the special case ‘ 0 of Theorem 2.3. In that case kernel (‘ ) = H, so0 0
(2.8) becomes (2.2). In addition, the case V =f0g of (2.9) yields the reverse
P 1inequality dim (H) p dim (H ). Thus the hypotheses of Theorem 2.3jj jHOLDER-BRASCAMP-LIEB MULTILINEAR INEQUALITIES 10005
imply those of Theorem 2.1 when ‘ 0. The converse implication also holds,0
as was pointed out in the discussion of Theorem 2.2.
Our next result is one of several possible discrete analogues. Recall [13] that
rany nitely generated Abelian group G is isomorphic toZ H for some integer
r and some nite Abelian group H; r is uniquely determined and is called the
rank of G.
Theorem 2.4. LetG andfG : 1jmg be nitely generated Abelian groups.j
Let ’ :G!G be homomorphisms. Let p 2 [1;1]. Thenj j j
X
1(2.10) rank (H) p rank (’ (H)) for every subgroup H of Gjj
j
if and only if there exists C <1 such that
mXY Y
p(2.11) (f ’ )(y)C kfk j for all f :G ! [0;1).j j j ‘ (G ) j jj
y2Gj=1 j
pjHere the ‘ norms are de ned with respect to counting measure.
d djA special case arises whenG is isomorphic toZ ,G is isomorphic toZ forj
all j, and each ’ is represented by a matrix with integer entries. The generalj
case of Theorem 2.4 can be deduced directly from this special case, using the
disomorphisms between e.g. G andZ H for some nite group H, and the fact
pthat all ‘ norms are mutually equivalent on nite sets.
d d dA related variant is as follows. InR , for each n2Z de ne Q =fx2R :np
p 1 d 1 djx nj dg. The space ‘ (L )(R ) is the space of all f2L (R ) for whichP p 1=p( kfk ) is nite.d 1n2Z L (Q )n
Theorem 2.5. Let m 1 be a positive integer, and for each j2f1; 2; ;mg
d djlet ‘ :R !R be a surjective linear transformation. Let p 2 [1;1]. Thenj j
X
1 d(2.12) dim (V ) p dim (‘ (V )) for every subspace V Rjj
j
if and only if there exists C <1 such that
Z mY Y
(2.13) (f ‘ )(y)dyC kfk p d1j j j j j‘ (L )(R )
dR j=1 j
djfor all measurable f :R ! [0;1).j
A related result is Corollary 8.11 of [3].
Yet another variant of our results, based on Cartesian product rather than
linear algebraic or group theoretic structure, has been obtained earlier by Finner
[10]; see also [11] for a discussion of some special cases from another point of
view. Letf(X ; ) g be a nite collection of measure spaces, and let ( X;) =i i i2IQ
(X ; ) be their product. LetJ be another nite index set. For each j2J,i ii2I Q
let S be some nonempty subset of I. Let Y = X , equipped with thej j ii2Sj
associated product measure, and let :X!Y be the natural projection map.j j10006 Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao
Letf :Y ! [0;1] be measurable. To avoid trivialities, we assume throughoutj j
the discussion that I;J are nonempty and that (X) is strictly positive. De ne
Z Y
(2.14) ( f ) = f d :j j2J j j
X j2J
Denote byjj the cardinality of a nite set.
Let p 2 [1;1] for each j2J. Finner’s theorem then asserts that ifj
X
1(2.15) 1 = p for all i2Ij
j:i2Sj
then
Y
(2.16) ( f ) kfk p :j j2J j jL (Y )j
j2J
A modest generalization of Finner’s theorem is discussed inx7.
The hypothesis (2.15) can be equivalently restated as
X
1
(2.17) jKj = p jS \Kj for every subset KI,jj
j2J
P P1 1or again as the conjunction ofjIj = p jSj andjKj p jS \Kjj jj2J j j2J j
for every K I. When each space X is some Euclidean space equipped withi
Lebesgue measure, the hypotheses in this last form are precisely those of Theo-
rem 2.1, specialized to this limited class of linear mappings. The analogue of a
subspace is now a subset K I, and the analogue of criticality is (2.17); thus
(2.16) holds if and only if every subsetK is critical. This contrasts with the sit-
uations treated by Barthe [1], by Carlen, Lieb, and Loss [7], and in Theorem 2.1,
where generic subspaces will be subcritical even if critical subspaces exist.
