Abstract Abstract. Recently, Kohn constructed examples of sums of squares of complex vector fields satisfying Hörmander's condition that lose derivatives, but are nevertheless hypoelliptic. He also demonstrated optimal L 2 regularity. In this paper, we construct parametricies for Kohn's operators, which lead to the corresponding L p (1 < p < ∞) and Lipschitz regularity. In fact, our parametrix construction generalizes to a somewhat larger class of operators, yielding some new examples of operators which are hypoelliptic, but lose derivatives. Introduction Suppose X 0 , X 1 , . . . , X n are real vector fields. In a seminal paper, [Hör67] , Hörmander proved that an operator of the form Introduction is hypoelliptic provided X 0 , . . . , X n , X i , X j , X i , X j , X k , . . . span the tangent space at every point. That is, if the X i s along with commutators of all orders span the tangent space at every point. By hypoelliptic, we mean that if u is a distribution and Lu is C ∞ near a point ξ, then u is C ∞ near ξ. In fact, he proved that L is subelliptic, that is, there exists an > 0 (depending on ξ) such that if Lu ∈ L 2 s near ξ, then u ∈ L 2 s+ near ξ (here, L 2 s is the standard L 2 Sobolev space). The , above, only depends on the number of commutators it takes to span the tangent space near ξ. Following Hörmander's result, Rothschild and Stein [RS76] constructed parametricies for the above operators of Hörmander (and for more general operators). These parametrices yield proofs of the optimal regularity for Lin Lipschitz and L p Sobolev spaces (1 < p < ∞). Introduction Recently, Kohn [Koh05] studied the analogous question with complex vector fields in place of real vector fields. He showed that the subellipticity does not hold for general complex vector fields. More specifically, he showed: Theorem 1.1. For k > 0 an integer, there exist complex vector fields X 1,k and X 2 on a neighborhood of 0 ∈ R 3 such that X 1,k , X 2 , and their commutators of order k + 1 span the tangent space near 0, and such that if we set A = −X * 1,k X 1,k − X * 2 X 2 , then A is not subelliptic near 0, however it is hypoelliptic near 0. Here, * denotes the L 2 adjoint. Moreover, A "loses precisely k − 1 derivatives near 0 in L 2 Sobolev spaces"; that is, there exists a neighborhood U of 0 such that if u is a distribution, ξ ∈ U, and Au is in L 2 s on a neighborhood of ξ, then u is in L 2 s−k+1 on a neighborhood of ξ-and one can say no better in general. See Section 1.4 for the definition we will use of loss of derivatives. Introduction In the appendix to [Koh05] , Derridj and Tartakoff prove that Kohn's operator is analytic hypoelliptic. Soon after Kohn's paper appeared, Bove, Derridj, Kohn, and Tartakoff [BDKT06] used the same methods to prove hypoellipticity, analytic hypoellipticity, and optimal L 2 Sobolev regularity for a larger class of operators. At around the same time as Kohn's paper appeared, Parenti and Parmeggiani [PP05] constructed parametrices for a certain class of pseudodifferential operators that were hypoelliptic but lost derivatives. Subsequently, they showed ( [PP06] ) that their results could be applied to operators related to Kohn's example, yielding a parametrix construction and a proof of hypoellipticity. Their methods use the pseudodifferential calculus of [BdM74] (and also involve conjugation by a Fourier integral operator), and so do not seem to yield any sort of L p or Lipschitz regularity (we discuss the relationship between their results and ours in Section 5). Christ [Chr05] showed that a sum of squares of complex vector fields whose commutators span the tangent space at every point may not be denisephillips@example.com. In a paper which is seemingly unrelated to the above papers, Journé and Trépreau [JT06] study systems of complex vector fields that are hypoelliptic but lose derivatives. They prove regularity results in L 2 Sobolev spaces. Introduction Prior to the above results, Heller (in his thesis, [Hel86] ) expanded on an idea of Stein ([Ste82] ) and proved L p regularity (and analytic hypoellipticity) by a parametrix construction for certain left invariant differential operators on the Heisenberg group, that lose derivatives. His results do not apply to Kohn's operator, since Kohn's operator is not left invariant. Introduction Some of the results of this paper were announced in [Str06] , though the methods involved here are more complicated than those that were discussed in that paper. Moreover, we present here methods which prove Lipschitz regularity, give a more precise form of the parametrix, and yield better understanding of what terms are "lower order." Statement of results The purpose of this paper is to construct a parametrix for Kohn's operator, which proves optimal regularity in L p Sobolev spaces and Lipschitz spaces. In fact, we will be able to construct our parametrix for a somewhat more general class of operators (see below), which are not necessarily a sum of squares (indeed, not even necessarily positive operators on L 2 ). This parametrix demonstrates what sort of terms can be considered "lower order", as well as proves Lipschitz and L p Sobolev regularity (which we will show to be optimal for a subclass of our operators). In