On the Oberlin affine curvature condition
In this paper we generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension in , . We show that a canonical affine invariant measure exists and that, modulo sufficient regularity assumptions on the submanifold, the measure satisfies the affine curvature condition of D. Oberlin with an exponent which is best possible. The proof combines aspects of Geometric Invariant Theory, convex geometry, and frame theory. A significant new element of the proof is a generalization to higher dimensions of an earlier result [gressman2009] concerning inequalities of reverse Sobolev type for polynomials on arbitrary measurable subsets of the real line.
Many of the deep questions in harmonic analysis, including Fourier restriction, decoupling theory, or -improving estimates for geometric averages, deal with certain operators associated to submanifolds of Euclidean space. In most cases, the “nicest possible” submanifolds are, informally, as far as possible from lying in any affine hyperplane. Many of these problems also exhibit natural affine invariance, meaning that when the underlying Euclidean space is transformed by a measure-preserving affine linear mapping, the the relevant quantities (norms, etc.) are unchanged. This simple observation leads naturally to the question of how in general to properly quantify this sort of well-curvedness in a way that respects affine invariance. Of the many approaches to this question, one particularly successful strategy has been the use of the so-called affine arclength measure for curves and the analogous notion of affine hypersurface measures (sometimes called the equiaffine measure). In the former case, affine arclength is defined on a curve parametrized by by
while equiaffine measure on the graph over is given by
where is the Hessian matrix of second derivatives of . Though these measures were well-known outside harmonic analysis for quite some time (see, for example, [guggenheimer1963, lutwak1991]), their first appearances within the field are somewhat more recent, in work of Sölin [sjolin1974] (in two dimensions, generalized later by Drury and Marshall [dm1985]) and Carbery and Ziesler [cz2002], respectively. Both measures have the property that they are independent of the parametrization and that they are unchanged when the curve or surface is transformed by a measure-preserving affine mapping. These measures and certain “variable coefficient” generalizations to families of curves and hypersurfaces have played a central role in the Fourier restriction problem as well as the problem of characterizing the – mapping properties of geometrically-constructed convolution operators, two problems which have been of sustained interest for many years [drury1990, choi1999, oberlin1999II, dw2010, stovall2016, dlw2009, stovall2014, gressman2013, oberlin2012].
The deep connections between analysis and geometry enjoyed by affine arclength and hypersurface measures naturally lead to the problem of generalizing these objects to manifolds of arbitrary dimension or even to abstract measure-theoretic settings. One particularly interesting approach is due to D. Oberlin [oberlin2000II] (which generalizes an earlier observation of Graham, Hare, and Ritter [ghr1989] in one dimension), who introduced the following condition on nonnegative measures associated to submanifolds: a measure on a -dimensional immersed submanifold of will be said to satisfy the Oberlin condition with exponent when there exists a finite positive constant such that for every in the set of compact convex subsets of ,
where represents the usual Lebesgue measure of in . When restricted to the class of balls with respect to the standard metric on , the condition (1) becomes a familiar inequality from geometric measure theory. Unlike in that setting, here the exponent measures not just dimension of the measure, but also a certain kind of curvature (for the simple reason that (1) cannot hold for any when is supported on a hyperplane, which can be seen by taking to be increasingly thin in the direction transverse to such a hyperplane). Oberlin observed that this condition is necessary for Fourier restriction or -improving estimates to hold; in particular,
where is the Fourier transform, is convolution. Here and throughout the paper, the notation means that there is a finite positive constant such that and this constant is independent of the relevant variables (functions and sets in this case) appearing in the expressions or quantities and . By virtue of known results for these two problems, the affine arclength and hypersurface measures must satisfy (1) for appropriate exponents when suitable regularity hypotheses on the submanifolds are imposed.
The significance of the Oberlin condition (1) for curves and hypersurfaces in is that, up to a constant factor, the affine arclength and affine hypersurface measures on an immersed submanifold are the unique largest measures on the manifolds satisfying (1) when and , respectively. More precisely, in the case of hypersurfaces (first established by Oberlin [oberlin2000II]), if is any nonnegative measure on an immersed hypersurface , if satisfies (1) with , then for affine hypersurface measure (where here means uniformly for all Borel sets ). Moreover, subject to certain algebraic limits on the complexity of the immersion, itself satisfies (1) for this same exponent. The condition (1) also turns out to be equivalent to the boundedness of certain geometrically-constructed multilinear determinant functionals [gressman2010] and leads to a natural affine generalization of the classical Hausdorff measure [oberlin2003].
