On certain trilinear oscillatory integral inequalities

1.1 Some background

Inequalities of the form

with γ > 0 and C < ∞ dependent on ϕ and on η, have been analyzed in various works. Hörmander [1] established the fundamental upper bound O | λ | − d / 2 f 1 2 f 2 2 for the bilinear case |J| = 2, with the mixed Hessian ∂ 2 ϕ ( x , y ) ∂ x ∂ y everywhere nonsingular. The bilinear case, with x, y in spaces of unequal dimensions, has been intensively studied in connection with Fourier restriction inequalities. Likewise, in connection with Fourier restriction, multilinear forms have been investigated, in which each function f _j is individually acted upon by a linear oscillatory integral operator, and the product of the resulting functions is integrated. Such multilinear forms are not studied here.

Forms | T λ ϕ ( f ) | and |S _λ (f)|, with |J| ≥ 3, have been studied by Phong-Stein-Sturm [2], Gilula-Gressman-Xiao [3], and others. An introductory treatment can be found in Ref. [4]. The works [3], [2] deal with general phase functions ϕ, and seek optimal relationships between ϕ, decay exponents γ, and Lebesgue exponents p _j [2]. Also emphasizes the issue of stability – whether the optimal exponent γ is a lower semicontinuous function of ϕ.

The regime |J| > D/d is singular, in the sense that the integral extends only over a positive codimension subvariety of the Cartesian product of the domains of the functions f _j . Variants of the form (1.2), in the singular regime and with all mappings φ _j linear, were investigated by Li, Tao, Thiele and the present author [5]. They established conditions under which there exists an exponent γ > 0 for which a corresponding inequality holds. One of the results of the present paper relaxes the assumption of linearity. Another treats certain cases with φ _j linear that were not treated in Ref. [5], and provides an alternative proof of one of the main results of that work.

Results of this type under lower bounds for certain partial derivatives of the phase function, but with no upper bounds at all, have been investigated by Carbery and Wright [6] for |J| ≥ 3, building on earlier work [7] for the bilinear case |J| = 2. This thread is not developed further in the present paper, in which upper bounds are implicit through smoothness hypotheses on phase functions.

For certain ranges of exponent tuples p = (p _j : j ∈ J), the works [3], [2] establish upper bounds for (1.1) in terms of products of L p j norms of the factors f _j , with optimal exponents γ as |λ| → ∞, up to powers of log(1 + |λ|), where “optimal” means largest possible under indicated hypotheses on ϕ for given p. We will not review the hypotheses of those works precisely, but their general form is significant for our discussion. For each α (with α _j ≠ 0 for at least two distinct indices j) there are certain parameters p, γ for which nonvanishing of ∂ ^α ϕ/∂x ^α at x ₀ implies validity of (1.3), with B = B(x ₀, r) for sufficiently small r > 0. This conclusion is independent of other coefficients in the Taylor expansion of ϕ about x ₀. Thus any other nonvanishing coefficients imply corresponding inequalities, and interpolation of the resulting inequalities yields further inequalities. All bounds obtained in these cited works are consequences of bounds obtained in this way, together with inclusions among L ^p spaces resulting from Hölder’s inequality.

1.2 Four questions

Thus we ask for the very best exponent γ, rather than the best for a bound in terms of ∏ j f j L p j .

For multilinear forms of the more general type (1.2), a less precise question is at present appropriate.

Roughly speaking, the difficulty in establishing inequalities (1.5) increases as the ratio d/D increases.

In the formulation (1.4), the main structural hypothesis is that the nonoscillatory part of the integrand is a product of factors f _j (x _j ). No smoothness is required of these factors, and the strongest possible size restriction, L ^∞, is imposed. As a refinement, one could ask to what degree the L ^∞ norms could be replaced by weaker L p j norms without reducing γ. One focus of the present paper is on obtaining a comparatively large exponent γ, rather than on weakening the hypotheses under which it is obtained. Although we are not able to determine optimal exponents γ in Question 1.1, we improve on the largest exponent previously known for generic real analytic phases in the trilinear case with d = 1. We find that interactions between monomial terms can give rise to upper bounds not obtainable from monomial-based inequalities.

A second focus, for related functionals, is on obtaining some decay inequality of power law type, for situations in which no decay bound was previously known, without attention to the value of the exponent γ. Theorem 4.3 is one result in that direction.

Oscillation can arise implicitly through the presence of high frequency Fourier components in the factors f _j , instead of explicitly through the presence of overt oscillatory factors e^iλϕ . This suggests a third question.