the particular case of Kohn's operator, the parametrix shows that A loses precisely k − 1 derivatives in L p Sobolev spaces (1 < p < ∞), and precisely k − 1 derivatives in Lipschitz spaces. Statement of results Let Z L = ∂ z + iz∂ t andZ L = ∂z − iz∂ t (as operators on R 3 ; these are actually left invariant vector fields on the Heisenberg group-see Section 2). Fix integers a and b such that 0 ≤ b ≤ a and 0 < a and fix t 0 ∈ R. Define the operators: Statement of results where h ∈ C ∞ H 1 , h (0, t 0 ) = 0, and A l will be a "lower order" term whose form will be made precise below. We will construct a left parametrix for A near (0, t 0 ). This parametrix will imply: Theorem 1.2. A is hypoelliptic on a neighborhood of (0, t 0 ). Moreover, A loses at most In fact, our parametrix will prove sharper regularity results, which we will make precise later on (see Section 3.3 and Section 3.4). The proof of optimality can be found in Section 4. It is worth noting that when a = b, h = 1, and A l = 0, this result implies Kohn's result-see Section 1.3. Statement of results To define A l , define the operators: Statement of results and Statement of results where g (1) and the g (i) Statement of results d ,e are arbitrary C ∞ functions. The terms A 3 , A 4 , A 5 , and A 6 should be each considered as "lower order". Indeed, if J is the parametrix we will construct for A 1 + A 2 , we will see that JA 3 , JA 4 , and JA 5 are all smoothing of order 1 − (see Section 2.4 for definitions of orders of smoothing), and therefore these terms are genuinely lower order. JA 6 will be smoothing of order 0 − but will vanish as z → 0, and so will be lower order in a much more delicate sense, which we describe in Section 3.4. Remark 1.3. These terms are lower order with respect to a left parametrix. If we were considering a right parametrix, we would have to take the adjoint of each of the above terms. For instance, the reader may note that A * 3 = 0 on the kernel of A 1 , which would come into play in the construction of the right parametrix. In this paper, we will be using the adjoint of this fact. Statement of results Our parametrix construction happens entirely at the operator level. We do not take any Fourier transforms, or microlocalize in any way. All of our localizations will be trivial, as the operators we are using are pseudolocal. This allows us to avoid most of the difficulties of Kohn's proof (see Section 1.3). Perhaps the greatest difficulty that arises in our proof is the explicit inversion of a certain Toeplitz operator (see Section 3.2); and generalizing our method seems dependent on finding a better way of dealing with such operators. Basic notation We offer the reader a quick introduction to our notation. H 1 will denote the 3 dimensional Heisenberg group (see Section 2). For coordinates in H 1 , we will use (z, t) ∈ H 1 , with z ∈ C and t ∈ R. z andz will be complex numbers, while t will be a real number, and ξ, η, and ζ will be elements of H 1 . We will write L p s (s ∈ R) for the standard L p Sobolev spaces, and Λ α (α > 0) for the standard Lipschitz spaces (for a background on Lipschitz spaces, see [Ste70] ). L (j ) will stand for a homogeneous (of order j ) left invariant convolution operator, whose convolution kernel is smooth away from 0 (see Section 2.1). Similarly, R (j ) will stand for a homogeneous right invariant convolution operator. Usually, a number in a superscript (like E (j ) ) will signify the degree of smoothing of the operator (as defined in Section 2.4). A problem arises, with (for instance) operators like L (0) which are smoothing of order 0 − (also defined in Section 2.4). We will be explicit when this difference matters. Terms like L (j ) or R (j ) will stand for arbitrary elements of that form. Thus, for instance, Basic notation really means a product of an arbitrary pair of such operators, not one operator squared. For φ, ψ ∈ C ∞ 0 , φ ≺ ψ means that ψ is equal to 1 on a neighborhood of the support of φ. When we write functions in our formulas, we will (almost always) mean them as multiplication operators. When we do not mean them as operators (and instead mean them as being in the domain of our operator) we will usually use u or v (which will usually mean an arbitrary distribution with compact support); in any case, it will be clear from context whether we mean to use the functions as multiplication operators or not. Basic notation For operators that do not necessarily commute, we will use ordered multi-index notation. For instance, if A and B do not commute, and I is a sequence of 1s and 2s, say I = (1, 1, 2, 1), then we have: Basic notation (A, B) I = AABA and we write |I | = 4, i.e. the length of the sequence. Also, if A j , j = 1, . . . , n, is a sequence of noncommuting operators, we write: Basic notation For two operators A and B (taking C ∞ 0 to distributions), we will use the notation A = B to mean equality on C ∞ 0 . This equality will extend (by continuity) to distributions with compact support in essentially every place we use it. If A and B are two operators taking distributions with compact support to distributions, we will write A ≡ B if A − B maps to C ∞ , i.e. the Schwartz kernel of A − B is C ∞ . Basic notation We will have