This paper examines the Oberlin condition for arbitrary -dimensional submanifolds of (where ) and characterizes it in the case of maximal nondegeneracy. Specifically, the analogous results to those just mentioned above are established in all dimensions and codimensions: an affine invariant measure is constructed which is essentially the largest-possible measure satisfying the Oberlin condition for the largest nontrivial choice of . To say that is nontrivial means simply that there is a nonzero measure, satisfying (1) for this , on some immersed submanifold of the given dimension and codimension. As in the case of curves and hypersurfaces, the largest nontrivial can be understood as a ratio of the intrinsic dimension of the submanifold and its “homogeneous dimension,” which captures information about scaling and curvature-like properties to be measured. The correct value of homogeneous dimension is defined as follows: when and are fixed, let the homogeneous dimension be defined to be the smallest positive integer which equals the sum of the degrees of some collection of distinct, nonconstant monomials in variables (see Figure 1). The main result of this paper is Theorem 1:
Suppose is an immersed -dimensional submanifold of equipped with a nonnegative measure . Then the following are true:
If (1) holds for with exponent , then is the zero measure.
There is a nonnegative measure on such that if (1) holds for when , then .
Under certain algebraic constraints on the immersion of in (which are satisfied globally for polynomial embeddings and locally for real analytic embeddings), satisfies the Oberlin condition (1) with .
There is a -dimensional submanifold of for which has everywhere strictly positive density with respect to Lebesgue measure on and satisfies (1) with .
Condition 2 shows that the measure to be constructed (which is intrinsic, invariant under measure-preserving affine linear transformations of , and agrees up to normalization with affine arclength and equiaffine measure when , respectively) is, up to a constant factor, the unique maximal measure on which satisfies (1) for . This extends the result of Oberlin for equiaffine measure [oberlin2000II] to submanifolds of any dimension.
The structure of the rest of this paper is as follows. In Section 2, the measure is constructed by combining ideas of Kempf and Ness [kn1979] from Geometric Invariant Theory together with a simple but far-reaching observation that any covariant tensor field on a manifold can be used to construct an associated measure on that manifold in a way that generalizes the relationship between the Riemanninan metric tensor and the Riemannian volume. In particular, the measure will be the measure associated to an “affine curvature tensor” on the manifold immersed in . After these constructions are complete, Parts 1 and 2 of Theorem 1 follow in a rather immediate way.
Section 3 is devoted to the proof of Parts 3 and 4 of Theorem 1. Part 3 is proved by first generalizing Theorem 1 of [gressman2009] to higher dimensions. The result, Lemma 4, is interesting in its own right and will have important implications for the theory of -improving estimates for averages over submanifolds in much the same way that Theorem 1 of [gressman2009] formed the basis for a new proof of a restricted version Tao and Wright’s result [tw2003] for averages over curves. The final part of Section 3 shows that the measure is not trivial by constructing submanifolds on which it is possible to say with certainty that the density of with respect to Lebesgue measure is never zero. Finally, Section 4 establishes uniform estimates for the number of nondegenerate solutions of certain systems of equations. These estimates are important for Part 3 of Theorem 1.
2 Affine geometry and necessity
2.1 Geometric Invariant Theory
The main ideas and results from Geometric Invariant Theory that will be used in this paper come from the seminal work of Kempf and Ness [kn1979] and its subsequent extension to real reductive algebraic groups by Richardson and Slodowsky [rs1990]. The idea of interest is that, for suitable representations of such groups, one can study group orbits by understanding the infimum over the orbit of a certain vector space norm. For the purposes of this paper, it suffices to consider only representations of or on vector spaces of tensors. In this context, the associated minimum vectors can be understood as normal forms of tensors and the actual numerical value of the infimum carries meaningful and important quantitative information about these tensors (in contrast to the usual situation in GIT in which one cares only about whether the infimum is zero or nonzero and whether or not it is attained).
To begin the construction, suppose that is any -linear functional on a real vector space of dimension . Appropriating the Kempf-Ness minimum vector calculations of GIT, it becomes possible to canonically associate a density functional to any such . Specifically, for any such and any vectors , let be the quantity given by
Before showing that the quantity (2) is a density functional, it is worthwhile to acknowledge the algebraic structure that lies behind it. When the -tuple of vectors are linearly independent, one may define a representation by setting
for each and then extending to all of by linearity. This representation extends to act on -linear functionals by duality, i.e.,
where is the transpose of . If we further define a norm on the space of -linear functionals by means of the formula
then the formula (2) becomes
To see that is a density as promised, the first step is to demonstrate that when are linearly dependent. In this case, there must exist an invertible matrix such that , and without loss of generality, one may assume that this matrix has been normalized so as to belong to . Now for each , let be the transpose of the matrix obtained by scalar multiplying the first row of by and all remaining rows by . These matrices belong to for all , and
by multilinearlity of because
by homogeneity if each is not equal to one, and if any index does equal one, then both sides vanish, making the equality (4) true trivially. Taking shows that the infimum in (2) over all must vanish when are linearly dependent.