In analyzing these three questions concerning oscillatory integral forms, we are led to questions concerning sublevel sets. Let φ j : [ 0,1 ] 2 → R 1 and a j : [ 0,1 ] 2 → R 1 be smooth. To an ordered triple f of Lebesgue measurable f j : R → R , and to ɛ > 0, associate the sublevel set

Denote by |S(f, ɛ)| the Lebesgue measure of this set.

For such an inequality to hold, some condition on f is needed to exclude trivial solutions with f ≡ 0 or with every |f _j | small. Likewise, the existence of solutions f of ∑ _j a _j ⋅ (f _j ◦φ _j ) ≡ 0 must be excluded.

Situations in which there is a small family of such exact solutions, e.g. constant f or affine f, are also of interest. In such a situation, one asks instead whether the inequality (1.7) can fail to hold only for those f that are closely approximable by elements of the family of exact solutions.

One theme of this paper is the tapestry of interconnections between these questions.

1.3 Content of paper

We begin with remarks and examples placing Question 1.1 better in context. We then focus on the trilinear case, with d = 1. In all previous results for this trilinear one-dimensional case known to this author, the exponent γ in the upper bound (1.4) has been less than or equal to 1 2 . We introduce a property of the phase function ϕ, called rank one nondegeneracy. Its negation, rank one degeneracy, implies that (1.4) cannot hold for any γ > 1 2 . Our first main result is that under a mild auxiliary hypothesis, the inequality (1.4) does hold for some γ > 1 2 if ϕ is rank one nondegenerate. Building on this result, Zirui Zhou has relaxed, though not eliminated, the auxiliary hypothesis in her 2024 dissertation [8].

In a sense clarified in Section 15, generic C ^ω phase functions are rank one nondegenerate. We also explore the connection between multilinear oscillatory forms (1.2) and the multilinear oscillatory forms studied in Ref. [5].

Our main results concerning (1.4) and (1.5), respectively, are Theorems 4.1 and 4.2. Their proofs are based on decomposition in phase space, a dichotomy between structure and pseudo-randomness, a two scale analysis, and a connection with sublevel sets.

In Section 16 we use the same method to give an alternative proof of a theorem of Li, Tao, Thiele, and the author [5], and to establish an extension.

We utilize two term sublevel set inequalities (1.7) – meaning that only two general functions f _j appear in their formulation – for both scalar- and vector-valued functions as a tool in proving trilinear oscillatory inequalities. In Section 17 we reverse this flow of ideas to study special types of three term sublevel set inequalities. We consider sublevel sets of the type (1.6), with constant coefficients a _j . We use oscillatory inequalities established earlier in the paper to prove upper bounds for their Lebesgue measures under natural hypotheses on (φ _j : j ∈ {1, 2, 3}) and appropriate nonconstancy hypotheses on f. In Section 18 we develop a rather different method to study sublevel set inequalities for nonconstant coefficients a _j , in the special case in which the mappings φ _j are all linear. In Section 20 we construct an example, based on multiprogressions of rank greater than 1, concerning one of the simplest possible vector-valued sublevel set inequalities.

In most of the paper, we assume phase functions and mappings φ _j to be real analytic, rather than merely infinitely differentiable. This is done primarily because hypotheses can be formulated more simply in the C ^ω case, with its dichotomy between functions that vanish identically, and those that vanish to finite order.

There are connections between the results and methods in this paper, a much earlier work of Bourgain [9], and recent works of Peluse and Prendiville [10], [11], [12] and of Krause, Mirek, Peluse, and Wright [13] involving quantitative nonlinear analogues of Roth’s theorem, cut norms, and degree reduction. See also [14] for an exposition of some of these ideas.

The principal results of this paper were first announced in Ref. [15]. The techniques employed here have been further developed and applied in several sequels, jointly with Durcik and Roos [16] in work on multilinear singular integral operators, with Durcik, Kovač, and Roos [17] in work on averages associated to R -actions, and with Zhou [18] in work on certain bilinear maximal operators. An extension to data of limited regularity has been developed in Ref. [19]. More singular cases, such as |J| ≥ 4 in Theorem 4.2, will be treated in future work via a further extension of these techniques.

The author is indebted to Zirui Zhou for corrections and useful comments on the exposition, to Philip Gressman for a useful conversation, and to Terence Tao for pointing out the connection with the works of Bourgain, Peluse, and Prendiville. He thanks Craig Evans for serendipitously acquainting him with related work of Joly, Métivier, and Rauch [20], and for stimulating discussion.

