aa r X i v : . [ m a t h . C A ] F e b A Nonlinear Variant of Ball’s Inequality
Jennifer DuncanSchool of MathematicsUniversity of BirminghamFebruary 3, 2021
Abstract
We adapt a recent induction-on-scales argument of Bennett, Bez, Buschenhenke, Cowling,and Flock to establish a global near-monotonicity statement for the nonlinear Brascamp–Liebfunctional under a certain heat-flow, from which we establish some finiteness and stability resultsfor the associated nonlinear Brascamp–Lieb inequality.
For each j P t , ..., m u , let L j : R n Ñ R n j be a linear surjection and p j P r , s . The Brascamp–Liebinequality associated with the pair p L , p q : “ pp L j q mj “ , p p j q mj “ q is the following: ż R n m ź j “ p f j ˝ L j q p j ď C m ź j “ ˆż R nj f j ˙ p j @ f j P L p R n j q , f j ě . (1)Using the notation of [10], we refer to the pair p L , p q as a Brascamp–Lieb datum (we shall oftenabuse this terminology and refer to L as a datum as well, given that p may be regarded as fixed forthe entirety of this paper). We define the Brascamp–Lieb constant, BL( L , p ), to be the infimum overall constants C P p , for which the above inequality holds. This constant may be equivalentlydefined via the Brascamp–Lieb functional, which is itself defined as, for a given m -tuple of non–negative functions f “ p f j q mj “ P À mj “ L p R n j q such that ş R nj f j ą p L , p ; f q : “ ş R n ś mj “ p f j ˝ L j q p j ś mj “ `ş R nj f j ˘ p j . (2)We may then write BL p L , p q “ sup f BL p L , p ; f q . The Brascamp–Lieb inequalities are a naturalgeneralisation of many classical multilinear inequalities that commonly arise in analysis, examplesof which include H¨older’s inequality, Young’s convolution inequality, and the Loomis-Whitney in-equality. Connections with the Brascamp-Lieb inequalities have been found in a broad range ofareas, including convex geometry [1, 2], kinetic theory [16], and computer science [17], to namea few. Nonlinear Brascamp–Lieb inequalities are a relatively recent further generalisation, wherewe relax the linearity requirement on these defining maps. While most of the central questionsin the linear theory have been broadly addressed, in the nonlinear setting many interesting gen-eral questions remain open, and progress on global inequalities is still at the early stage of theanalysis of certain special cases. This paper represents a small step towards a general theory ofglobal nonlinear Brascamp–Lieb inequalities, in that we prove, under fairly general assumptions,a nonlinear variant of a powerful inequality due to Keith Ball that underpins much of the theoryof linear Brascamp-Lieb inequalities, from which one can establish that the finiteness of nonlinearBrascamp–Lieb inequalities is a property that is stable under L perturbations of the underlyingmaps; these results are stated in Section 2.2. We begin with a necessary background in the linearsetting. 1 .1 The Linear Inequalities: Finiteness and Extremisibility Naturally, the central problem in linear Brascamp–Lieb theory was finding the necessary and suf-ficient conditions for BL( L , p ) to be finite. We begin with the observation that, by an elementaryscaling argument, the following is a necessary condition for finiteness: m ÿ j “ p j n j “ n. (3)It was first proved by Barthe, later reproved by Carlen, Lieb, and Loss in [16], that this conditiontogether with a spanning condition on the L j forms a necessary and sufficient condition for finitenessin the rank-one case, i.e. when n j “ j P t , ..., m u [1].Another important and related question is that of finding necessary and sufficient conditions forthe extremisability of Brascamp–Lieb inequalities, and to find a characterisation of the extremisersshould they exist. As we shall discuss later on, if a Brascamp–Lieb inequality admits an extremiser,then it must admit a gaussian extremiser, a fact foreshadowed by the following celebrated theoremdue to Lieb. Theorem 1.1 (Lieb’s Theorem [22]) . Given any Brascamp–Lieb datum p L , p q , the set of gaussianinputs G , G : “ p G j q mj “ : G j p x q : “ exp p´ π x A j x, x yq , A j P R n j ˆ n j is positive definite @ j P t , ..., m u ( exhausts the associated Brascamp–Lieb inequality, that is to say sup G P G BL p L , p ; G q “ BL p L , p q . Only needing to test on gaussians makes the problem of establishing whether or not BL( L , p )is finite significantly more tractable, and upon this result, necessary and sufficient conditions forfiniteness were proved by Bennett, Carbery, Christ, and Tao in [10]. They also establish a necessaryand sufficient condition for gaussian-extremisability, and a necessary and sufficient condition forsuch a gaussian extremiser to be unique, but before we give a statement of their theorem, we shallneed to state some preliminary definitions. Definition 1.2.
Let p L , p q be a Brascamp–Lieb datum. We say that the datum p L , p q is feasible ifit satisfies the scaling condition (3), and that for all subspaces V ď R n , dim p V q ď m ÿ j “ p j dim p L j V q . (4) Definition 1.3.
Given p L , p q , we say that a proper non-trivial subspace V ď R n is critical if itsatisfies (4) with equality, and that the datum p L , p q is simple if it admits no critical subspaces. The significance of these subspaces is that if we were to restrict the domains of the surjections L j to V , and their codomains to L j V , then we would obtain a restricted datum that is itself feasible.As a result, the Brascamp–Lieb datum, in the presence of critical subspaces, exhibits a certainsplitting phenomenon. In this sense, simple data, similarly to the role that simple groups playin group theory, are treated as fundamental objects from which one builds a much larger class ofBrascamp–Lieb data. An in-depth discussion of such structural considerations can be found in [10]and [27]. Theorem 1.4 (Bennett, Carbery, Christ, Tao (2007) [10]) . A Brascamp–Lieb datum p L , p q isfinite if and only if it is feasible, and is gaussian-extremisable if it is simple. Moreover, gaussianextremisers for simple data are unique up to rescaling. Theorem 1.5 (Bennett, Bez, Cowling, Flock (2017) [7]) . The Brascamp–Lieb constant BL p L , p q is continuous, but not differentiable, on the set of L for which p L , p q is feasible. Fixing dimensions and exponents, let G : L ÞÑ G p L q be the unique (up to normalisation) mapthat sends a simple Brascamp–Lieb datum L to an m -tuple of gaussians such that BL p L , p ; G p L qq “ BL p L , p q . A natural question to then ask is how smooth G is, as whatever smoothness this maphas must then transfer over to the Brascamp–Lieb constant. Theorem 1.6 (Valdimarsson (2010) [27]) . G is smooth on the set of simple data, hence theBrascamp–Lieb constant is also smooth on this set.