occasion to use holomorphic maps from an open set in C to a Fréchet space. We will say such a map, f , has a pole of order j at z 0 if (z − z 0 ) j f extends to a holomorphic function on a neighborhood of z 0 (and does so for no lesser j ). We will call a map whose set of poles is discrete, a meromorphic function. All of the poles in the functions we consider will be simple (of order 1), so there is no need to discuss essential singularities. Basic notation We will call a map, f , mapping an open subset of R n to a Fréchet space, smooth if it is infinitely differentiable (in the sense of taking limits of difference quotients in the topology of the Fréchet space). Thus, holomorphic maps are smooth. Basic notation Finally, in our estimates, we will use the notation u −∞ , where u is a distribution of compact support. If we write: Basic notation we mean, for every N ∈ N, there exists a C > 0 such that: Basic notation Since every distribution with compact support is in L 2 −N for some N , this is the best sort of error term we can hope for. Kohn's proof If we define Z L = ∂ z + iz∂ t andZ L = ∂z − iz∂ t (as in Section 1.1) then Kohn's operator takes the form: Kohn's proof Loss of derivatives complicates the usual proofs of hypoellipticity. Indeed, if one were to want to apply Hörmander's methods (see [Hör67] ) to Kohn's operator, then one would start with an a priori estimate like: Kohn's proof for all u ∈ C ∞ with support in some small, but fixed, compact neighborhood around 0 (see [BDKT06] , Proposition 1.1). Then one would like to localize this estimate. That is, we replace u with φu, and try and commute φ past A with error terms that we can control. Unfortunately, that leaves error terms like: Kohn's proof where φ ≺ φ . This is much worse that what we were trying to bound. One thing Kohn realized was that φ u L 2 k−1 was far too coarse an estimate. In fact, if we imagine that φ was only localizing in the z variable, we see that we need only estimate a distribution that vanishes as z → 0 (since [A, φ] is zero on a neighborhood of 0 in the z variable). Kohn proved that, in this case, one can trade vanishing in z at 0 for regularity. Our analog of this concept can be found in Proposition 3.4. Kohn's proof A similar problem occurs when one wishes to raise the estimate (1.1) to higher Sobolev norms. I.e., if we wish to show an estimate like: Kohn's proof usually, one would replace u in Equation (1.1) by something like Kohn's proof Again, commuting the pseudodifferential operator (1 − ) s/2 past A leaves errors that are worse than what we were originally trying to bound. To take care of this problem, Kohn uses microlocalization techniques. Kohn's proof We will not be too detailed in the description of Kohn's argument, since it is quite complicated. We refer the reader to [BDKT06] and [Koh05] for details. However, we will briefly discuss Kohn's microlocalization techniques. Kohn's proof Let τ be dual to t and σ dual to (x, y). Kohn creates three microlocalizations: Kohn's proof • The 0 microlocalization: |σ| |τ | • The + microlocalization: τ |σ| • The − microlocalization: −τ |σ| Kohn's proof Kohn proves that on the 0 microlocalization, A is elliptic, and on the − microlocalization, A is subelliptic. Our analog of this result can be found in Theorem 3.3 (neither result implies the other directly, but they are similar). These results are analogous since the operator π (L) (the Cauchy-Szegö projection) is micro-supported on the + microlocalization. On the + microlocalization, the L 2 s norm can be defined in terms of the pseudodifferential opera- Kohn's proof s (which is a genuine pseudodifferential operator on the + microlocalization). Kohn's proof s commutes with A, thereby removing the problem discussed above. Loss of derivatives There are many possible definitions of loss of derivatives. We take the following definition: Definition 1.4. Let A be a differential operator on a manifold M , let U ⊂ M be an open set, and fix 1 < p < ∞ and s 0 ∈ R. We say that A loses at most r derivatives on U in L p s 0 if for every φ ≺ φ with φ, φ ∈ C ∞ 0 (U) and for every distribution u on U, we have: Loss of derivatives (where the LHS being infinite implies the RHS is infinite, and the RHS being finite implies the LHS is finite). If A loses at most r derivatives on U in L For ξ ∈ M , we use "loses derivatives near ξ" to denote the existence of an appropriate U. For instance "A loses precisely r derivatives near ξ in L p Sobolev spaces," means that there exists a neighborhood U with ξ ∈ U, such that A loses precisely r derivatives on U in L p Sobolev spaces. Loss of derivatives For loss in Lipschitz spaces, merely replace L p s with Λ s in the above (with the restriction that the parameter in Λ s must be > 0, wherever it is used). Where we said "L p Sobolev spaces," we change to "Lipschitz spaces." Remark 1.5. The above definition is not universally used. Some authors replace our r with r + m, where m is the degree of A. Loss of derivatives By the Sobolev imbedding theorem, if U is a fixed open set and if for every s, there exists r(s) such that A loses at most r(s) derivatives on U in L p s , then A is hypoelliptic on U. In particular, if A loses at most r derivatives in L p Sobolev