Sobolev norm estimates for a class of bilinear multipliers

We consider bilinear multipliers that appeared as a distinguished particular case in the classification of two-dimensional bilinear Hilbert transforms by Demeter and Thiele [9]. In this note we investigate their boundedness on Sobolev spaces. Furthermore, we study structurally similar operators with symbols that also depend on the spatial variables. The new results build on the existing L^p estimates for a paraproduct-like operator previously studied by the authors in [5] and [10]. Our primary intention is to emphasize the analogies with Coifman-Meyer multipliers and with bilinear pseudodifferential operators of order 0.


Introduction
Multilinear Fourier multipliers that are singular along linear subspaces of relatively high codimension have attracted some attention since the breakthrough papers of Lacey and Thiele [11], [12] on L p boundedness of the bilinear Hilbert transform, πi sgn(η − ξ)e 2πix(ξ+η) f (ξ) g(η)dξdη. (1.1) A more general result for multilinear singular multipliers obtained using similar techniques can be found for instance in a paper by Muscalu, Tao, and Thiele [13]. Recently, Demeter and Thiele discovered certain new phenomena in higher dimensions in an attempt to classify and study two-dimensional analogues of (1.1). Their paper [9] is concerned with L p estimates for two-dimensional bilinear multipliers T µ (f, g)(x, y) := R 4 µ(ξ 1 , ξ 2 , η 1 , η 2 )e 2πi(x(ξ 1 +η 1 )+y(ξ 2 +η 2 )) f (ξ 1 , ξ 2 ) g(η 1 , η 2 )dξ 1 dξ 2 dη 1 dη 2 , (1.2) with a symbol µ(ξ 1 , ξ 2 , η 1 , where A, B ∈ M 2 (R) are linear maps on R 2 and m ∈ C ∞ R 2 \{(0, 0)} satisfies Note that the singularity of µ can be a subspace of R 4 of codimension 2, making (1.2) substantially more singular than Coifman-Meyer multipliers [7]. An additional structural difficulty observed in [9] are degeneracies in the way the singular subspace is positioned with respect to the planes ξ 1 = ξ 2 = 0 and η 1 = η 2 = 0. Some of those degeneracies had to be handled with a mixture of one-dimensional and two-dimensional techniques, while the particular case A = 1 0 0 0 and B = 0 0 0 1 (1.4) could not be resolved using wave-packet analysis. Moreover, (1.4) was identified as the only such case (modulo trivial transformations) and left out from the discussion. If one further chooses m associated with a single "cone" in the frequency domain, then T µ will become a paraproduct-like operator studied by each of the authors in [5] and [10] respectively. More precisely, take two Schwartz functions ϕ and ψ such that ψ(ξ) is supported in 1 2 ≤ |ξ| ≤ 2 and consider the symbol Its continuous-scale variant can be written as and there is no significant difference between these two models. Operator (1.5) is sometimes called the twisted paraproduct, due to its paraproduct-like structure. It satisfies a certain range of L p estimates, with constants depending on the first several Schwartz seminorms of ϕ and ψ. We refer to [10,Theorem 1] for the precise formulation of that result. From the known L p bounds for (1.5) one can then handle L p estimates for the general multipliers T µ with A, B as in (1.4) and with m satisfying (1.3), as was briefly commented in [10]. Some details of that reduction can also be found in Section 3. On the other hand, it is a classical fact that properly supported Coifman-Meyer multipliers satisfy a range of Sobolev norm estimates. Following the analogy between ordinary paraproducts and the twisted paraproduct we believe it is of interest to formulate and prove Sobolev estimates for (1.5) and for the more general class of multipliers to which it belongs. In order to state our results we denote the two-dimensional non-homogeneous Sobolev norm of f by f W s,p , and we also introduce a particular case of the mixed (or partial) Sobolev norm f L p Let us agree to write A B whenever A ≤ CB holds with some constant C > 0. The set of parameters on which C depends will usually be clear from the context; otherwise we will state it in the text or denote it in the subscript. We also write A ∼ B when both A B and B A are satisfied.
As we said, we consider bilinear operators of the form defined for Schwartz functions f, g ∈ S(R 2 ). It turns out that the simplest nontrivial Sobolev estimate one can hope for is a bound from L p × W s,q to L r y (W s,r x ), at least in the particular case of (1.5). Before stating our positive results, let us formulate a simple proposition, which shows the necessity of imposing some support conditions on general m.
holds for some choice of 1 < p, q, r < ∞, 1 p + 1 q = 1 r and s > 0, then every super-level set of m has to lie inside a region of the form |τ 1 | 1 + |τ 2 |. More precisely, for each γ > 0 there exists a constant c γ > 0 such that The above proposition also explains the need to break the symmetry between the roles of f and g in order to have nontrivial estimates for T m (f, g). Since we imposed a homogeneous condition on m, we find it natural to discuss only symbols that are actually supported in a half-plane |τ 1 | |τ 2 |. Now we can formulate the first boundedness result.
Note that the symbol of the operator (1.5) fulfills requirement (1.7) if ϕ is compactly supported. Indeed, the proof of Theorem 2 will begin by establishing (1.8) for a more general variant of (1.5). Restricting to the subregion r < 2 is required in order to be able to apply the results from [10], but we do not know if it is necessary for the estimate to hold.
Finally we turn to (x, y)-dependent symbols σ ∈ C ∞ (R 4 ) satisfying non-homogeneous estimates of order 0, (1.10) Theorem 3. Suppose that σ ∈ C ∞ R 4 satisfies Estimates (1.9) and define bilinear multiplier operator T σ by (1.10). If there exists a constant c > 0 such that for every (x, y) ∈ R 2 holds for s ≥ 0 and the exponents 1 < p, q, r < ∞ such that 1 No extra work will be needed for the proof of Theorem 3 as it is derived from the previous result using a variant the Coifman-Meyer "freezing argument" from [6].
Let us comment on our motivation for formulating Theorems 2 and 3. We wanted to point out analogies between bilinear operators defined by (1.6) and (1.10) on the one side, and the Coifman-Meyer multipliers [7], [8] and bilinear pseudodifferential operators [2], [3], [4] on the other. Even though most of the reductions we perform can already be found in the literature, the auxiliary results we rely on were established by somewhat different techniques in [5] and [10], so the remarked connections might not be instantly visible.