By Lieb’s theorem, we know that for any δ ą δ -near extremiser g , i.e. BL p L , p ; g q ě p ´ δ q BL p L , p q , however we do not have any a prioriinformation about its eccentricity or whether or not this choice may be made smoothly in L . Aswe shall be dealing with data that is in general non-extremisible, we shall find ourselves in need ofmore quantified regularity statements for families of near-extremisers. That is to say, for a given δ ą
0, can we construct a map g δ : L ÞÑ g δ p L q that both sends a given feasible Brascamp-Liebdatum to a δ -near gaussian extremiser and satisfies good W , bounds in L , and how best can wecontrol how these bounds blow up as δ approaches 0? Answering this question shall be the contentof the forthcoming Theorem 2.7. Our brief exposition of the linear theory now complete, in thenext section we turn our attention to the main focus of this paper. It is perfectly natural to ask the question of nonlinear variants, where the linear surjections L j arereplaced with submersions B j : M Ñ M j , and M and M j are Riemannian manifolds of dimensions n and n j respectively, equipped with the induced volume measure. Given an m -tuple of exponents p “ p p j q mj “ , we shall consider the corresponding inequality: ż M m ź j “ p f j ˝ B j q p j ď C m ź j “ ˜ż M j f j ¸ p j . We shall refer to the pair p B , p q as a nonlinear Brascamp–Lieb datum. Inequalities of this typearise quite naturally in PDE and restriction theory contexts, as evidenced in [3, 4, 7, 11, 20]. Earlyresults of significance include a Sobolev variant of the nonlinear Brascamp–Lieb inequality [8] and anonlinear perturbation of the Loomis-Whitney inequality [11], which was later improved by Carbery,H¨anninen, and Valdimarsson via multilinear factorisation [14]. Significant progress in this area wasmade recently by Bennett, Bez, Buschenhenke, Cowling, and Flock in [6], where they employ atight induction-on-scales that utilises techniques from convex optimisation to prove the followingvery general local nonlinear Brascamp–Lieb inequality.3 heorem 1.7 (Local Nonlinear Brascamp–Lieb Inequality (2018) [6]) . Let ε ą , and supposethat p B , p q is a C nonlinear Brascamp–Lieb datum defined over some neighbourhood r U of a point x P R n . There exists a neighbourhood U Ă r U of x such that the following inequality holds for all f j P L p R n j q : ż U m ź j “ f j ˝ B j p x q p j dx ď p ` ε q BL p dB p x q , p q m ź j “ ˆż R nj f j ˙ p j . (5)A common feature of the inequalities mentioned is that they are local, in the sense that ourdomain of integration on the left-hand side must be taken to be small, although of course theseresults generalise in an elementary manner to compact domains, at the cost of sharpness in theconstant. Sharp results on the sphere were established by Carlen, Lieb and Loss in [16], generalisedby Bramati in [13], who has also established some results for compact homogeneous spaces [12].Some interesting results for compact domains depart from the usual transversality assumptionsof the aforementioned authors, and instead hinge on a curvature condition, including L p boundsfor certain multilinear Radon-like transforms, explored by Tao and Wright in the bilinear settingin [26] then generalised by Stovall to the fully multilinear setting in [24], although we shall notinvestigate curvature considerations of this type. There has been some progress in the direction ofglobal Brascamp–Lieb inequalities, including inequalities for certain homogeneous data of degreeone [9], a global weighted nonlinear Loomis-Whitney inequality in R [20], and some results in theabstract contexts of integration spaces [15]. Our global results may be viewed as a general nonlinearalternative to the heat-flow monotonicity properties of the linear Brascamp-Lieb inequality, whichwe shall discuss in the next section. Generally speaking, the typical manner in which a ‘heat-flow’ argument in this context works isthat if one wishes to prove an inequality of the form A p f q ď B p f q for all f in some class offunctions, where A and B are functionals on this class, it is enough to prove that there exists asemigroup S t acting on this class such that A p f q ď lim inf t Ñ A p S t f q , A p S t f q is increasing in t , and thatlim sup t Ñ8 A p S t f q ď B p f q . Carlen, Lieb and Loss exploit heat-flow monotonicity to great effect intheir proof of the rank-one case of the Brascamp–Lieb inequality [16], generalisations of which canbe found in [10,12,13], and a systematic study of the generation of monotone quantities for the heatequation can be found in [5]. Heat-flow techniques were also used to great effect by in a variety ofrelated multilinear settings by Tao in [25]. Methods that exploit heat-flow monotonicity are oftenreferred to as ‘semigroup interpolation’ methods, see an article by Ledoux for further reading [21].An interesting manifestation of heat-flow monotonicity for the Brascamp–Lieb functional arises fromthe following inequality due to Keith Ball. Lemma 1.8 (Ball’s inequality [2]) . Let p L , p q be a Brascamp–Lieb datum and let f “ p f j q mj “ , g “p g j q mj “ P À mj “ L p R n j q . Given x P R n , we define h x : “ p f j p¨q g j p L j p x q ´ ¨qq mj “ . For all choices ofinputs f and g , the following inequality holds. BL p L , p ; f q BL p L , p ; g q ď sup x P R n BL p L , p ; h x q BL p L , p ; f ˚ g q If we assume that BL p L , p q ă 8 and that g is an extremising input, i.e. BL p L , p ; g q “ BL p L , p q ,4hen this inequality implies the following two statements:BL p L , p ; f q ď BL p L , p ; f ˚ g q (6)BL p L , p ; f q ď sup x P R n BL p L , p ; h x q (7)An important consequence is that, if we further suppose that f is an extremiser, then (6) impliesthat the set of extremisers is closed under convolution. This, along with the the topological closureof extremisers, guarantees the existence of a gaussian extremiser, as we may convolve a givenextremiser with itself iteratively and apply the central limit theorem to the resulting sequence tofind that the limiting extremiser must be gaussian [10].Suppose that g is a gaussian extremiser, and define its associated family of rescalings as g τ : “p τ ´ n j { g j p τ ´ { x qq mj “ where τ ą
0. By the scale-invariance of the Brascamp–Lieb inequality, each g τ is also an extremiser, hence if we now substitute g τ into (6) then we see that (6) then statesthat the Brascamp–Lieb functional is monotone increasing as the inputs flow under the followingdiffusion equation: B t f j “ ∇ ¨ p A ´ j ∇ f j q where A j is the positive definite matrix such that g j : “ exp p´ π x A j x, x yq . We shall now run thescheme outlined at the beginning of this subsection to derive the sharp finiteness and extremisabilityof the Brascamp–Lieb inequality from (6), as was carried out in a special case in [10]. Lemma 1.9.
Let p L , p q be a Brascamp–Lieb datum and assume that (3) holds.Let g p x q : “ p g j p x qq mj “ : “ p exp p´ π x A j x, x yqq mj “ for all x P R n , where A j P R n j ˆ n j is positivedefinite. If (6) holds for all inputs f , then BL p L , p q ă 8 , furthermore g extremises the Brascamp–Lieb functional BL p L , p ; ¨q .Proof. By homogeneity and scale-invariance of the Brascamp–Lieb functional, we may assume with-out loss of generality that ş R nj f j “ ş R nj g j “ j P t , ..., m u . Given τ ą
0, we definean anisotropic heat kernel g τ as follows: g τ p x q : “ p g j,τ p x qq mj “ “ p τ ´ n j exp p´ πτ ´ { x A j x, x yqq mj “ . Observe that for all τ ą τ n j { f j ˚ g τ,j p L j p τ { x qq “ ż R nj f j p z q exp p´ πτ ´ x A j p τ { L j p x q ´ z q , τ { L j p x q ´ z yq dz “ ż R nj f j p z q exp p´ π | A { j L j p x q| ` πτ ´ { x A j L j p x q , z y ´ πτ ´ | z | q dz ÝÑ τ Ñ8 exp p´ π | A { j L j p x q| q ż R nj f j p z q dz “ g j ˝ L j p x q . p L , p ; f q “ ż R n m ź j “ f j ˝ L j p x q p j dx ď ż R n m ź j “ p f j ˚ g j,τ q ˝ L j p x q p j dx “ ż R n m ź j “ p f j ˚ g j,τ q ˝ L j p τ x q p j τ n { dx “ ż R n m ź j “ τ p j n j { p f j ˚ g j,τ q ˝ L j p τ x q p j dx ÝÑ τ Ñ8 ż R n m ź j “ g j ˝ L j p x q p j dx “ BL p L , p ; g q . Taking the supremum in all f with unit mass, then implies that BL p L , p q “ sup f BL p L , p ; f q ď BL p L , p ; g q ď BL p L , p q , hence g is an extremiser, from which we may read off the sharp constantBL p L , p q “ det ´ř mj “ L ˚ j A j L j ¯ ´ { .Observing this equivalence between heat-flow monotonicity and (gaussian) extremisability, it isthen natural to consider whether or not, for some suitable choice of nonlinear Brascamp-Lieb datum,there exists some variable coefficient heat-flow for which the associated nonlinear Brascamp–Liebfunctional is monotone, and if so whether or not this would imply that the inequality holds withfinite constant. Indeed, this is the approach that was taken in both [12] and [16] to prove nonlinearBrascamp-Lieb inequalities in certain geometrically rigid settings, so it is natural to suppose thatsuch a method could work in a more general context. Admittedly, we do not prove a such a perfectmonotonicity statement, or even a near-monotonicity statement that holds for large times, howeverone may certainly view our result as evidence for such a conjecture.The inequalities (6) and (7) express an amenability of the linear Brascamp–Lieb functional totwo distinct processes, the former being smoothing via heat-flow and the latter being localisationvia gaussian extremisers, as we may think of h xj as an essentially truncated version of f j , whoseessential support is contained within a ball centred at L j p x q . The proof strategy of [6] was to finda nonlinear version of (7) that would serve as a way to bound the left-hand side of (5) above by asupremum of similar integrals over smaller domains, so that if used recursively this would form theengine of an induction-on-scales argument. In this paper we find a corresponding nonlinear versionof (6), although admittedly we only establish heat-flow near-monotonicity for small times. At itscore it is still an induction argument, where we tightly bound the possible error between times thatare close to one another so that when we string these inequalities together we are left with an errorthat is still well-controlled.This paper was funded by a grant from the EPSRC, and will form part of the authors PhDthesis. She would like to thank her supervisor Jonathan Bennett for his invaluable guidance andpatience, without which this work could not have been produced. The author would also like tothank Alessio Martini for his help regarding some of the more geometric aspects of the paper.6 Main Section
In this paper, we shall consider fixed complete Riemannian manifolds (without boundary) M , M ,..., M m of dimensions n , n ,..., n m . We shall refer to the exponential map based at a point x ona manifold e x : T x N Ñ N . The injectivity radius of a point x P N is the largest number ρ x ą e x restricts to a diffeomorphism on the ball of radius ρ x around 0 P T x N . We shallassume that the manifolds we consider have bounded geometry, by which we mean that they haveinjectivity radii uniformly bounded below, by a number ρ ą x P M of radius r ą M by U r p x q ,and refer to a ball centred at a point v P T x M of radius r ą V r p v q (the tangent space that thisball belongs to should always be clear from context, if it is not stated explicitly). We shall considersubmersions B j : M Ñ M j p j P t , ..., m uq that may be viewed as fixed for the entirety, and areassumed to have at least L bounded derivative maps. Noting this, it shall prove useful to refer to aball centred at z P M j of radius r } dB j } L simply by U r,j p z q , and similarly a ball centred at w P T z M j of radius r ą V r,j p w q , simply for the technical reason that then Ş mj “ dB j p x q ´ p V r,j p qq Ă V r p q ,a property that shall prove to be useful later on. Similarly to the linear case, we refer to the pair p B , p q as a nonlinear Brascamp-Lieb datum. We shall also make use of a fixed parameter γ P p , q close to 1, The exact choice of value here is not particularly important, the reader may even take γ to be 0 .