spaces, then A is hypoelliptic. The Heisenberg group Kohn's example takes place on the so-called three dimensional Heisenberg group, H 1 . H 1 is a 3-dimensional, step 2, nilpotent Lie group, which is diffeomorphic to C × R as a manifold. We will let (z, t) ∈ C × R denote coordinates on the Heisenberg group. In these coordinates, the multiplication is given by: The Heisenberg group (z 1 , t 1 ) (z 2 , t 2 ) = (z 1 + z 2 , t 1 + t 2 + 2Im (z 1z2 )) . Thus, the identity of H 1 is 0 ∈ C × R. If we write z = x + iy, then the Lie algebra of H 1 is spanned by: The Heisenberg group here we have used the subscript L to denote that these are left-invariant vector fields. We will later need both right and left invariant vector fields. The Heisenberg group (these are the same vector fields as in Section 1.1), we see that Kohn's operator is of the form: The Heisenberg group Freezing coefficients" in the above, leaves the operator The Heisenberg group where β ∈ C. A β is a left invariant differential operator on H 1 , and in this section we will review the relevant theories of such operators. For more background on H 1 and its higher dimensional analogs, we refer the reader to [Ste93]. The Heisenberg group [Ste93] and [Fol75] together contain almost all the results that will be presented in this section, and we refer the reader to those two sources for more details and proofs. H 1 has a natural one parameter family of dilations, given by γ r (z, t) = rz, r 2 t here 0 < r ∈ R and (z, t) ∈ H 1 . It is easy to see that, for each 0 < r ∈ R, γ r defines an automorphism of H 1 . Differentiating this map, for each 0 < r ∈ R, we get an automorphism of the Lie algebra (which we will also denote by γ r ), given by The Heisenberg group For convenience, we will sometimes denote elements of H 1 by (z, t) = ζ ∈ H 1 . Euclidean measure on C × R is Haar measure on H 1 , and satisfies: The Heisenberg group we therefore call 4 the "homogeneous dimension" of H 1 . To accent its role in the theorems that follow, we will denote it by Q. Almost all of the results in this section hold for more general nilpotent Lie groups, and details may be found in [Fol75] . Left invariant convolution operators In this section, we will review the theory of convolution operators on H 1 . We will be concerned with convolution by certain "homogeneous distributions", and the theory of such operators was laid out in great generality by Folland in [Fol75] . Left invariant convolution operators If τ is given by integration against an L 1 loc function f, these two definitions agree. Definition 2.2. A distribution which is homogeneous of degree λ − Q and which is C ∞ away from 0 will be called a kernel of type λ. The vector space of kernels of type λ will be denoted by O λ . Definition 2.2. We will only be concerned with kernels of type λ, where λ ≥ 0. We therefore restrict our attention to these operators. One may topologize O λ in the following way: If λ > 0 we identify K ∈ O λ with the function f ∈ C ∞ H 1 \ {0} that it agrees with away from 0. We restrict this function to 1 ≤ |ζ| ≤ 2 (which uniquely defines the distribution by homogeneity) and therefore can identify O λ with a subset of C ∞ (1 ≤ |ζ| ≤ 2), and give O λ the subspace topology. When λ = 0, we do the same thing, but now we have to account for the fact that K = PV(f ) + Cδ (here, PV stands for the integral taken in the principle value sense, see [Fol75] ). Thus, we identify Definition 2.2. Again we give O λ the subspace topology. In either case, O λ becomes a Fréchet space. We will only use this topology briefly in the sequel. Definition 2.2. Here, u * K is the group convolution. We choose the notation Op L since this is a left invariant operator. We will also define Op R (K) u = K * u. We will write Op L (O α ) for the vector space Op L (K) : K ∈ O α , and topologize this space so that Op L is a homeomorphism. We define Op R (O α ) in a similar manner. All of the results of this section work equally well with Op R replacing Op L , provided one replaces X L and Y L with their right invariant analogs. For convenience (for the rest of this paper), we will denote by L (α) an arbitrary element of Op L (O α ), and by R (α) an arbitrary element of Op R (O α ). The following propositions are well known, and mostly contained in [Fol75] . Definition 2.2. where Definition 2.2. Remark 2.7. We used the notation L (L) 0 in Proposition 2.6 since we will define more general operators in Section 2.3. Definition 2.2. An easy consequence of Proposition 2.6 is the following: Definition 2.2. A similar result holds for Y L T . Operators of Rothschild and Stein We will now need to review a certain class of operators from [RS76] due to Rothschild and Stein (though what we introduce here will not be their operators in full generality). The operators below are also special cases of those covered in [CGGP92] . Definition 2.9. A distribution kernel K (ξ, η) on H 1 × H 1 is said to be a pseudodifferential kernel of type λ, λ ≥ 0, if for any positive integer l, we can write: Operators of Rothschild and Stein where Operators of Rothschild and Stein And we call an operator whose distribution kernel is a pseudodifferential kernel of type λ, a pseudodifferential convolution operator of