Proof of Proposition 1
We begin with a quite expected L ∞ bound on the symbol required for any L p estimates.
Lemma 4. A necessary condition for a bilinear Fourier multiplier T µ defined by (1.2) to be bounded from L p × L q to L r for some choice of 1 < p, q, r < ∞ such that 1 r = 1 p + 1 q is that its symbol µ is a.e. bounded. Moreover, we have Proof of Lemma 4. Let us fix two points ξ 0 = (ξ 0 1 , ξ 0 2 ) and η 0 = (η 0 1 , η 0 2 ) in R 2 . Choose a Schwartz function Φ with Fourier transform compactly supported in [−1, 1] 2 and satisfying Φ(0, 0) = 1. Also choose another Schwartz function Ψ such that Ψ = 1 on [−2, 2] 2 . In particular, Consider the functions defined by so the assumed boundedness of T µ implies the estimate uniformly in ǫ > 0. However, using Plancherel's identity we obtain which in a combination with (2.2) proves that for almost every From (2.3) we finally obtain (2.1).
Proof of Proposition 1. Observe that boundedness of T m from L p ×W s,q to L r y (W s,r x ) is equivalent to boundedness of T ν from L p × L q to L r , where T ν is a multiplier with the new symbol Applying Lemma 4 we get that In particular, for ξ 1 , η 2 satisfying |m(ξ 1 , η 2 )| ≥ γ > 0 by choosing η 1 = 0 we obtain which yields the desired support condition.
Lemma 5. Suppose that ϕ(ξ) is supported in |ξ| ≤ c for some c > 0 and that ψ(ξ) is supported whenever 1 < p, q, r < ∞, 1 p + 1 q = 1 r > 1 2 , s ≥ 0. We will present two slightly different proofs. The first one will be a bit more elegant but it will prefer integer values of s, while the second one will address general s ≥ 0 directly and avoid the use of interpolation.
First proof of Lemma 5. By linearity of Expression (3.1) in ϕ, ψ, and λ we may assume that ϕ α,β , ψ α,β ≤ 1 for 0 ≤ α, β ≤ 3 and that |λ k | ≤ 1 for k ∈ Z. Next, we make use of non-isotropic dilations D a h(x, y) := h(2 −a x, y) to reduce to the case when c = 1. Indeed, if a is an integer such that 2 a ≥ c, then by writing we effectively replace ϕ(t) with ϕ −a (t) = 2 −a ϕ(2 −a t) in the desired estimate. The Sobolev norms in question can change up to a constant that depends on s and c, but this is allowed. It remains to observe that ϕ −a (ξ) is now supported in |ξ| ≤ 1.
Because of s ≥ 0 we have so one actually needs to bound L r norms of Π ϕ,ψ,λ (f, g) and ∂ s x Π ϕ,ψ,λ (f, g). In this proof we find convenient to work with integer values of s. Arguments very similar to the complex interpolation of Sobolev spaces can then extend the inequality to general nonnegative s; for example see [1].
The estimate actually follows from [10, Theorem 1]. The only difference is the existence of coefficients λ k in the definition of (3.1). However, one can insert arbitrary bounded coefficients in Equation (6.6) of [10, Section 6] and hence arbitrary bounded coefficients can also appear in the definition of T c in [10, Section 1]. We turn to bounding the higher derivatives and for this we apply the product rule, The term for β = 0 is precisely Π ϕ,ψ,λ (f, ∂ s x g) and (3.2) implies Π ϕ,ψ,λ (f, ∂ s x g) L r p,q,r f L p ∂ s x g L q s f L p g W s,q .