9, say, however we refrain from doing this for the sakes of clarity and book-keeping.Given the importance of gaussians in the context of Brascamp–Lieb inequalities, it shall at timesbe useful to tailor our notation specifically for them. Let p L , p q be a Brascamp–Lieb datum andlet G “ p G j q mj “ be an m -tuple of gaussians of the form G j p x q : “ exp p´ π x A j x, x yq , where each A j P R n j ˆ n j is a positive-definite matrix. We then define BL g p L , p ; A q : “ BL p L , p ; G q . We shallrefer to such an m -tuple of symmetric positive definite matrices as a gaussian input . Of course,since integrals of gaussians may be computed in terms of their underlying matrices, we have accessto the following explicit formula:BL g p L , p ; A q “ ś mj “ det p A j q p j { det ´ř mj “ p j L ˚ j A j L j ¯ { We shall always use a single bar to denote a finite dimensional norm, usually a 2-norm, and doublebars to denote an infinite dimensional norm, which we shall always specify with a subscript. In thecase where we are taking a norm of a matrix, we shall assume that this is the induced 2-norm unlessstated otherwise. Since we may assume that the underlying dimensions, exponents, manifolds, andthe nonlinear Brascamp–Lieb datum p B , p q are all fixed throughout, we shall use the relation A À B to denote that there exists a constant C ą A ď CB ,and the relation A » B to denote that A À B À A . Similarly, if y is some variable, Q is a normedspace valued function of y , and f is a real valued function of y , then we shall use the the notation Q p y q “ O p f p y qq to denote that } Q p y q} À f p y q . Before we state our nonlinear version of (6), we must first preliminarily define our ‘heat-flow’. Theconstruction thereof is rather involved, however the resulting flow operator H x,τ,j may nonetheless7e written essentially as a convolution with a gaussian kernel G x,τ,j : T B j p x q M j Ñ R , the keyproperties of which we now state as a proposition. Proposition 2.1.
Suppose that p B , p q is a nonlinear Brascamp-Lieb datum such that each B j : M Ñ M j is C and that there exists C ą such that } dB } W , , } BL p dB , p q} L ď C . Then, thereexists an ε ą such that,for τ ą sufficiently small, there exists a smooth family of gaussian inputs G x,τ : “ p G x,τ,j q mj “ parametrised by x P M satisfying the following properties:1. Each gaussian G x,τ,j is of unit mass and is defined by a corresponding τ -dependent positive def-inite matrix A τ,j p x q , in the sense that G x,τ,j p z q : “ τ ´ n j det p A τ,j p x qq { exp p´ πτ ´ x A τ,j p x q z, z yq .2. G x,τ is a τ ε -near extremiser for the datum p dB p x q , p q .3. } A τ,j } W , p M q , } det A τ,j } W , p M q ď τ ´ ε for all j P t , ..., m u . The construction of this G x,τ,j is carried out in detail in Section 2.4, whence (1) automaticallyfollows, see the end of Section 2.5 and the remark after Lemma 3.1 for the proof of properties (2)and (3) respectively. We may now define the corresponding flow operator, wherein we include sometruncation to allow us to map locally to the tangent space on which G x,τ,j is defined. H x,τ,j : L p M j q Ñ L p U ρ ´ τ γ p B j p x qqq H x,τ,j f j p z q : “ ż U τγ,j p z q f j p w q G x,τ,j p e ´ B j p x q p z q ´ e ´ B j p x q p w qq dw We now state our near-monotonicity result, which is the main theorem of this paper.
Theorem 2.2 (Nonlinear Ball’s Inequality) . Suppose that p B , p q is a nonlinear Brascamp-Liebdatum such that each B j : M Ñ M j is C and that there exists C ą such that } dB } W , , } BL p dB , p q} L ď C . Then, there exists a β ą such that for τ ą sufficiently small,for all non-negative f j P L p M j q , ż M m ź j “ f j ˝ B j p x q p j dx ď p ` τ β q ż M m ź j “ H x,τ,j f j ˝ B j p x q p j dx. (8)Of course, in the euclidean case we may identify our domain with each of the tangent spaces viatranslation, and so (8) then takes the following more familiar form: ż R n m ź j “ f j ˝ B j p x q p j dx ď p ` τ β q ż R n m ź j “ f j ˚ p G x,τ,j χ U τγ,j p q q ˝ B j p x q p j dx. which of course implies a non-truncated, genuine heat-flow near-monotonicity statement. ż R n m ź j “ f j ˝ B j p x q p j dx ď p ` τ β q ż R n m ź j “ f j ˚ G x,τ,j ˝ B j p x q p j dx. The main upshot of Theorem 2.2 is that one may use the local-constancy of H x,τ,j f j to perturbthe argument in the right-hand side of (8), either at small scales as in Corollary 2.3, which yieldsa slightly improved version of the local nonlinear Brascamp–Lieb inequality first proved in [6], andat large scales as in Corollary 2.4, which states that finiteness is stable under L perturbations.8 orollary 2.3. Let p B , p q be a nonlinear Brascamp–Lieb datum satisfying the same conditionsas in Theorem 2.2, then there exists a number β ą such that for each x P M and all τ ą sufficiently small, ż U τ p x q m ź j “ f j ˝ B j p x q p j dx ď p ` τ β q BL p dB r x s , p q m ź j “ ˜ż M j f j ¸ p j Corollary 2.4.
Suppose that B j , r B j : R n Ñ R n j for all j P t , ...m u and p B , p q is a C nonlineardatum satisfying the conditions of Theorem 2.2, that the inequality associated with p r B , p q holds withfinite constant, and that } B ´ r B } L ă 8 , then the inequality associated with p B , p q holds with finiteconstant. It would be reasonable to suggest that a similar result would hold in the non-euclidean setting,however, due to certain technical geometric considerations, this appears falls beyond the scope ofthis paper.
Let C p s, t q denote the best constant C P p , for the following inequality. ż M m ź j “ H x,s,j f j ˝ B j p x q p j dx ď C p s, t q ż M m ź j “ H x,t,j f j ˝ B j p x q p j dx (9)It is easy to see that C p s, t q enjoys the submultiplicative property C p r, t q ď C p r, s q C p s, t q . We claimthat this together with the following proposition is sufficient to prove Theorem 2.2. Proposition 2.5.
There exist β, ν ą such that, for all τ P p , ν q , C p τ, { τ q ď p ` τ β q .Proof of Theorem 2.2 given Proposition 2.5. Setting τ “ τ , define the geometric sequence τ k : “ ´ k { τ and let K P N . We can split the constant C p τ K , τ q into pieces that can be dealt with byProposition 2.5. C p τ K , τ q ď C p τ K , τ K ´ q C p τ K ´ , τ qď C p τ K , τ K ´ q C p τ K ´ , τ K ´ q C p τ K ´ , τ qď ... ď K ź k “ C p τ k , τ k ´ q ď K ź k “ p ` τ βk q Taking logarithms of the above inequality, we obtain thatlog p C p τ K , τ qq ď K ÿ k “ log p ` τ βk qď ÿ k “ τ βk “ τ β β { ´ τ accordingly smaller if necessary, that C p τ K , τ q ď exp p τ β β ´ q ďp ` τ β { q . For each j P t , ..., m u , let f j P C p M j q be a non-negative function. By the forthcoming9emma 3.4, we know that H x,τ,j f j ˝ B j p x q Ñ f j ˝ B j p x q as τ Ñ x P M , hence we may applyFatou’s lemma and consider (9) with s “ τ K and t “ τ , taking the limit as K Ñ 8 . ż M m ź j “ f j ˝ B j p x q p j dx ď lim inf K Ñ8 ż M m ź j “ H x,τ K ,j f j ˝ B j p x q p j dx ď lim inf K Ñ8 C p τ K , τ q ż M m ź j “ H x,τ,j f j ˝ B j p x q p j dx ď p ` τ β { q ż M m ź j “ H x,τ,j f j ˝ B j p x q p j dx (10)This implies the theorem since we may extend this inequality by density to general non-negative f j P L p M j q .This initial reduction complete, we now turn our attention to the task of constructing the familyof near-extremising gaussians G x,τ,j , but, as we discussed at the end of Section 1.1, in order to dothis we shall first need to establish a slight improvement of the effective version of Lieb’s theoremfirst proved in [6]. An issue with constructing a suitable heat-flow outside of the case where p dB p x q , p q is simple isthat we do not then have a natural choice of gaussian extremiser to use as our heat kernel, in fact,generally speaking p dB p x q , p q may not admit a gaussian extremiser at all. Lieb’s theorem doeshowever guarantee the existence of a δ -near gaussian extremiser for any δ ą
0, i.e. there existsa gaussian input A such that BL g p dB p x q , p ; A q ě p ´ δ q BL p dB p x q , p q . The authors of [6] hadsimilar issues, and the solution they found was to establish an effective version of Lieb’s theoremthat tracks how the family of δ -near extremisers for a given Brascamp–Lieb datum degenerates as δ Ñ
0; we now state a simplified version of their result.