Now let be any linear transformation of . When are linearly dependent or when is not invertible, will be linearly dependent, so it must hold that
Otherwise, when are linearly independent and is invertible, there is a matrix with such that for each . Factor as for some with in general and when is odd. Once again, by multilinearity of ,
Since is a group, the set of matrices of the form when is itself exactly assuming that . If , then the matrices for are exactly those matrices which belong to after the first two rows of are interchanged. Since (5) is invariant under permutations of the rows of , it follows in both cases () that
which gives the desired identity
for any and any linear transformation , as asserted.
It is illuminating to compute in the specal case when is a symmetric bilinear form. Fix linearly independent vectors and define the matrix by . It follows that
Now is symmetric and positive semidefinite, so its eigenvalues are all nonnegative. Thus the AM-GM inequality implies that
with equality when all eigenvalues are equal (which, when is invertible, can be attained for some by building from a basis of unit-length eigenvectors of with respect to some inner product and then rescaling the eigenvectors appropriately). Because , and therefore
In particular, on a Riemannian manifold, setting equal to the metric tensor yields a tensor density which is exactly equal to a dimensional constant times the corresponding Riemannian volume density.
At this point, the reader may be somewhat understandably disappointed by the abstract nature of the infimum appearing in (2) since it is not immediately apparent how to compute the infimum in finitely many operations. However, the abstract nature of the definition (2) turns out to be a blessing rather than a curse, because it effectively allows the analysis to entirely sidestep the very deep and rich algebraic question of what (2) computes. It turns out that (2) is deeply connected, both algebraically and analytically, to the problem of finding polynomials in the entries of the tensor which are invariant under the action of the representation . Hilbert [hilbert1893] showed that, when the group is replaced by , there are finitely many polynomials invariant under the action of which generate the algebra of all such invariant polynomials. From this fact it is easy to see that the same result must be true for itself. A vast body of literature shows (via Weyl’s unitarian trick) shows that the same result holds for representations of any group which is a real reductive algebraic group (which, as far as the present paper is concerned, is a class which includes and is closed under Cartesian products). It is possible in principle to compute these polynomials explicitly in finite time (see Sturmfels [sturmfelsbook]), but in general going about the calculation in this way is somewhat unwieldy and akin to the computation of the determinant via its permutation expansion rather than by more efficient, symmetry-exploiting techniques. In any case, the density (2) simultaneously captures the behavior of all invariant polynomials at once, as demonstrated by the following lemma:
Suppose that is a real reductive algebraic group and that is a -representation on some finite-dimensional real vector space equipped with a norm . Let be any collection of homogeneous polynomials of positive degree in which generate the algebra of all -invariant polynomials. Then there exist constants such that
To prove the first inequality, observe by scaling that
for all and all , where is the supremum of on the unit sphere of . Moreover, because each is invariant under ,
so taking an infimum in and a supremum in gives
To prove the reverse inequality, suppose for the sake of contradiction that it does not hold for any finite . Because the inequality is homogeneous in the norm , its failure would imply that one could find a sequence with for all such that
Moreover, by replacing by for suitable and taking a subsequence by compactness, it may be assumed that converges to some in the unit sphere as . By continuity of the polynomials , for all . Therefore belongs to the so-called nullcone of the representation and by the real Hilbert-Mumford criterion, first proved by Birkes [birkes1971], there must exist a one-parameter subgroup of such that as . This, of course, implies that . However for all implies that for all and all , which means by continuity that for all , so must be contradicted. ∎
The inequality (6) shows that the numerical value of is, in rough analogy with the symmetric bilinear form example just examined, comparable to the maximum of appropriate powers of the invariant polynomials applied to . It is also worth observing that when many invariant polynomials exist (which, unlike the symmetric bilinear form case, is generally the more common situation), the nullcone of tensors for which will have codimension greater than one. In terms of affine curvature, this will mean that for general submanifolds of dimension in , it is typically “easier” to have nonvanishing affine curvature than it is in the case of hypersurfaces because the space of “flat” Taylor polynomial jets which must be avoided is often of codimension greater than one.
2.2 Construction of the affine curvature tensor and associated measure
We move now to the construction of a covariant tensor which captures the affine curvature we are interested in. This tensor will be given an associated density using the formula (2) which can be integrated to give a canonical measure on immersed submanifolds .
Suppose that is a manifold of dimension which is equipped with a smooth immersion . For convenience, let the values of be regarded as column vectors. For any positive integer , let be the smallest integer such that the dimension of the space of real polynomials of degree in variables has dimension at least , and let be the index set
The index set is represented pictorially in Figure 2 as the first columns of boxes. The cardinality of is exactly the homogeneous dimension defined in the introduction. We are going to define a -linear covariant tensor at each point which captures the affine geometry of the immersion . We will denote the action of on -tuples of vectors by either
depending on which approach is most convenient at the moment (where we lexicographically order the elements of when such an order is not otherwise specified).