(3.4)
In the rest we work with a fixed integer β such that 1 ≤ β ≤ s. We have where the functionsψ k,j are defined by Observe that the Fourier transform ofψ k,j is a smooth function adapted to the annulus 2 k+j−1 ≤ |ξ| ≤ 2 k+j+1 . After expanding we apply the Cauchy-Schwarz inequality in k ∈ Z. Using the classical Littlewood-Paley inequalities in x and y variables respectively, we obtain so summing in j finally gives Alternative proof of Lemma 5. Begin with the same normalizations and reductions as before. This time we let ∆ k denote a general smooth 1-dimensional Littlewood-Paley truncation at frequency scale 2 k . We do not insist on the fixed choice of truncations from the beginning of this section, but rather allow the smooth cutoffs to change from expression to expression and from line to line, and still use the generic notation ∆ k . Whenever we have an equality that holds up to the change of the Littlewood-Paley cutoffs, we write it using the sign ≃.
Because of s ≥ 0, we have .
Since the L r norm of Π ϕ,ψ,λ (f, g) is controlled by (3.2), one actually needs to study ∆ ℓ,x Π ϕ,ψ,λ (f, g). As usually, for two functions h 1 , h 2 we have Applying the above decomposition to ∆ ℓ,x Π ϕ,ψ,λ (f, g) we obtain these three terms: and Term I 1 ℓ . We observe that ∆ j 1 ,x P ϕ k x ≃ ∆ j 1 ,x for j 1 ≤ k and that it vanishes if j 1 > k. Thus, the sum in I 1 may be reduced to the following model sum, Here we use the generic notation for φ and the associated convolution operator. By usual estimates involving the Hardy-Littlewood maximal function M we get Hence for s > 0 by Young's inequality, . Using the ℓ 2 -valued inequality for the maximal function we obtain Term I 2 ℓ . By similar considerations we observe that this term can be reduced to the following model sum, Computing the sum over j 1 and ℓ it may be reduced to a twisted paraproduct applied to f and ∆ ℓ,x g, I 2 ℓ ≃ Π ϕ,ψ,λ (f, ∆ ℓ,x g).
Using the L p × L q → L r boundedness of Π ϕ,ψ,λ we also know that it admits ℓ 2 -valued estimates. Hence, Term I 3 ℓ . By similar considerations we observe that this term can be reduced to the following model sum, Computing the sum over k we obtain Reasoning as for the first term gives This completes the proof of Sobolev estimates for the three terms and thus also establishes Lemma 5. Now we are ready to prove the general result by a somewhat standard decomposition of (1.6) into a rapidly convergent series of twisted paraproducts (3.1).

Proof of Theorem 3
In this section we only list the main ingredients and give references, as the proof closely follows excerpts from the existing literature.
Proof of Theorem 3. The way to get boundedness for symbols with smooth dependence on spatial variables is now well-known and it uses the Coifman-Meyer "freezing argument" [6]. Therefore we only sketch the proof and refer the reader to [4, Theorem 2.4] for a very similar argument adapted to some other bilinear Sobolev estimates. The procedure relies on local estimates and we only have to check the following local version of (1.8): For every two-dimensional ball B of radius 1 and every (x, y)-independent symbol σ(x, y, τ 1 , τ 2 ) = m(τ 1 , τ 2 ) satisfying (1.9) one has T m (f, g) L r y (W s,r x )(B) ≤ j 1 ,j 2 ≥0 γ j 1 ,j 2 f L p (2 j 1 B) g W s,q (2 j 2 B) , (4.1) with some rapidly decreasing sequence of coefficients γ j 1 ,j 2 . Following the reduction in the proof of Theorem 2 it is sufficient to check (4.1) for modified twisted paraproducts of the form Π ϕ,ψ,λ , which correspond to the symbols m(ξ 1 , η 2 ) = k≥0 λ k ϕ(2 −k ξ 1 ) ψ(2 −k η 2 ).
Here we restrict the sum to nonnegative indices k, corresponding to frequencies at scales larger than 1, since we are only concerned with non-homogeneous regularity condition on the symbol. However, the kernel of this operator (given by the inverse Fourier transform of the symbol) is of the form K m (y 1 , y 2 ), (z 1 , z 2 ) = K(y 1 , z 2 ) and it satisfies |K(y 1 , z 2 )| (|y 1 | + |z 2 |) −N for an arbitrarily large exponent N. Thus it is easy to check that the bilinear operator satisfies (4.1) with coefficients |γ(j 1 , j 2 )| ∼ (2 j 1 + 2 j 2 ) −N . Indeed, the diagonal part j 1 , j 2 ≤ 4 is resolved by Theorem 2 and then the non-diagonal part max{j 1 , j 2 } > 4 is an easy consequence of the pointwise decay of the kernel.
As a consequence, it is not difficult to see that (4.1) holds for every (x, y)-independent symbol satisfying (1.9). Therefore this estimate also holds for (x, y)-dependent symbols, due to the Coifman-Meyer freezing argument, as we have already commented. Finally, summing these local estimates over a covering of the whole space by balls of radius 1, we obtain the required global estimate.