Theorem 2.6 (Effective Lieb’s theorem [6]) . There exists N P N depending only on the dimensionsand exponents such that the following holds: For any given D ą there exists δ ą such that forevery δ P p , δ q and any feasible datum p L , p q such that BL p L , p q , | L | ď D , sup | A | , | A ´ |ď δ ´ N BL g p L , p ; A q ě p ´ δ q BL p L , p q . (11)This theorem, in other words, establishes the existence of a function Y δ from the set of feasibleBrascamp–Lieb data to the set of gaussian inputs such that Y δ p L q is a δ -near extremiser for p L , p q and both } Y δ } L and } Y ´ δ } L are bounded above by δ ´ N (to clarify, Y ´ δ p L q refers to the gaussianinput whose j th entry is the inverse of the j th entry of Y δ p L q ). It says nothing however about theexistence of a smooth, let alone continuous, function with such properties. Unfortunately, we require Y δ to be W , bounded for our analysis, moreover, we require that the W , norm is boundedpolynomially in δ ´ . Theorem 2.7.
For fixed dimensions and exponents, let BL denote the set of feasible Brascamp–Lieb data, and G denote the set of gaussian inputs. There exists an N P N depending only on thedimensions and exponents such that the following holds: For all open Ω Ť BL there exists a ν ą such that for all δ P p , ν q , there exists a smooth function Y δ : Ω Ñ G such that det p Y δ p L q j q “ for all j P t , ..., m u , } Y δ } W , p Ω q , } Y ´ δ } L p Ω q ď δ ´ N and, for each L P Ω , Y δ p L q is a δ -nearextremiser for p L , p q , i.e., that BL g p L , p ; Y δ p L qq ě p ´ δ q BL p L , p q . Proposition 2.8 ( [6]) . There exists a number θ P p , q and a constant C depending on thedimensions p n j q mj “ and exponents p p j q mj “ such that the following holds: Given data L , L such that | L | , | L | ď C and BL p L , p q , BL p L , p q ď C , then we have | BL p L , p q ´ BL p L , p q| ď C C n ` θ p n ´ q C | L ´ L | θ . (12) Proof of Theorem 2.7.
The proof strategy is to locally average the potentially discontinuous func-tion given by Theorem 2.6 in such a way that we both preserve its good properties and impose onit some additional regularity. We will be averaging via a discrete cover of Ω, which we shall nowdefine. Let θ P p , q be an exponent to be determined later, and let E Ă Ω be the following discretegrid of points: E : “ Ω X ˜ˆ δ ˙ θ m à j “ Z n j ˆ n ¸ . (13)Now, let I be an indexing set for E so that we may write E “ t L i u i P I , then let Q : “ t Q i u i P I be acover of Ω via axis-parallel cubes of width equal to p δ { q θ , with each Q i centred at L i . One shouldnote as a matter of technicality that we may need to take δ to be very small for Q to genuinely bea cover.By Theorem 2.6, there exists an N P N such that for sufficiently small δ ą Y δ : Ω Ñ G such that } Y δ } L p Ω q , }p Y δ q ´ } L p Ω q ď δ ´ N and Y δ p L q is a δ { p L , p q P Ω. We begin by showing that, for a suitable choice of θ and provided that δ is chosen to be sufficiently small, for all i P I , Y δ p L i q is also a δ -near extremiser for any p L , p q such that L P Q i X Ω. By compactness of Ω and smoothness of the Brascamp–Lieb functional in L on BL , there exists a ν P p , q such that for η P p , ν q and all L , L P Ω satisfying | L ´ L | ď η ,we have BL g p L , p ; Y δ p L qq ě p ´ η q BL g p L , p ; Y δ p L qq . (14)The presence of the bound η here is merely for absorbing constants (we shall freely use this kindof device throughout this paper). By Proposition 2.8, we may choose θ P p , { q such that thefollowing holds: There exists ν P p , q such that for η P p , ν q and | L ´ L | ď η θ , we have thatBL p L , p q ě p ´ η q BL p L , p q . (15)Again we have used some freedom in our choice in θ to absorb the constants that arise in (12).Choose δ such that 0 ă δ ď min t ν , ν , u , then for all i P I and all L P Q i X Ω, since | L ´ L i | ă δ θ { θ ď δ { Y δ p L i q is a δ { p L i , p q to prove the claim.BL g p L , p ; Y δ p L i qq ě p ´ δ { q BL g p L i , p ; Y δ p L i qqě p ´ δ { qp ´ δ { q BL g p L i , p qě p ´ δ { qp ´ δ { q BL g p L , p qě p ´ δ q BL p L , p q (16)11ow, let t ρ i u i P I be a smooth partition of unity subordinate to Q such that } dρ i } L À δ ´ θ (this caneasily be constructed by translation and rescaling), and define the function Y δ : Ω Ñ G . Y δ p L q : “ ˜ÿ i P I ρ i p L q Y δ p L i q ´ ¸ ´ (17)We claim that, for any L P Ω, Y δ p L q is an O p δ q -near extremiser for p L , p q . Firstly, by the homo-geneity of the Brascamp–Lieb functional and (16), each ρ i p L q ´ Y δ p L i q is a δ -near extremiser forall p L , p q such that L P Q i X Ω. Consider now a generic δ -near and δ -near extremiser, call them A and A respectively, for some generic linear datum p L , p q , then by Ball’s inequality,BL g p L , p ; A q BL g p L , p ; A q ď BL p L , p q BL g p L , p ; p A ´ ` A ´ q ´ qùñ p ´ δ qp ´ δ q BL p L , p q ď BL p L , p q BL g p L , p ; p A ´ ` A ´ q ´ qùñ p ´ δ qp ´ δ q BL p L , p q ď BL g p L , p ; p A ´ ` A ´ q ´ qùñ p ´ δ ´ δ q BL p L , p q ď BL g p L , p ; p A ´ ` A ´ q ´ q (18)hence p A ´ ` A ´ q ´ is a p δ ` δ q -near extremiser for p L , p q . Since we are pointwise only eversumming boundedly many contributions in (17), by iterating (18), we find that Y δ p L q is an O p δ q -near extremiser for p L , p q (similar observations about the closure of extremisers under harmonicaddition were made in [10]). We may of course remove the implicit constant here by a simplesubstitution, so we shall proceed assuming that Y δ p L q is a δ -near extremiser for p L , p q , for all L P Ω.It remains to prove that Y δ satisfies the necessary L and W , bounds. We shall start withthe L bounds. One bound is trivial, namely that | Y δ p L q ´ | ď max α : L P Q α | Y δ p L i q ´ | ď δ ´ N . The other requires the elementary fact that, for all symmetric positive definite matrices
A, B P R n ˆ n , |p A ´ ` B ´ q ´ | À | A | ` | B | , which follows from the fact that, for all | v | “ |p A ´ ` B ´ q v | ě p| A ´ v | ` | B ´ v | q { Á | A | ´ ` | B | ´ ě p| A | ` | B |q ´ (19)which then gives us that | Y δ p L q| À max i : L P Q i | Y δ p L i q| ď δ ´ N . It remains to prove the L bound on the derivative dY δ : Ω Ñ L p À mj “ R n j ˆ n ; À mj “ R n j ˆ n j q . Bythe chain rule, we may write dY δ explicitly as dY δ r L sp W q “ r Y δ p L q , d pp Y δ q ´ qr L sp W qs , where r¨ , ¨s denotes a component-wise commutator and W is an arbitrary vector. By linearity, wemay write this as a sum of boundedly many terms. dY δ r L sp W q “ ÿ i P I r Y δ p L q , dρ i r L sp W q Y δ p L i q ´ s