1.4 Dedication

This paper is dedicated to Robert Fefferman. In the Fall of 1977, the author, as a beginning PhD student, attended Professor Fefferman’s course on harmonic analysis and singular integral operators at the University of Chicago. The brilliance of the lectures, the depth of the lecturer’s insight, his passion for ideas, and his kind encouragement inspired this pupil, laying a firm foundation within a few months for a career that has extended over nearly half a century, and continues. I was, am, and will remain deeply grateful.

4.1 First main theorem

The first main result of this paper is concerned with multilinear expressions

We restrict attention here to three functions f j : [ 0,1 ] → C , and integrate over [0, 1]³ rather than over a ball.

The condition that a single partial derivative ∂ 2 ϕ ∂ x 1 ∂ x 2 vanishes nowhere suffices, for C ^∞ phases ϕ without other hypotheses, to ensure that

uniformly in x ₃ [1]. Consequently

The content of Theorem 4.1 is the improvement, with appropriate norms on the right-hand side, of the exponent beyond this benchmark 1 2 .

The set of all ϕ that satisfy the hypotheses of Theorem 4.1 is nonempty, and is open with respect to the C ³ topology. The set of all 3-jets for ϕ at x ₀ that guarantee validity of the hypotheses in some small neighborhood of x ₀ is open and dense. Moreover, its complement is contained in a C ^ω variety of positive codimension in the space of jets. This is shown in Section 15.

The theorem is not valid for C ^∞ phases ϕ as stated. If ϕ were merely C ^∞, then ϕ could vanish to infinite order at a single point, without any equivalent phase ϕ ̃ satisfying ∇ ϕ ̃ | H ≡ 0 for any hypersurface H. Infinite order degeneracy at a point implies that (4.2) does not hold for any γ > 0, even with L ^∞ norms on the right-hand side of the inequality. Corresponding remarks apply to other results formulated in this paper.

The norms appearing on the right-hand side of (4.2) are L ² norms, rather than L ^∞. Thus phases that satisfy the hypotheses of the theorem enjoy stronger bounds on L ² × L ² × L ² than does the example ϕ(x) = x ₃(x ₁ + x ₂), which attains the largest possible exponent, γ = 1, on L ^∞ × L ^∞ × L ^∞, but only γ = 1 2 on L ² × L ² × L ². This phase satisfies the main hypothesis of rank one nondegeneracy, but fails to satisfy the auxiliary hypothesis of three nonvanishing mixed second partial derivatives.

We believe that under the rank one nondegeneracy hypothesis, the conclusion holds if one of the three mixed second partial derivatives vanishes nowhere, but the other two are merely assumed not to vanish identically. Theorem 4.4, below, supports this belief. A partial result in this direction has been obtained by Zhou [8].

Functions associated to ϕ by solutions of certain implicit equations arise naturally in our analysis, so it is not natural to restrict attention to polynomial phases in the formulation of the theorems, as is done in some works on oscillatory integral inequalities. Example 2.6 also demonstrates that for polynomial phases, it is not always natural to restrict to polynomial functions h _j in formulating the equivalence relation between phases or the notion of rank one degeneracy.

4.2 Second main theorem

Oscillatory factors do not appear explicitly in the formulation of our second main result, which is concerned with conditions under which the integral of ∏ _j∈J (f _j ◦φ _j ) is well-defined. If η ∈ C ⁰ has compact support in R 2 , and if ∇φ _j and ∇φ _k are linearly independent at each point in the support of η for every pair of distinct indices j, k ∈ {1, 2, 3}, and if each f j ∈ L 3 / 2 ( R 1 ) , then the product η ( x ) ∏ j = 1 3 f j ◦ φ j belongs to L 1 ( R 2 ) . This is a simple consequence of complex interpolation, since the product belongs to L ¹ whenever two of the three functions belong to L 1 ( R 1 ) and the third belongs to L ^∞. The exponent 3 2 is optimal in this respect. This leaves open the possibility that the integral might be well-defined when the f _j belong to certain Sobolev spaces of negative orders.

For p ∈ (1, ∞) and s ∈ R , denote by W ^s,p the Sobolev space of all distributions having s derivatives in L ^p .

The answer is negative without further hypotheses. In particular, it is negative whenever all φ _j are linear. But inequalities (4.3) do hold under suitable conditions, as expressed by our second main theorem.

The assumption that f ∈ L ^3/2 guarantees absolute convergence of the integral. The particular instance of Theorem 4.2 with the ordered triple ( x 1 , x 2 ) ↦ x 1 , x 1 + x 2 , x 1 + x 2 2 of mappings was treated by Bourgain [9]. An alternative proof of Theorem 4.2 has subsequently been developed by Evans [25].