A special case treated by Calder on [6] is as follows: Let 1 k < n. Let
nx = (x ; ;x ) be coordinates for R . For each subset Sf1; 2; 3; ;ng of1 n
k k n kcardinalityk letR be a copy ofR , with coordinates (x ) . Let :R !Ri i2S SS
be the natural projections. Then for arbitrary nonnegative measurable functions,
Z Y Y
(2.18) f ( (x))dx kf k kS S S pL (R )SnR S S
n 1where p = . A particular instance of Calder on’s theorem is the Loomis-
k 1
Whitney inequality
Z n nY Y
(2.19) f (x)dx kfk ;j j j n 1L
nR j=1 j=1
n n 1where :R !R is the mapping that forgets the j-th coordinate.jHOLDER-BRASCAMP-LIEB MULTILINEAR INEQUALITIES 10007
Two quite distinct investigations motivated our interest in these problems.
One derives from work [2] of three of us on multilinear versions of the Kakeya-
Nikodym maximal functions. A second motivator was work [9] on multilinear op-
erators with additional oscillatory factors; see Proposition 3.1 and Corollary 3.2
below.
3. An application to oscillatory integrals
n njProposition 3.1. Let m > 1. For 1 j m let ‘ : R ! R be surjec-j
n 1 ntive linear mappings. Let P : R ! R be a polynomial. Let ’2 C (R ) be0
a compactly supported, continuously di erentiable cuto function. For 2 RR Qmp n i P (x)j jand f 2 L (R ) de ne (f ; ;f ) = e f (‘ (x))’(x)dx.j 1 m n j jj=1R
1Suppose that there exist > 0 and C <1 such that for all functions f 2Lj
and all 2R
mY
(3.1) j (f ; ;f )jCjj kfk : 1 m j 1L
j=1
m nLet (p ; ;p )2 [1;1] , and suppose that for every proper subspace V R ,1 m
X
1
n n(3.2) codim (V )> p codim j (‘ (V )):R jRj
j
Then there exist > 0 and C <1, depending on (p ; ;p ), such that1 m
mY
(3.3) j (f ; ;f )jCjj kfk p 1 m j jL
j=1
p nj jfor all parameters 2R and functions f 2L (R ).j
P 1
nBy Theorem 2.2, the condition that codim n(V ) p codim (‘ (V ))jR R jj j
nfor every subspaceV R guarantees that the integral de ning (f ; ;f ) 1 mQpj pconverges absolutely for all functionsf 2L , and is bounded byC kfk j .j j Lj
The conclusion of Proposition 3.1 then follows directly from this inequality and
the hypothesis by complex interpolation.
A polynomialP is said [9] to be nondegenerate, relative to the collectionf‘gjP
of mappings, if P cannot be expressed as P = P ‘ for any ofj jj
njpolynomials P :R !R.j
Corollary 3.2. Letf‘g;P;’ be as in Proposition 3.1. Suppose that P is non-j
degenerate relative tof‘g. Suppose that either (i)n = 1 for allj,m< 2n, andj j
the familyf‘g of mappings is in general position, or (ii) n =n 1 for all j.j j
m nLet (p ; ;p )2 [1;1] and suppose that for every proper subspace V R ,1 mP 1
ncodim n(V ) > p codim j (‘ (V )). Then there exists > 0 such that forR R jj j
1 p nj jany ’2C there exists C <1 such that for all functions f 2L (R ),j0
mY
j (f ; ;f )jCjj kfk p : 1 m j jL
j=110008 Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao
Here general position means that for any subsetSf1; 2; ;mg of cardinality
jSjn,\ kernel (‘ ) has dimension nj Sj.j2S j
By Theorems 2.1 and 2.2 of [9], the hypotheses imply (3.1). Proposition 3.1
then implies the Corollary.