Now for any finite sequence of vector fields indexed by , let
Here the determinant of an -fold wedge of vectors in is understood to equal the determinant of the matrix whose columns are the factors of the wedge (technically these factors are not unique, but the antisymmetry of the determinant and of wedge products guarantees the same determinant for any factorization). In other words, (7) equals the determinant of an matrix whose -th column is the column vector . (Note also that the lexicographic order on corresponds exactly to the order that each appears in the above formula when moving from left to right; with respect to Figure 2, the order is left-to-right followed by bottom-to-top.)
This object will be called the affine curvature tensor at . First observe that it is certainly linear in for each . To see that depends only on the pointwise values of the at and not any derivatives of these vector fields, it suffices to show that any single one of the vector fields may be replaced by any other vector field agreeing with at without changing the value of . For any indices such that , this invariance under replacement follows immediately from the fact that these vector fields appear alone in their own column (i.e., the formula (7) contains no derivatives of to begin with). For any with , the identity
(where indicates omission of in its usual place) shows that vanishes when is replaced by : the number of columns of the matrix in (7) for which is differentiated to some order between and is strictly greater than the dimension of the vector space generated by such operators, so there must be linearly dependent columns in the matrix, which forces to vanish.
We begin with the following elementary lemma which gives an estimate for the volume of the convex hull of certain sets :
Suppose is a compact set containing the origin, and let be its convex hull. There exist such that the sets
Let be the unique vector subspace of of smallest dimension which contains (where uniqueness holds because the intersection of two subspaces containing would be a subspace of smaller dimension also containing ). Let denote the dimension of , and let be any nontrivial alternating -linear form on . Let be any -tuple at which the maximum of the function
is attained. Since is not contained in any subspace of smaller dimension, unless (in which case and the lemma is trivial). Now by Cramer’s rule, for any ,
where, in this case, the circumflex indicates that a vector is to be omitted from the determinant. In the particular case when , the -tuple belongs to the set over which the supremum of was taken; therefore each numerator has magnitude less than or equal to the denominator. Thus belongs to the parallelepiped
Since is convex and contains , it must contain as well. To establish the lemma, we extend the sequence to a sequence of length by fixing for . Trivially for this choice, so the containment must hold. For the remaining containment, observe that must belong to since they belong to . Therefore, by convexity of , the set must be contained in . The volume inequality (9) follows from the elementary calculation of the volumes of and . ∎
With Lemma 2 in place, we turn now to the proof of Parts 1 and 2 of Theorem 1. Pick any point and fix any smooth coordinate system near so that the immersion may be regarded in these coordinates as a function from a neighborhood of the origin (chosen so that are the coordinates of ) into . By Taylor’s formula, for all with ,
for any finite , where each remainder term is continuous on and equals when . (For most of what follows, will be regarded as a fixed but otherwise arbitrary point with .) For definiteness, let , i.e., equals the highest order of differentiation that appears in a column of the matrix whose determinant forms (or equivalently, is the number of boxes in column of the diagram given in Figure 2). This choice of implies that the dimension of the space of polynomials of degree with no constant term is at least equal to . For any , let be the compact subset of given by
and let be the convex hull of . Now each term in either sum on the right-hand side of (10) belongs to whenever and . Because the total number of summands on the right-hand side is at most some constant depending only on and , the difference vector must belong to the dilated set whenever and . In particular, this implies that the translated set must contain the vector whenever and .
By virtue of (9), the Lebesgue measure of the set is as since it is dominated by a constant depending on and times a determinant for some and since is by definition the smallest integer which it is possible to express as a sum of degrees of distinct, nonconstant monomials in variables (thus corresponds to the smallest possible factor of which will appear via scaling in such determinants). In fact, a slightly stronger result is also true: namely, that it is possible to quantify the implied constant in this estimate in terms of the affine curvature tensor at . For any collection of monomials such that , it is possible to find indices for each (these indices being obtained by “expanding” each as a composition of first-order coordinate derivatives) so that
Therefore it follows from (9) that when , the image is contained in , which is a compact convex set with volume no greater than
as , where is some new constant depending only on and . Consequently, if is any measure on satisfying the restricted Oberlin condition (1) with exponent and constant , then
for any with with an implied constant depending only on and . If , this implies that must be absolutely continuous with respect to Lebesgue measure on on a -neighborhood of the chosen origin point , and if , it further implies that must be the zero measure on that neighborhood (since the Radon-Nykodym derivative of with respect to Lebesgue measure must vanish at every Lebesgue point, which is almost every point in the neighborhood), thus establishing Part 1 of Theorem 1. When , because must be absolutely continuous with respect to Lebesgue measure, it must follow that