In this paper, we derive Theorem 4.2 as a straightforward consequence of a variant of Theorem 4.1, illustrating the relationship between the two theorems. The validity of the inequality (4.2) for some exponent strictly greater than 1 2 is the crucial ingredient in this derivation. It is convenient in this derivation to have the strong transversality hypothesis that det(∇φ _j (x), ∇φ _k (x)) vanishes nowhere whenever j ≠ k, but this hypothesis has been relaxed in subsequent work [18] to the condition that det(∇φ _j (x), ∇φ _k (x)) does not vanish identically.

It is significant that the deduction relies on the appearance of L ² norms, rather than merely L ^∞ norms, on the right-hand side of (4.2). The tuple Φ = (φ ₁, φ ₂, φ ₃) = (x ₁, x ₂, x ₁ + x ₂) illustrates this relatively delicate distinction. Being linear, Φ does not satisfy the inequality (4.4). When the analysis used below to reduce Theorem 4.2 to (4.2) is applied to it, the phase that arises is ϕ(x ₁, x ₂, x ₃) = x ₃(x ₁ + x ₂). This is Example 2.1, for which the L ^∞ inequality holds with γ = 1, but the L ² inequality (4.2) holds only for γ = 1 2 , not for any larger exponent.

This paper is organized so that Theorem 4.2 is proved along with related results, including Theorems 4.1 and 4.4. A more direct and somewhat simpler proof of Theorem 4.2 can be extracted from the discussion; see [18], where the auxiliary hypotheses are relaxed; det(∇φ _j , ∇φ _k ) and the curvature of the web are allowed in that extension to vanish on analytic varieties of positive codimension.

4.3 Third main theorem

Our third main result is concerned with functionals of the form

with both explicit oscillatory factors e^iλψ and mappings φ j : R 2 → R 1 .

If the Jacobian determinant of x ↦ (φ _j (x), φ _k (x)) vanishes nowhere for each pair of distinct indices j, k, then | S λ ( f ) | = O f i 1 f j 1 f k ∞ for any permutation (i, j, k) of (1, 2, 3). Then the stronger conclusion (4.8) follows from the conclusion (4.7) by interpolation.

For linear mappings φ _j , two different generalizations of this inequality were proved by Li, Tao, Thiele, and this author [5]. For this linear case, and for any tuple (φ _j : j ∈ {1, 2, 3}) reducible to a linear tuple by a change of variables, Theorem 4.3 is a special case of results obtained in that work. While the method of analysis in Ref. [5] exploited linearity of φ _j , in Section 16 we sketch an alternative proof of one of the two main results of Ref. [5] by the method developed here, which allows an extension to the nonlinear case.

4.4 Variants and extensions

We next formulate a result for the special case in which ϕ is an affine function of x ₃; thus ϕ(x) = x ₃ φ(x ₁, x ₂) + ψ(x ₁, x ₂).

An example for which the conclusion (4.11) fails to hold is (φ, ψ) = (x ₁ + x ₂, x ₁ x ₂). This datum violates the hypothesis (4.10), since x 1 x 2 = − 1 2 x 1 2 − 1 2 x 1 2 + 1 2 ( x 1 + x 2 ) 2 .

Theorem 4.4 is not quite a special case of Theorem 4.1, because the hypotheses of the latter need not be satisfied; it is not assumed in Theorem 4.4 that ∂ 2 ψ ∂ x 1 ∂ x 2 is nonzero, and therefore ∂ 2 ϕ ∂ x 1 ∂ x 2 ( 0 ) could vanish.

The next result combines an explicit oscillatory factor with negative order Sobolev norms of the factors f _j in the context of Theorem 4.2.

The proofs of these theorems are organized as follows. We begin the proofs with Theorem 4.4, reducing it in Section 5 to a special bandlimited case of Theorem 4.3. We then treat that bandlimited case in Sections 6–8 and 10.

Theorem 4.1 is proved in Sections 11 and 12 by elaborating on that analysis. In Section 13 we complete the proof of Theorem 4.3, and derive Theorems 4.2 and 4.5 from the same methods and results. In Section 14 we enunciate and prove extensions to a more general framework considered by Joly-Métivier-Rauch [20], in which the condition that the factors f _j be constant along leaves of foliations is relaxed to mild smoothness along those leaves. Section 15 contains remarks concerning the hypotheses, demonstrating that these are satisfied generically, in an appropriate sense.