12e then apply the triangle inequality, and observe that the loss incurred is at most polynomial. | dY δ r L sp W q| ď ÿ i : L P Q i | Y δ p L q|| dρ i r L sp W q|| Y δ p L i q ´ |À max i : L P Q i t| Y δ p L q||| dρ i r L s|| Y δ p L i q ´ |u| W |À δ ´ p N ` θ q| W | Finally, we obtain the desired function Y δ by renormalising the determinant, which, by the homo-geneity of the Brascamp–Lieb functional, does not affect the property of being a δ -near extremiser.We shall use the polynomial bounds for Y δ “ : p Y δ,j q mj “ to help us establish polynomial bounds for Y δ “ : p Y δ,j q mj “ , which we define below. Y δ p L q : “ p det p Y δ,j p L qq ´ { n j Y δ,j p L qq mj “ Since the 2-norm of a symmetric matrix is its maximal eigenvalue, and the determinant of a ma-trix is the product of its eigenvalues, we know that det Y δ p L q ď | Y δ p L q| n j ď δ ´ n j N , similarly,det Y δ p L q ´ ď | Y δ p L q ´ | n j ď δ ´ n j N . Finally, we must bound the derivative dY δ r L s . Letting L P Ω, by the chain rule, d p det Y δ,j qr L sp W q “ Y δ,j p L q : dY δ,j r L sp W q , where the colon denotes thefrobenius inner product, so, taking | W | “ | dY δ r L sp W q| ď | d p det p Y δ,j q ´ { n j qqr L sp W q Y δ p L q| ` | det p Y δ,j r L sq ´ { n j dY δ,j r L sp W q|ď δ ´ N ´ | d p det p Y δ,j q ´ { n j qr L s| ` δ ´ N ¯ ď δ ´ N ˆ n j | d p det p Y δ,j qr L s| ´ { n j q ` δ ´ N ˙ ď δ ´ N ˆ n j |p Y δ q j | ´ { n j F rob | d p Y δ q j r L s| ´ { n j F rob ` δ ´ N ˙ À δ ´ N . G x,τ,j We shall now define the gaussian arising in the statement of Theorem 2.2. First, recall the definitionof Y δ from Theorem 2.7, where here we take Ω to be an open set containing dB p M q . Let α P p , q be a small exponent to be defined later, and for each τ ą x P M , define the gaussian input a τ p x q : “ Y τ α p dB r x sq , whose j th component we refer to by a τ,j p x q . We use the exponent α here tohelp us control the polynomial blow-up of a τ and its inverse under various norms. We then definethe gaussian g x,τ,j : T B j p x q M j Ñ R as g x,τ,j p z q “ τ ´ n j exp ´ ´ πτ x a τ,j p x q z, z y ¯ . We may view this gaussian as the fundamental solution of the following anisotropic heat equationat time t “ τ . B t u p z, t q “ ∇ z ¨ p a τ,j p x q ´ ∇ z u p z, t qq We define our gaussian as the following infinite convolution: G x,τ,j : “ ˚ k “ g x, ´ k { τ,j
13e shall now show that G x,τ,j is well-defined if α ă N ´ , where this N P N is the one that arisesin Theorem 2.7. To see this, we consider the partial convolution G p K q x,τ,j : “ g x, ´ { τ,j ˚ ... ˚ g x, ´ K { τ,j “ τ ´ n j det p C K q ´ { exp p´ πτ x C K v, v yq , where C K : “ p ř k “ Kk “ ´ k a ´ k { τ,j p x q ´ q ´ . We now just need to show that C K converges as K Ñ 8 ,since then G p K q x,τ,j converges pointwise. Let l P N , then by the fact that } a ´ k { τ,j } L ď ´ kαN { τ αN for all k ą | C ´ K ` l ´ C ´ K | ď K ` l ÿ k “ K ` ´ k | a ´ k { τ,j p x q| ´ ď K ` l ÿ k “ K ` ´ k kαN { τ ´ αN ď p αN { ´ q K τ ´ αN l ÿ k “ p αN { ´ q k By our choice of α , | C ´ K ` l ´ C ´ K | Ñ K Ñ 8 uniformly in l , so C ´ K is Cauchy, and thereforeconverges. We now only need to check that this limit is invertible, which follows from applying (19)and the L bound on a τ to see that C K is uniformly bounded in K , and so must converge, whence G p K q x,τ,j Ñ G x,τ,j pointwise. | C K | À K ÿ k “ ´ k | a ´ k { τ,j p x q| ď ÿ k “ p αN { ´ q k τ ´ αN À α,N τ ´ αN ă 8 , If we denote the limit of C K by A τ,j p x q , then G x,τ,j p z q “ τ ´ n j det p A τ,j p x qq { exp p´ πτ ´ x A τ,j p x q z, z yq .It is worth noting that by using infinitely many applications of (18), we see that A τ p x q : “ p A τ,j p x qq mj “ is an O p τ α q -near extremiser for p dB r x s , p q , establishing property (2) of Proposition 2.1.BL g p dB r x s , p ; A τ p x qq BL p L , p q ě ´ τ α ÿ k “ kα { “ ´ τ α p α { ´ q ´ Observe that in the case where p dB p x q , p q is simple we may forego Theorem 2.7 and use an exactextremiser for our definition of g x,τ,j , in which case a τ is constant in τ ą
0, and we would then havethe identifications A τ “ a τ and G x,τ,j “ g x,τ,j . Of course, if the reader wanted to run our argumentin the simple case with exact extremisers, then they would need to take care to ensure that theysatisfy appropriate W , and L boundedness of the type we prove for our near-extremisers in thegeneral case. This section is, for the most part, dedicated to establishing the properties we require of our gaussians G x,τ,j and g x,τ,j in order to prove Proposition 2.5, which, as we have shown in Section 2.3, impliesTheorem 2.2. We need to quantify how these gaussians behave under small perturbations in anumber of variables, and for this purpose we shall first need to prove various bounds on norms theunderlying (gaussian) input-valued functions a τ and A τ .14 emma 3.1. For any ε P p , ´ γ q , provided α is chosen such that α ă ε N , there exists a ν ą such that for every τ P p , ν q , } A τ } W , , } det A τ } W , , } A ´ τ } L ď τ ´ ε .Proof. The proof is similar to that of Lemma 2.7, as it amounts to a straightforward application ofthe triangle inequality and the bounds on a ´ k { τ,j p x q , taking ν ą τ αN ´ ε { for all τ P p , ν q . | A τ,j p x q ´ | ď ÿ k “ ´ k | a ´ k { τ,j p x q| ´ ď ÿ k “ p αN { ´ q k τ ´ αN ď τ ´ ε { | A τ,j p x q| À ÿ k “ ´ k | a ´ k { τ,j p x q| ď ÿ k “ p αN { ´ q k τ ´ αN ď τ ´ ε { Now, take W P T B j p x q M j such that | W | “ | dA τ,j r x sp W q| ď ÿ k “ ´ k |r A τ,j p x q , d x a ´ k { τ,j p x qp W qs|ď | A τ,j p x q| ÿ k “ ´ k | d x a ´ k { τ,j p x q|ď | A τ,j p x q| ÿ k “ ´ k } dY ´ αk { τ α } L } d B j } L ď Cτ ´ ε { ´ αN ÿ k “ ´ k p ´ αN { q ď τ ´ ε First of all, | det A τ,j p x q| ď | A τ,j p x q| n j ď τ ´ εn j for all τ P p , ν q , we have the bound | A τ,j p x q| ď τ ´ ε ,similarly | A τ,j p x q ´ | ď τ ´ ε for all such τ . It remains to establish the L bound on d p det A τ,j q .Taking any x P M and W P T x M such that | W | “
1, by the chain rule, the Cauchy-Schwartzinequality, and the equivalence of finite dimensional norms, | d p det A τ,j qr x sp W q| “ | A τ,j p x q : dA τ,j p x qp W q| (20) ď | A τ,j p x q| F rob | dA τ,j p x q| F rob À τ ´ ε (21)This proves the claim, provided we adjust our choice of ν accordingly.We shall henceforth consider ε P p , p ´ γ q{ q and α P p , ε { N q as fixed parameters, and wealso note at this point that we have now proved property (3) of Proposition 2.1, completing itsproof. Lemma 3.2.
For all η P p , min t γ ´ ε, . γ ´ ε, γ ´ ´ ε uq , there exists a ν ą such thatthe following holds: for all τ P p , ν q and x, y P M such that d p x, y q ď τ γ , and z P M j such that d p z, B j p x qq ď τ γ , for all f j P L p M j q , H y,τ,j f j p z q ď p ` τ η q H x,τ,j f j p z q (22) Proof.
Let τ ą x, y P M satisfy d p x, y q ď τ γ , and take some z P M j suchthat d p z, B j p x qq ď τ γ . First of all, by the chain rule, for any v P T x M , d p e ´ B j p y q ˝ e B j p x q qr v s “ p e ´ B j p y q qr e B j p x q p v qs de B j p x q r v s . Given w P U τ,j p z q , by Taylor’s theorem, we may approximate v y : “ e ´ B j p y q p z q ´ e ´ B j p y q p w q in terms of v x : “ e ´ B j p x q p z q ´ e ´ B j p x q p w q in the following manner: v y “ e ´ B j p y q ˝ e B j p x q ˝ e ´ B j p x q p z q ´ e ´ B j p y q ˝ e B j p x q ˝ e ´ B j p x q p w q“ d p e ´ B j p y q ˝ e B j p x q qr e ´ B j p x q p z qsp v x q ` O p| v x | q“ d p e ´ B j p y q qr z s de B j p x q r e ´ B j p x q p z qsp v x q ` O p| v x | q . (23)Above, we implicitly state that the norms of the higher-order derivatives are bounded uniformlyabove, a fact we prove in Lemma 6.1, located at the back of the paper. Define the linear map T x,y : “ d p e ´ B j p y q qr z s de B j p x q r e ´ B j p x q p z qs , then it follows that | A τ,j p y q { v y | “ | A τ,j p y q { p T x,y v x ` O p| v x | qq| ď | A τ,j p y q { p T x,y v x q| ` τ . γ ´ ε , (24)for sufficiently small τ ą
0. Now, by the uniform bounds on det A τ,j established in Lemma 3.1, wehave that | log p det A τ,j p x qq ´ log p det A τ,j p y qq| ď t| det A τ,j p x q| , | det A τ,j p y q|u | det A τ,j p x q ´ det A τ,j p y q|ď τ ´ ε } d p det A τ,j q} L d p x, y qď τ γ ´ ε . Together with (24), this implies the bound G y,τ,j p v y q ď p ` τ η q G x,τ,j p v x q for sufficiently small τ ą G y,τ,j p v y q G x,τ,j p v x q “ det p A τ,j p x qq det p A τ,j p y qq exp p πτ ´ p| A τ,j p x q { v x | ´ | A τ,j p y q { v y |qqď exp p τ γ ´ ε ` πτ . γ ´ ε ` πτ ´ p| A τ,j p x q { v x | ´ | A τ,j p y q { T x,y v x | qqď exp p τ γ ´ ε ` πτ . γ ´ ε ` πτ ´ xp A τ,j p x q ´ T ˚ x,y A τ,j p y q T x,y q v x , v x yqď exp p τ γ ´ ε ` πτ . γ ´ ε ` πτ ´ | A τ,j p x q ´ T ˚ x,y A τ,j p y q T x,y || v x | qď exp p τ γ ´ ε ` πτ . γ ´ ε ` πτ ´ } dA τ,j } L τ γ qď exp p τ γ ´ ε ` πτ . γ ´ ε ` πτ γ ´ ´ ε q ď ` τ η In the penultimate line we applied the mean value theorem in to obtain | A τ,j p x q ´ T ˚ x,y A τ,j p y q T x,y | ď } dA τ,j } L d p x, y q . The claim then easily follows from the definition of H x,τ,j . H y,τ,j f j p z q : “ ż U τ,j p z q f j p w q G x,τ,j p e ´ B j p x q p z q ´ e ´ B j p x q p w qq dw ď p ` τ η q ż U τ,j p z q f j p w q G x,τ,j p e ´ B j p x q p z q ´ e B j p x q ´ p w q dw “ p ` τ η q H y,τ,j f j p z q (25) Lemma 3.3 (General Truncation of Gaussians) . Let m, n P N , κ » , and for each τ ą let A τ P R n ˆ n be a positive definite matrix. Define g τ : R n Ñ R to be the gaussian g τ p x q : “ τ ´ n exp p´ πτ ´ x A τ x, x yq . Let γ ą , and suppose that | det p A τ q| ´ , | A ´ τ | ď τ ´ ε for some ε Pp , p ´ γ q{ q . There exists a ν ą depending only on n , m , ε , and γ such that for all τ P p , ν q det p A τ q ´ { “ ż R n g τ ď p ` τ ε q ż U κτγ p q g τ . (26)16 roof. By the freedom of choice we have in γ , if the claim holds for κ “ γ “ γ ´ η forsome small η ą
0, we obtain the full result by increasing the domain of integration on the right-handside of (26) from U τ γ to U κτ γ p q , taking τ ď κ ´ { η . It is sufficient to show that there exists ν ą τ P p , ν q , ż R n z U τγ p q g τ ď cτ ε . (27)for some c »
1. To see this we simply split the integral of g τ into U τ γ p q and R n z U τ γ p q .det p A τ q ´ { “ ż R n z U τγ p q g τ ` ż U τγ p q g τ ď cτ ε ` ż U τγ p q g τ det p A τ q ´ { ´ cτ ε ď ż U τγ p q g τ det p A τ q ´ { ď p ´ c det p A τ q ´ { τ ε q ´ ż U τγ p q g τ ď p ´ cτ ε { q ´ ż U τγ p q g τ Which of course implies (26) if τ is taken to be sufficiently small. To estimate the left hand sideof (27), we shall partition the domain of integration R n z U τ p q into annuli, and bound the resultinginfinite sum above by a lacunary series. We take τ P p , ν q , where ν P p , q is chosen such that π ν ε ` γ ´ ě ε . ż R n z U τ p q g τ “ ż | x |ě τ γ ´ exp p´ π | A { τ x | q dx “ ÿ k “ ż k τ γ ´ ď| x |ď k ` τ γ ´ exp p´ π | A { τ x | q dx ď ÿ k “ sup k log p { τ q“| x | p exp p´ π | A τ x | qq V ol pt k τ γ ´ ď | x | ď k ` τ γ ´ uqď σ n ´ τ n p ´ γ q 8 ÿ k “ nk exp p´ π | A ´ τ | ´ k τ p γ ´ q qÀ ÿ k “ τ π τ ε k τ γ ´ À ÿ k “ τ π ν ε ` γ ´ k À ÿ k “ τ ε k À τ ε , We may now prove the pointwise convergence to initial data for H x,τ,j f j ˝ B j , a fact the readerwill recall that we needed to prove that Proposition 2.5 implied Theorem 2.2. Lemma 3.4 (Pointwise convergence to initial data) . For each j P t , ..., m u , let f j P C p M j q and x P M , then, lim τ Ñ H x,τ,j f j ˝ B j p x q “ f j ˝ B j p x q . (28) Proof.
Let τ ą f j is uniformly continuous, given ε ą
0, there exists a δ ą z, z P M such that d p z, z q ď δ , we have that | f j p z q ´ f j p z q| ď ε . Therefore, provided17 τ γ ď δ , we may bound | f j ˝ B j p x q ´ H x,τ,j f j ˝ B j p x q| in the following way: | f j ˝ B j p x q ´ H x,τ,j f j ˝ B j p x q| “ ˇˇˇˇˇ f j ˝ B j p x q ´ ż U τγ,j p q f j p w q G x,τ,j p e ´ B j p x q p w qq dw ˇˇˇˇˇ ď f j ˝ B j p x q ˜ ´ ż U τγ,j p B j p x qq G x,τ,j ˝ e ´ B j p x q ¸ ` ˇˇˇˇˇż V τγ,j p q p f j ˝ B j p x qq ´ f j p w qq G x,τ,j p e ´ B j p x q p w qq dw ˇˇˇˇˇ . By the uniform boundedness of the second derivative of the exponential map e B j p x q established inLemma 6.1, provided that τ ą x P M , j P t , ..., m u , and v P V τ γ ,j p q Ă T B j p x q M j , we have p ` τ η q ´ ď det p de B j p x q qr v s ď ` τ η . (29)We may then apply Lemma 3.3 to bound the first term by a power of τ . For the second term, weapply the triangle inequality and bound the resulting gaussian integral similarly. | f j ˝ B j p x q ´ H x,τ,j f j ˝ B j p x q|ď f j ˝ B j p x q ˜ ´ p ` τ η q ´ ż V τγ,j p q G x,τ,j ¸ ` ż U τγ,j p B j p x qq | f j ˝ B j p x q ´ f j p w q| G x,τ,j ˝ e ´ B j p x q p w q dw ď p ` τ η q ´ p f j ˝ B j p x q τ η ` ε q This of course implies the claim of the lemma.
Lemma 3.5 (Switching) . For all η P p , α q , there exists ν ą such that for τ P p , ν q and x, y P M such that d p x, y q ď τ γ , p dB r y s , p q m ź j “ g y,τ,j ˝ dB j r y sp e ´ y p x qq p j ď ` τ η BL p dB r x s , p q m ź j “ g x,τ,j ˝ dB j r x sp e ´ x p y qq p j . (30) Proof.