type λ. Operators of Rothschild and Stein Remark 2.10. Note that, in light of the definition, the distribution K has compact support in H 1 × H 1 . Also, it is clear from the definition that if T is a pseudodifferential convolution operator of type λ ≥ 1, then X L T and Y L T are pseudodifferential convolution operators of type λ − 1. Operators of Rothschild and Stein It is shown in [RS76] that the class of pseudodifferential kernels of type λ is closed under adjoints. We will review the regularity properties of such operators in Section 2.4. From the calculus of [CGGP92] , we easily see: Operators of Rothschild and Stein Proposition 2.11. Suppose T is a pseudodifferential convolution operator of type λ, and Operators of Rothschild and Stein Proof. One can either use the calculus of [CGGP92] or see the bottom of page 568 of [CG84] for a direct proof. Operators of Rothschild and Stein Example 2.12. In this example, we will discuss the argument in the proof of Theorem 10 of [RS76] . Operators of Rothschild and Stein is a homogeneous noncommutative polynomial of degree m that is hypoelliptic and depends smoothly on α (namely its coefficients depend smoothly on α, and we think of the coefficients as being to the left of the vector fields). Since Operators of Rothschild and Stein If β : H 1 → R n is smooth, and ψ, ψ ∈ C ∞ 0 we may define the integral kernel: Operators of Rothschild and Stein If we consider X L and Y L acting in the ξ variable, and we consider the expression: Operators of Rothschild and Stein There are two types of terms. If any of the vector fields land on ψ (ξ) or on β (ξ), we are left with pseudodifferential kernel of type at least 1. This is clear when the vector fields land on ψ (ξ), so let us consider (as an example) the case when X L lands on β (ξ). We are then left with the term: Operators of Rothschild and Stein and this is again a pseudodifferential kernel of type m. Thus we may imagine all of our vector fields land on the ξ of φ β η −1 ξ so long as we are willing to work modulo error terms that are pseudodifferential kernels of type 1. But, when all the vector fields land on the ξ of φ α η −1 ξ , we may imagine β is fixed, and use the formula P β (X L , Y L ) φ β = δ 0 to see that if T is the operator with Schwartz kernel K we have Operators of Rothschild and Stein where E (1) is a pseudodifferential convolution operator of type 1. Mix of left and right convolution operators In Section 2.1, we reviewed the properties of operators defined by f → f * K where K ∈ O λ for some λ, yielding a left invariant convolution operator. The entire theory, there reviewed, works exactly the same on operators given by: Mix of left and right convolution operators provided we replace our left invariant vector fields X L , Y L , with their right invariant analogs: Mix of left and right convolution operators This then leaves us with a right invariant convolution operator. In this section, we will review the basic properties of an operator given by a composition of such a left invariant operator with a right invariant one. The author has no doubt that these results are well known, but was unable to find a suitable reference. All of the results in this section follow by standard methods, and we leave the details to the reader. Most of the results follow from the methods used in [Fol75] , with slight modifications. Mix of left and right convolution operators Remark 2.13. The results in this section are not actually necessary for our main results, since we are just constructing a local parametrix, and R (j ) φL (j ) (φ ∈ C ∞ 0 ) is an easy operator to understand: it is clearly pseudolocal, and its mapping properties follow from those of R (j ) and L (j ) . However, it will be nice for some side issues and for general convenience to understand operators like Mix of left and right convolution operators The results we need about R (β) L (α) are summarized below: Mix of left and right convolution operators • X L and Y L commute with R (β) , while X R and Y R commute with L (α) . Mix of left and right convolution operators The same thing would happen if we replaced, say, Some special operators In the beginning of this section, we introduced the left invariant vector fields Some special operators Essentially all the results in this section can be found in [Ste93], but many were introduced in [FS74] . We state the results we need without proof, leaving the details to the reader. When α is not an odd integer (here an odd integer can be either positive or negative), L Some special operators α has a fundamental solution given by: Some special operators Moreover, it is clear from Equation (2.1) that the map α → O 2 is meromorphic in α, and has simple poles at the odd integers. Some special operators For Some special operators for all α ∈ C and n an odd integer, and Some special operators for all n 1 , n 2 odd integers. Some special operators Since it will be of great importance to us, we fix the notation Some special operators 1 . All of the results of this section move over without change to the right invariant vector fields Z R ,Z R (defined byZ R = 1 2 (X R + iY R ) and similarly for Z R ). Really, these results are related to the others by the change of variables t → −t. Whence, we have P Some special operators It is well known, and easily seen, that the range of π (L) 1 is supported on D t > 0 and the range of π Degrees of smoothing The L p regularity properties of pseudodifferential convolution operators, and of left invariant convolution operators, is best understood in terms of nonisotropic L p Sobolev spaces, which we now define. Definitions of these spaces (and background on them) can be found in [Fol75, Koe02, NS79, RS76] and many other sources. Degrees of smoothing We Degrees of smoothing We use the standard notation of I L , where I denotes a sequences of 1s and 2s (an ordered multi-index, see Section 1.2). |I | denotes the length of the sequence. The non-isotropic Sobolev spaces NL p k (for k ∈ N) are given by the completion of C ∞ 0 H 1 in the norm Degrees of smoothing One may define NL p s for 0 ≤ s ∈ R by complex interpolation (see [Fol75] for a detailed account) and all of our results extend to these spaces (by interpolation), but we will not bother mentioning such details. Degrees of smoothing Definition 2.15. Suppose T is an operator taking C ∞ 0 to distributions. We say that T is smoothing of order λ − (λ ≥ 0) if T is pseudolocal and: Degrees of smoothing (1) T extends to a continuous map NL Degrees of smoothing In the above definition, the − in λ − really just refers to the available smoothing in Lipschitz spaces. These are so defined because L (0) is smoothing of order 0 − in this sense. L (j ) for j > 0 is better than smoothing of order j − , though, so we make a definition to accommodate that: Degrees of smoothing Definition 2.16. Suppose T is an operator taking C ∞ 0 to distributions. We say that T is smoothing of order λ (λ ≥ 0) if T is pseudolocal and: Degrees of smoothing (1) T extends to a continuous map NL Degrees of smoothing Similarly for Y L T . Degrees of smoothing is of the form Degrees of smoothing where f is a C ∞ function that is nonzero at 0. Degrees of smoothing Many of the technical difficulties come from the A 6 term. Indeed, the parametrix construction is far easier when A 6 = 0, and when A 6 = 0 and is of the general form above, we have to use C L functions in our construction (instead of only C ∞ functions), where L is any fixed integer we choose. This does not affect our regularity results, as we may pick an L large enough for any regularity result we wish to prove, but does make the construction much less elegant. We, therefore, proceed with the construction under the assumption A 6 = 0, and we will return to the more general case in Section 3.4. Operators which vanish at 0 This section is devoted to proving the following result: Operators which vanish at 0 Theorem 3.3. Let φ ≺ φ be a cut off functions with support sufficiently close to 0 ∈ H 1 . Then there exists a pseudodifferential convolution operator U (2) of type 2, such that: Operators which vanish at 0 where κ (1) is a pseudodifferential convolution operator of type 1. Operators which vanish at 0 In this section, A 3 , A 4 , and A 5 will be considered as lower order terms. Define A 0 = A 1 +A 2 . Note that if we have Theorem 3.3 with A 0 in place of A, then we have the theorem for A. Thus, we proceed with proving Theorem 3.3 with A 0 in place of A. Define Operators which vanish at 0 Proposition 3.4. Suppose φ is a cut off function supported sufficiently close to 0 ∈ H 1 . Then, there exists a pseudodifferential convolution operator S (2) , of type 2, such that Operators which vanish at 0 where κ (1) is a pseudodifferential convolution operator of type 1. Operators which vanish at 0 Proof. Consider, without worrying about dividing by zero for the moment, Operators which vanish at 0 where Operators which vanish at 0 Note that β (0) = 1, and so β(ζ) is as close to 1 as we like, so long as we keep ζ close to 0. Recall θ α from (2.1). θ α is meromorphic in α with a simple pole at α = 1 and satisfies Operators which vanish at 0 Thus, (1 − α) θ α is a holomorphic map α → O 2 for α near 1. Let φ ≺ φ ∈ C ∞ 0 equal 1 on a neighborhood of 0 ∈ H 1 , and 0 outside a small neighborhood of 0. Namely, we wish the neighborhood to be so small that (1 − α) θ α is holomorphic where α ranges over a neighborhood of β supp φ , and F(ζ) stays away from −1 on supp φ . Whence, on the support of φ , our considerations in (3.1) become rigorous. Operators which vanish at 0 Define the kernel K (ξ, η) (mapping from the η variable to the ξ variable) by Operators which vanish at 0 Let T (2) be the pseudodifferential convolution operator of type 2 with kernel K. By an argument similar to that in Example 2.12 we see that: Operators which vanish at 0 where E (1) is a pseudodifferential convolution operator of type 1. But, Operators which vanish at 0 . Operators which vanish at 0 T (2) , the claim follows. Operators which vanish at 0 Remark 3.5. We can now easily construct a parametrix for Kohn's operator away from z = 0. Indeed, for Kohn's operator, F(ζ) = |z| 2k , and so by Equation (3.1), we see: Operators which vanish at 0 and so −1 < β (ζ) < 1 for z = 0. Thus, if we are concerned with regularity near a point 0 = ξ 0 ∈ H 1 , we let ψ ∈ C ∞ 0 be a bump function that is 1 on a neighborhood of ξ 0 and 0 on a neighborhood of 0 and we let S be the operator with