4. Proof of su ciency in Theorem 2.1
We begin with the proof of of the hypotheses (2.1), (2.2) for the
niteness of the supremum in (1.2). Necessity will be established in the next
section.
The next de nition is made for the purposes of the discussion of Theorem 2.1;
alternative notions of criticality are appropriate for the other theorems.
De nition 4.1. Relative to a set of exponentsfpg, a subspace V H is saidj
to be critical if
X
1(4.1) dim (V ) = p dim (‘ (V ));jj
j
to be supercritical if the right-hand side is less than dim (V ), and to be subcritical
if the right-hand side is greater than dim (V ).
In this language, the hypothesis (2.1) states thatV =H is critical relative to
fpg, while (2.2) states that no subspace of H is supercritical.j
Proof of su ciency in Theorem 2.1. The proof proceeds by induction on the di-
mension of H. When dim (H) = 1, necessarily dim (H ) = 1 for all j. ThejP 1hypothesis of the theorem in this case is that p = 1, and the conclusion isj j
p 1jsimply a restatement of H older’s inequality for functions in L (R ).
Suppose now that dim (H) > 1. There are two cases. Case 1 arises when
there exists some proper nonzero critical subspace W H. The analysis then
relies on a factorization procedure visible in the work of Calder on [6], Finner
? ?[10], and Carlen, Lieb, and Loss [7]. Express H = W W where W is
0 00 ?the orthocomplement of W , with coordinates y = (y;y )2 W W ; we will
0 0 00 00identify (y; 0) with y and (0;y ) with y . De ne U H to bej j
(4.2) U =‘ (W ):j j
0 ?~De ne ‘ = ‘j : W ! U , which is surjective. For y 2 W and x 2 Uj j W j j j
de ne
0
0(4.3) g (x ) =f (x +‘ (y )):j;y j j j j
Then
0 00 0 00 00~ ~0(4.4) f (‘ (y;y )) =f (‘ (y ) +‘ (y )) =g (‘ (y )):j j j j j j;y jHOLDER-BRASCAMP-LIEB MULTILINEAR INEQUALITIES 10009
Now
Z Z Y
0 00 00 0 ( f ; ;f ) = f (‘ (y;y ))dy dy1 m j j
?W W j
Z Z
Y
00 00 0~= g 0(‘ (y ))dy dy;j;y j
?W W j
so
Z
0~(4.5) ( f ; ;f ) = ( g 0; ;g 0)dy1 m 1;y m;y
?W
where
Z Y
00 00~ ~(4.6) ( g ; ;g ) = g (‘ (y ))dy :1 m j j
W j
We claim that
Y
~(4.7) ( g ; ;g )C kgk :1 m j pj
j
SinceW has dimension strictly less than dim (H), this follows from the induction
hypothesis provided that W is critical and no subspace V W is supercritical,
~ ~relative to the mappings‘ and exponentsp . But since‘ is the restriction of‘j j j j
toW , this condition is simply the specialization of the original hypothesis from
arbitrary subspaces of H to those subspaces contained in W , together with the
criticality of W hypothesized in Case 1. Thus
(4.8) Z Z
Y
0 0~ 0 0 0 p ( f ; ;f ) = ( g ; ;g )dy C kg k j dy:1 m 1;y m;y j;y L (U )j
? ?W W j
We will next show how this last integral is another instance of the original
?problem, with H replaced by the lower-dimensional vector space W . For z 2j
?U de nej
Z
1=pjpj(4.9) F (z ) = f (x +z ) dx ;j j j j j j
Uj
4recalling that f 0, with F (z ) = ess sup f (x +z ) if p =1. Thusj j j x2U j j j jj j
p p(4.10) kFk ? =kfk :j jj j L (H )L (U ) jj
?Denote by ? : H ! U and : H ! U the orthogonal projections.j U j jU j jj
? ?De ne L :W !U byj j
(4.11) L = ?‘ :j jU
j
4If U =f0g then the domain of F is H , and F f . If U =H then the domain of Fj j j j j j j j
isf0g, andkFk is by de nition F (0) =kfk .j p j j pj j
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