Let 0 ă τ ă ρ {
10, and define the positive-definite symmetric matrix field M τ P Γ p T M b T ˚ M q . M τ p x q : “ m ÿ j “ p j dB j r x s ˚ a τ,j p x q dB j r x s It follows from the definition of M τ that m ź j “ g x,τ,j ˝ dB j r x sp v q p j “ m ź j “ exp p p j x a τ,j p x q dB j r x s v, dB j r x s v yq“ exp p´ πτ ´ | M τ p x q { v | q Hence, by the fact that a τ,j p x q is a τ α -near extremiser for p dB r x s , p q , p ´ τ α q BL p dB r x s , p q ď BL g p dB r x s , p ; a τ,j p x qq “ det p M τ p x qq ´ { ď BL p dB r x s , p q , p ´ τ α q det p M τ p x qq { τ ´ n exp ´ ´ πτ | M τ p x q { v | ¯ ď ś mj “ g x,τ,j ˝ dB j r x sp v q p j BL p dB r x s , p qď det p M τ p x qq { τ ´ n exp ´ ´ πτ | M τ p x q { v | ¯ Taking logarithms of the ratio of the two quantities arising on either side of (30) reveals that thelogarithm of the error factor in (30) is polynomial in τ .log ˜ BL p dB r x s , p q ś mj “ g y,τ,j ˝ dB j r y sp e ´ y p x qq p j BL p dB r y s , p q ś mj “ g x,τ,j ˝ dB j r x sp e ´ x p y qq p j ¸ ď log ˜ exp ` ´ πτ ´ | M τ p y q { e ´ y p x q| ˘ det p M τ p y qq { p ´ τ α q exp ` ´ πτ ´ | M τ p x q { e ´ x p y q| ˘ det p M τ p x qq { ¸ ď πτ ´ ´ | M τ p y q { e ´ y p x q| ´ | M τ,j p x q { e ´ x p y q| ¯ ` log p det p M τ p y q M τ p x q ´ qq ´ log p ´ τ α q Let σ : I Ñ M be a geodesic such that σ p q “ x and σ p q “ y . Let P σ : T x M Ñ T y M denoteparallel transport along σ . It is straightforward to check that e ´ y p x q : “ ´ P σ e ´ x p y q , hence we maycollate the two squares in the first term, allowing us to bound the resulting quantity using the meanvalue theorem.log ˜ BL p dB r x s , p q ś mj “ g y,τ,j ˝ dB j r y sp e ´ y p x qq p j BL p dB r y s , p q ś mj “ g x,τ,j ˝ dB j r x sp e ´ x p y qq p j ¸ ď πτ ´ xp P ´ σ M τ p y q P σ ´ M τ p x qq e ´ x p y q , e ´ x p y qy ` log p det p M τ p y q M τ p x q ´ qq ´ log p ´ τ α qÀ τ γ ´ } dM τ } L | e ´ x p y q| ` τ γ ´ ε ` τ α À τ γ ´ ´ ε ` τ γ ´ ε ` τ α À τ α . Hence, provided τ is taken to be sufficiently small, we obtain the desired upper bound.BL p dB r x s , p q ś mj “ g y,τ,j ˝ dB j r y sp e ´ y p x qq p j BL p dB r y s , p q ś mj “ g x,j,τ ˝ dB j r x sp e ´ x p y qq p j ď exp p cτ α q ď ` τ η , where c » τ γ for the radius of truncation. We shall also need to define a minor modification of H x,τ,j ,where the domain of integration is increased by a factor of 1.1. This factor is of course chosenarbitrarily, but since this consideration is a minor technicality we simply choose a value for thesakes of concreteness. H . x,τ,j : L p M j q Ñ L p U ρ ´ . τ γ p B j p x qqq H . x,τ,j f j p z q : “ ż U . τγ,j p z q f j p w q G x,τ,j p e ´ B j p x q p z q ´ e ´ B j p x q p w qq dw Lemma 3.6 (Local-constancy) . For any η P p , γ ´ ε q , there exists a ν ą such that the followingholds for all τ P p , ν q : Let x P M , then given z, ˜ z P U τ,j p B j p x qq such that d p z, ˜ z q À τ we have thatfor all f j P L p M j q , H x,τ,j f j p z q ď p ` τ η q H . x,τ,j f j p ˜ z q (31)19 roof. First of all we need to prove a similar claim for the kernel G x,τ,j . Suppose that v, w P T B j p x q M j are such that | v ´ w | À τ and v, w P V τ γ ,j p q . G x,τ,j p v q G x,τ,j p w q “ exp p πτ ´ p| A τ,j p x q { v | ´ | A τ,j p x q { w | qq“ exp p πτ ´ x A τ,j p x qp v ´ w q , v ` w yqď exp p C πτ ´ } A τ,j }| v ´ w || v ` w |qď exp p C κπτ ´ τ ´ ε τ τ γ qď exp p C κπτ γ ´ ε q where κ »
1. Hence it follows that, for all τ ą d p z, ˜ z q À τ , and w P U τ,j p z q , G x,τ,j p e ´ B j p x q p z q ´ e ´ B j p x q p w qq ď p ` τ η q G x,τ,j p e ´ B j p x q p ˜ z q ´ e ´ B j p x q p w qq . The lemma then follows from applying this bound directly to the definition of H . x,τ,j f j . H x,τ,j f j p z q “ ż U τγ,j p z q f j p w q G x,τ,j p e ´ B j p x q p z q ´ e ´ B j p x q p w qq dw ď p ` τ η q ż U . τγ ,j p z q f j p w q G x,τ,j p e ´ B j p x q p ˜ z q ´ e ´ B j p x q p w qq dw “ p ` τ η q H . x,τ,j f j p ˜ z q Our proof strategy is to use the near-extremising gaussians g x,τ,j to construct a partition of unityfor the integral on the left-hand side of (8), subordinate to balls of scale τ γ . At this scale, we mayapply our lemmas from the previous section to perturb the integral, so that we may then applythe linear Brascamp–Lieb inequality locally, thereby obtaining the desired form on the right-handside. Gaussian partitions of unity were also used in [6], and, notably, more recently in the contextof decoupling for the parabola by Guth, Maldague, and Wang [18]. Proof.
For each j P t , ..., m u , take some arbitrary f j P L p M j q . Let η P p , min t α, . γ ´ ε, γ ´ ε, γ ´ ´ ε uq and choose ν ą τ P p , ν q .Consider the following collection of truncated gaussians. χ V . τγ p q ś mj “ g p j y,τ,j ˝ dB j r y s BL p dB r y s , p q + y P M (32)By Lemma 3.3 and the fact that a τ p y q is a τ α -near extremiser for p dB p y q , p q , we know that for τ ą g p dB r y s , p ; a τ p y qq ď p ` τ η q ż V . τγ p q m ź j “ g y,τ,j ˝ dB j r y sp v q p j BL p dB r y s , p q ď p ` τ η q ż V . τγ p q m ź j “ g y,τ,j ˝ dB j r y sp v q p j , ż M m ź j “ H y,τ,j f j ˝ B j p y q p j dy ď p ` τ η q ż M ż V . τγ p y q m ź j “ H y,τ,j f j ˝ B j p y q p j g y,τ,j ˝ dB j r y sp v q p j dv dy BL p dB r y s , p qď p ` τ η q ż M ż U . τγ p y q m ź j “ H y,τ,j f j ˝ B j p y q p j g y,τ,j ˝ dB j r y sp e ´ y p x qq p j dx dy BL p dB r y s , p q“ p ` τ η q ż M ż U . τγ p x q m ź j “ H y,τ,j f j ˝ B j p y q p j g y,j,τ ˝ dB j r y sp e ´ y p x qq p j dy BL p dB r y s , p q dx We want to perturb the inner integral to a linear Brascamp–Lieb inequality in y . To do this, we firstapply Lemma 3.2 and Lemma 3.5 to remove some of the unwanted y -dependence. Let P : “ ř mj “ p j ,then ż M m ź j “ H y,τ,j f j ˝ B j p y q p j dy ď p ` τ η q ` P ż M ż U . τγ p x q m ź j “ H x,τ,j f j ˝ B j p y q p j g y,τ,j ˝ dB r y sp e ´ y p x qq p j dy BL p dB r y s , p q dx ď p ` τ η q ` P ż M ż U . τγ p x q m ź j “ H x,τ,j f j ˝ B j p y q p j g x,τ,j ˝ dB r x sp e ´ x p y qq p j dy dx BL p dB r x s , p qď p ` τ η q ` P ż M ż V . τγ p x q m ź j “ H x,τ,j f j ˝ B j p e x p v qq p j g x,τ,j ˝ dB r x sp v q p j dv dx BL p dB r x s , p q . We may then use Lemma 3.6 to replace the instance of B j p e x p v qq with its affine approximationabout x , given by L xj p v q : “ e B j p x q p dB j r x s v q . ď p ` τ η q ` P ż M ż V . τγ p x q m ź j “ H . x,τ,j f j ˝ L xj p v q p j g x,τ,j ˝ dB j r x sp v q p j dv dx BL p dB r x s , p qď p ` τ η q ` P ż M ż T x M m ź j “ H . x,τ,j f j ˝ L xj p v q p j g x,τ,j χ V . τγ ,j p q ˝ dB j r x sp v q p j dv dx BL p dB r x s , p q . Above we used the fact that, for all x P M V . τ γ p q Ă Ş mj “ dB j r x s ´ V . τ γ ,j p q . At this point wemay apply the linear Brascamp–Lieb inequality p dB r x s , p q to the inner integral. ď p ` τ η q ` P ż M m ź j “ ˜ż V . τγ ,j p q H . x,τ,j f j p e B j p x q p v j qq g x,τ,j p v j q dv j ¸ p j dx (33)21he resulting integrals in (33) may be then be bounded by a convolution. ż V . τγ ,j p q H . x,τ,j f j p e B j p x q p v j qq g x,τ,j p v j q dv j “ ż V . τγ,j p q ż U . τγ,j p B j p x qq f j ˝ e x p w q G x,τ,j p v j ´ w q g x,τ,j p v j q dwdv j ď p ` τ η q ż V . τγ ,j p q ż U . τγ,j p B j p x qq f j p w q G x,τ,j p v j ´ w q g x,τ,j p v j q dwdv j “ G x,τ,j χ V . τγ p q ˚ g x,τ,j χ V . τγ p q ˚ f j ˝ e B j p x q p q (34)Now, G x,τ,j ˚ g x,τ,j “ G x, { τ,j by definition of G x,τ,j , and the support of χ V . τγ p q ˚ χ V . τγ p q is theball around the origin of radius 1 . τ γ , which is less than 2 γ { τ γ provided that γ ě p . q « . ... . This implies that supp p G x,τ,j χ V . τγ p q ˚ g x,τ,j χ V . τγ p q q Ă V γ { τ γ p q , hence G x,τ,j χ V . τγ p q ˚ g x,τ,j χ V . τγ p q ď p G x,τ,j ˚ g x,τ,j q χ V γτγ p q “ G x, { τ,j χ V γ { τγ p q We may then bound (34) as follows: ż V . τγ ,j p q H . x,τ,j f j p e B j p x q p v j qq g x,τ,j p v j q dv j ď H x, { τ,j f j ˝ B j p x q . (35)Finally, we complete the proof by combining (33) with (35) and taking β P p , η q . Proof of Corollary 2.3.