kernel (here, η = (w, s)): Operators which vanish at 0 and then by the argument in [RS76] (see Example 2.12) S will be locally (near ξ 0 ) a parametrix for A, and S is a pseudodifferential convolution operator of type 2. This proves maximal regularity in L p and Lipschitz spaces (see [RS76] ; this operator fits exactly into that framework for z away from 0). Because of this, we say that Kohn's operator is "maximally hypoelliptic" away from z = 0. Operators which vanish at 0 Remark 3.6. Proposition 3.4 implies that F(ζ) is a "subelliptic multiplier." Operators which vanish at 0 We now turn to proving Theorem 3.3. Let φ ≺ φ be nested cut off functions (of small support near 0 ∈ H 1 ), where φ plays the role of φ in Proposition 3.4. Define: Operators which vanish at 0 Then, consider, (here κ (1) will be a pseudodifferential convolution operator of type 1 that may change from line to line) Operators which vanish at 0 which completes the proof of Theorem 3.3. Operators which vanish at 0 Remark 3.7. The astute reader may note that U (2) is really the composition of two pseudodifferential convolution operators, one of type 2 and one of type 0. Indeed, this is again a pseudodifferential convolution operator, as claimed. This can be seen by the calculus of [CGGP92] . We do not really need this result however, since all we really need is that U (2) is smoothing of order 2. This comes from the fact that Operators which vanish at 0 and therefore is smoothing of order 0. Alternatively, one could see these operators in the calculus of [NS79] , and use the composition results there. Calculation of certain operators In light of Theorem 3.3, it suffices to find an operator T such that: Calculation of certain operators where E (1 − ) is smoothing of order 1 − . Let A 2 =Z L z azb Z L . Ie, A 2 is like A 2 , except with h = 1. We will achieve Equation (3.2) by constructing an M such that: Calculation of certain operators Here, we have used that we may commute A past φ with errors that are in E (1 − ) . Indeed, if any of the derivatives land on φ , we use the fact that φ ≺ φ , and that Mπ (L) is pseudolocal, to see that the error term is infinitely smoothing. Now, Calculation of certain operators is of the form covered by A 4 + A 5 and so we will have that Calculation of certain operators is smoothing of order 1 − as well. Hence, Calculation of certain operators But if φ ≺ ψ, we know from Proposition 2.11 that φπ (L) φ , ψh is smoothing of order 1, and therefore, Calculation of certain operators Actually, here we have used the fact that: Calculation of certain operators which we will also have for our specific choice of M (see Section 2.2 and Section 3.2.2). Since Mπ (L) A 2 is smoothing of order 0, and Calculation of certain operators is smoothing of order 1 (by Proposition 2.11), the third term in Equation (3.3) may be taken as part of our error term. It follows, if we set Calculation of certain operators (which is smoothing of order 0 since Mπ (L) A 2 is), we have: Calculation of certain operators We recall the operator U (2) from Theorem 3.3 (we pick φ ≺ φ in that theorem to be ψ ≺ ψ here). Hence, Calculation of certain operators where, in the last line, we have used the pseudolocality of everything involved to see that Formally calculating M In this section, we wish to formally calculate M. We will later show that these formal calculations can be made rigorous. Recall, we wish to have M such that Formally calculating M We first do our calculations for the example case when we replace A 2 by |z| 4 . This case contains essentially all of the ideas of the more complicated case for A 2 , and will offer the reader a road map for the more complicated book keeping that will arise later. Formally calculating M π (L) satisfies the equationZ L π (L) = 0. That is, (∂z + zD t ) π (L) = 0, and therefore, Formally calculating M and the adjoint result is: Formally calculating M Here we have used that Z R and π (L) commute, since left invariant convolution operators commute with right invariant differential operators. Thus, we compute: Formally calculating M (3.5) Formally calculating M This leads us to the calculation: Formally calculating M Combining Equations (3.5) and (3.6), we see (at least formally): Formally calculating M We therefore define M 0 = 16 D t P (R) 3 Formally calculating M t . This operator makes sense, since D t P (R) 1 and D t P (R) 3 are in Op R (O 0 ) and so compose well. Using that D t commutes with everything in sight, we see that (formally) Formally calculating M Here we have used that L We now turn to formally computing M in the case for A 2 with b > 0. By a computation completely analogous to the one above, Formally calculating M where the · · · denotes that from that point on, it follows as in the computation for |z| 4 . Hence, if we define Formally calculating M we see that (formally), Formally calculating M Here, as before M makes sense, since D t P (R) 2j −1 ∈ Op R (O 0 ) and so we may compose as many as we like. A similar computation shows that, when b = 0, Formally calculating M and in this case, we set Formally calculating M In either case, Formally calculating M (3.8) Formally calculating M Note that, by the results in Section 2.2, the composition Mπ (L) makes sense. Formally calculating M Remark 3.8. We now see why we needed the assumption a ≥ b in our operator. Rigorous calculations The reason the calculations in Section 3.2.1 may not be rigorous is that the operator Rigorous calculations may not make sense. To see this more simply, consider π (L) |z| 2M π (L) . The Schwartz kernel of |z| 2M π (L) grows as |z| → ∞ (for M sufficiently large) and it is therefore difficult to make sense of π (L) |z| 2M π (L) . In this section, we will discuss two ways around this problem. The first method is quite simple, and involves merely localizing all of our calculations so that no composition is hard to make sense of. The second method is a bit more complicated, but will imply (for instance) that Mπ (L) A 2 is smoothing of order 0, and so will be necessary for our purposes. Rigorous calculations Remark 3.9. This was no problem in [Str06] , as in that case Q (which is the analog of our π (L) ) had a kernel which vanished rapidly for x large, and therefore Qx 2k Q was easy to make sense of. The first method The first method involves replacing the result The first method with a local one. That is, let φ ≺ φ be nested cut off functions. We will show: The first method Using the results in Section 2.2, we see (computing now just in the case b > 0, the case b = 0 is similar): The first method (3.10) The first method Applying the RHS of Equation (3.10) to A 2 π (L) , we may now repeat the formal argument in Section 3.2.1, which is now rigorous. The only difference is that as we move differential operators around, they sometimes land on a φ -such terms are infinitely smoothing as all the operators involved are pseudolocal and φ ≺ φ . It follows that: The second method The problem we are facing in this section is that it is difficult to make sense of The second method The idea for this method is that, while we may have trouble making sense of π (L) A 2 π (L) , we will be able to make sense of: The second method where the parentheses denote the order in which we take the composition. As usual, R (j ) will denote an arbitrary element of Op R (O j ), and similarly L (j ) will denote an arbitrary element of Op L (O j ). Also, we will write R (j ) for any finite linear combination of terms of the form The second method We have already made sense of Mπ (L) on distributions of compact support (see the end of Section 3.2.1), and therefore we have made sense of Mπ (L) A 2 on distributions of compact support. We will show that Mπ (L) A 2 is a sum of terms of the form The second method , and D t R (1) L (1) . This will prove two things. First, it will make sense to compose Mπ (L) A 2 with π (L) (by the considerations of Section 2.2). More importantly, we will see in the proof that the L (0) s and R (0) s above are of the form D t L (2) and D t R (2) respectively, and therefore Mπ (L) A 2 is smoothing of order 0 by Remark 2.22 (and by Propositions 2.18 and 2.19). The reason this part is more important is that we could use the first method of this section to understand the compositions, but we need these considerations to understand the regularity properties of Mπ (L) A 2 . The second method Proposition 3.10. Mπ (L) A 2 can be written as a sum of terms of the form The second method To prove Proposition 3.10 we note (in either the case b = 0 or b > 0) that The second method We will write M (a, b) for an operator that is a sum of terms of that form. We will be inducting on a, b, a ≥ b. So consider, for a ≥ b, a > 0, The second method where J (R) ∈ Op R (O 2 ) is the right invariant analog of the operator from Proposition 2.6. Here, we have used: The second method (Here, again, we mean a sum of such terms in the above.) Consider (recalling that The second method (3.13) The second method We therefore separate Proposition 3.10 into the same result for each of the three terms on the RHS of (3.13) (when b = 0, the third term is 0). We prove these results in slightly greater generality needed here, for use in the next section. be written as a sum of terms of the form Proof. First we recall from Section 2.3 that π (L) = D t L (2) . We begin by proving the result for a = b = 1 and a = 2, b = 0 and then induct on a ≥ b. When a = b = 1, be written as a sum of terms of the form and when a = 2, b = 0, be written as a sum of terms of the form Now suppose a ≥ b and the result holds for (a, b). We prove the result for (a + 1, b), d + e = a + b − 1. We assume first that d > 0. be written as a sum of terms of the form is a sum of terms of the form be written as a sum of terms of the form Thus, we may replace be written as a sum of terms of the form in the RHS of (3.14) with a sum of such terms. But, be written as a sum of terms of the form Note, the composition be written as a sum of terms of the form actually makes sense by the equality zD t π (L) = − 1 2Z R π (L) . In a similar manner we see be written as a sum of terms of the form If instead d = 0 and e > 0 the proof follows just as above, except we use the equality: be written as a sum of terms of the form instead of the equality be written as a sum of terms of
{'paper_id': '241038', 'title': "A parametrix for Kohn's operator", 's2_url': 'https://api.semanticscholar.org/CorpusID:241038', 'source': 'S2ORC'}