Take some arbitrary f j P L p M j q for all j P t , ..., m u . By Theorem 2.2,there exists a β ą τ ą ż U τγ p x q m ź j “ f j ˝ B j p x q p j dx ď ż M m ź j “ f j χ U τγ,j p x q ˝ B j p x q p j dx ď p ` τ β q ż M m ź j “ H x,τ,j p f j χ U τγ,j p x q q ˝ B j p x q p j dx ď p ` τ β q ż U τγ p x q m ź j “ H x,τ,j f j ˝ B j p x q p j dx Take η and ν as in the proof of Proposition 2.5, if we take τ P p , ν q , then we may apply Lemma3.5 to perturb H x,τ,j to H x ,τ,j and Lemma 3.6 to perturb B j p x q to L x j p x q , at which point we mayapply the linear inequality to complete the proof. ď p ` τ β qp ` τ η q P ż U τγ p x q m ź j “ H x ,τ,j f j ˝ B j p x q p j dx ď p ` τ β qp ` τ η q P ż U τγ p x q m ź j “ H . x ,τ,j f j ˝ L x j p x q p j dx ď p ` τ β qp ` τ η q P BL p dB r x s , p q m ź j “ ˜ż U τγ,j p q H x ,τ,j f j ˝ e B j p x q ¸ p j ď p ` τ β qp ` τ η q P BL p dB r x s , p q m ź j “ ˜ż M j f j ¸ p j roof of Corollary 2.4. Fix some τ ą p B , p q . Let R : “ } B ´ r B } L p R n q and take some v P V τ γ p B j p x qq , then G x,τ,j p B j p x q ´ v q G x,τ,j p r B j p x q ´ v q ď exp p πτ ´ } A τ,j }p | r B j p x q ´ B j p x q|| B j p x q ´ v | ´ | r B j p x q ´ B j p x q| qqď exp p Rπτ γ ` ε ´ q . (36)Define the following convolution operator: H τ,j f j p y q : “ ż R n f j p z q exp p´ πτ ε ´ | y ´ z | q dz. (37)Since G x,τ,j p z q ď exp p´ πτ ε ´ | z | q by Lemma 3.1, H x,τ,j f j ď H τ,j f j , so combining this with (36),we may bound H x,τ,j f j ˝ B j p x q as follows, H x,τ,j f j ˝ B j p x q “ ż V τγ p B j p x qq f j p z q G x,τ,j p B j p x q ´ z q dz À R,τ ż V τγ p B j p x qq f j p z q G x,τ,j p r B j p x q ´ z q dz ď τ ´ n j ż R n f j p z q exp p´ πτ ε ´ | r B j p x q ´ z | q dz “ H τ,j f j ˝ r B j p x q . The finiteness of p B , p q then follows easily from (8) and the finiteness of p r B , p q . ż R n m ź j “ f j ˝ B j p x q p j dx ď p ` τ β q ż R n m ź j “ H x,τ,j f j ˝ B j p x q p j dx À τ ż R n m ź j “ H τ,j f j ˝ r B j p x q p j dx À r B m ź j “ ˆż R nj H τ,j f j ˙ p j À m ź j “ ˆż R nj f j ˙ p j Here we establish that our uniform boundedness assumptions from Section 2.1 imply good uniformcontrol of the first and second order derivatives of the exponential map. The proof uses somestandard ideas from the analysis of ODES and is fairly elementary, but we include it here for thesakes of completeness.
Lemma 6.1.
Suppose that M is a Riemannian manifold with bounded geometry, then given x P M ,then the norms of the covariant derivatives (up to second order) of the exponential map based at p P M may be bounded above uniformly in p in the open unit ball. roof. We should first clarify that, in this proof, double bars shall always denote L norms. Wefirst prove the case for derivatives of order 1. Let p P M and let X, Y P T p M , and | X | , | Y | ď J p t q : p ,
8q Ñ
T M defined over the curve parametrised by γ p t q : “ exp p tX q : J p t q : “ B s exp p p t p X ` sY qq| s “ . By definition of the exponential map, J is a Jacobi field with initial data J p q : “ J p q “ Y ,hence it satisfies the Jacobi equation: J ` R p J, γ q γ “ R denotes the Riemannian curvature endomorphism. Now, define the following quantity F p t q : “ | J p t q| ` | J p t q| . We shall aim to bound this quantity via bounding its derivative using(38) and the AM-GM inequality. F “ x J, J y ` x J , J y“ px J, J y ` x J , R p J, γ q γ yqď p| J || J | ` | J |} R }| J || X | qď p ` } R }q F Hence F p t q ď e t p `} R }q F p q , and so | d exp p p X q Y | “ J p q ď F p q { ď e p `} R }q{ F p q { “ e p `} R }q{ | Y | . We then bootstrap to the second order case via a similar method. Let Z P T p M , | Z | ď
1, andconsider the following family of variations of J : J ε p t q : “ B s exp p p t p X ` sY ` εZ qq| s “ Each such J ε is a Jacobi field for all ε ą
0, so we may then differentiate (38) in ε to find that B ε J ε ` B J R p J ε p t q , γ qp γ , B ε J ε q “ , @ t, ε ą . Where B J R refers to the partial covariant derivative of the Riemannian curvature tensor in thefirst argument. We now consider the quantity G p t q : “ |B ε J p t q| ` |B ε J p t q| , and apply a similarargument to last time G “ xB ε J , B ε J y ` xB ε J , B ε J y“ pxB ε J , B ε J y ` x J , B J R p J , γ qp γ , B ε J qyď p|B ε J ||B ε J | ` |B ε J |}B J R }| J ||B ε J || X | qď p ` }B ε R }| J | q G Hence G p t q ď e t p `}B J R }p sup ă l ă t | J | p l qqq G p q ď e t p `}B J R } e t p `} R }q | Y |q G p q , therefore, | d exp p X qp Y, Z q| “ B ε J p q ď G p q { ď e t p `}B J R } e t p `} R }q q{ G p q { “ e p `}B J R } e p `} R }q{ q{ | Z | By symmetry, we also have that | d exp p X qp Y, Z q| ď e p `}B J R } e t p `} R }q{ q{ | Y | , so we are done.24 eferences [1] Franck Barthe. On a reverse form of the Brascamp–Lieb inequality. Inventiones mathematicae ,134(2):335–361, 1998.[2] Franck Barthe. Optimal Young’s inequality and its converse: a simple proof.
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