A new class of pseudodifferential operators with mixed homogenities

A new class of pseudodifferential operators with mixed homogenities Po-Lam Yung University of Oxford Jan 20, 2014

Introduction Given a smooth distribution of hyperplanes on R N (or more generally on a smooth manifold), we find a class of pseudodifferential operators adapted to that. Many situations of this kind: e.g. the heat distribution on R n R +, or contact distribution on contact manifolds. Our motivation actually comes from several complex variables. On the boundaries of strongly pseudoconvex domains, there is often a need to compose two kinds of pseudodifferential operators, one with Euclidean homogeneities, another with Heisenberg homogeneities. Our new class of pseudodifferential operators is an algebra that is large enough to contain both of them, and yet small enough to consists only of pseudolocal operators. This is joint work with E. Stein.

Outline Review pseudodifferential operators on R N Review singular integrals on the Heisenberg group H n Our new class of pseudodifferential operators Geometric invariance, composition of operators, and regularity on various function spaces Applications to several complex variables

Pseudodifferential operators on R N A pseudodifferential operator is an operator of the form ˆ T a f (x) = a(x, ξ) f (ξ)e 2πix ξ dξ. R N Here a(x, ξ) is a smooth function on the cotangent bundle, called the symbol of the operator. e.g. If a(x, ξ) = 2πiξ j, then T a = x j ; if a(x, ξ) = x j, then T a is multiplication by x j. Studied in the early days by e.g. Kohn-Nirenberg and Hörmander.

A standard (elliptic / Euclidean / isotropic) symbol class of order m: a(x, ξ) (1 + ξ ) m J x α ξ a(x, ξ) J,α (1 + ξ ) m α. If a(x, ξ) is a standard symbol of order 0, then T a admits the following singular integral representation: ˆ T a f (x) = f (y)k(x, y)dy R N where K(x, y) satisfies estimates K(x, y) 1 x y N γ x δ y K(x, y) γ,δ 1 x y N+ γ + δ and some suitable cancellation conditions.

The Heisenberg group H n The Heisenberg group H n is a non-abelian Lie group diffeomorphic to C n R. It arises in several complex variables as the boundary of the upper half space in C n+1. We write a point x H n as x = (z, t) C n R. The group law on H n is then given by (z 1, t 1 ) (z 2, t 2 ) := (z 1 + z 2, t 1 + t 2 + 2Im(z 1 z 2 )). For y H n, the group inverse of y will be denoted y 1. Haar measure is the Lebesgue measure dy on R N.

Two different homogeneities on H n Automorphic dilation: δ λ (z, t) := (λz, λ 2 t), λ > 0. Automorphic dimension Q := 2n + 2. Non-isotropic norm: (z, t) := z + t 1/2. Euclidean dilation: λ(z, t) := (λz, λt), λ > 0. Euclidean dimension N := 2n + 1. Euclidean norm: (z, t) := z + t. We will write ξ = (η, τ) C n R for the dual variables to x = (z, t). We will study two classes of singular integral operators on H n.

Two classes of singular integral operators on H n First, identifying H n with R N, we have the standard Calderon-Zygmund operators on R N. These are operators of the form ˆ Tf (x) = f (y)k(x y)dy, R N where K(z, t) is a compactly supported kernel that satisfies K(z, t) 1 (z, t) N γ z δ t K(z, t) γ,δ 1 (z, t) N+ γ +δ and some suitable cancellation conditions. We call these singular integral operators with Euclidean (or isotropic) homogeneities.

Second, we have the following automorphic class of singular integral operators. These are operators of the form ˆ Tf (x) = f (y)k(y 1 x)dy, H n where K(z, t) is a compactly supported kernel that satisfies K(z, t) 1 (z, t) Q γ z δ t K(z, t) γ,δ 1 (z, t) Q+ γ +2δ and some suitable cancellation conditions. (Note the distinction between the good derivatives z and the bad derivative t.) We call these singular integral operators with Heisenberg (or non-isotropic) homogeneities.

Either class of singular integral operators form an algebra under composition. However, there are situations when one wants to compose a singular integral operator with one homogeneity with another operator with a different homogeneity. e.g. Solution of -Neumann problem. In studying such compositions, flag kernels have been introduced in the 90 s; c.f. Muller-Ricci-Stein (1995-96), Nagel-Ricci-Stein (2001). These flag kernels form an algebra, in which the singular integrals with different homogeneities can be composed, but the class of flag kernels is too big - they fail to be pseudolocal. In another attempt, Nagel-Ricci-Stein-Wainger (2011) introduced the following class of pseudolocal singular integral operators on H n, where the kernels have mixed homogeneities:

A class of singular integral operators on H n with mixed homogeneities This is the class of operators of the form ˆ Tf (x) = f (y)k(y 1 x)dy, H n where K(z, t) is a compactly supported kernel that satisfies K(z, t) 1 (z, t) 2n (z, t) 2 γ z δ t K(z, t) γ,δ 1 (z, t) 2n+ γ (z, t) 2+2δ and some suitable cancellation conditions. It contains the two classes of singular integral operators introduced earlier, and forms an algebra under composition. Furthermore, operators in this class are pseudolocal.

Two Questions Q1. For each of the above three algebras of singular integral operators, is there a corresponding variable coefficient theory? This means a theory where one replaces the kernels K(x y) (or K(y 1 x)) above by a more general kernel K(x, y), that satisfies the same kind of estimates. Certainly doable for the class of singular integral operators with Euclidean homogeneities. But being able to do so is important in applications since we would like to adapt these singular integrals to boundaries of strongly pseudoconvex domains, and a variable coefficient theory is necessary for such applications. Beals-Greiner (1988) answered this question for the class of singular integral operators with Heisenberg homogeneities.

Q2. If one could find a variable coefficient theory for these classes of singular integral operators, can one find a pseudodifferential realization of these operators? This means a theory where one rewrites such singular integral operators as pseudodifferential operators, namely ˆ T a f (x) = a(x, ξ) f (ξ)e 2πix ξ dξ, R N and characterizes all the symbols a(x, ξ) that arises this way. Again well-known for the class of singular integral operators with Euclidean homogeneities; these are just the standard pseudodifferential operators on R N we recalled earlier. Beals-Greiner (1988) answered this question for the class of singular integral operators with Heisenberg homogeneities.

Our goal: to answer both questions above for the class of singular integral operators with mixed homogeneities. In fact, our theory will allow treatment of pseudodifferential operators that are of different orders (not just 0th order ones) We will study these operators in a more general set-up, that applies not only to the Heisenberg group, but to a situation where a distribution of hyperplanes is given on R N. We will also see some geometric invariance of our class of operators as we proceed.

Our set-up Suppose on R N, we are given a (global) frame of tangent vectors, namely X 1,..., X N, with X i = N j=1 A j i (x) x j. We assume that all A j i (x) are C functions, and that J x A j i (x) L (R N ) for all multiindices J. We also assume that det(a j i (x)) is uniformly bounded from below on R N. Let D be the distribution of tangent subspaces on R N given by the span of {X 1,..., X N 1 }. Our constructions below seem to depend on the choice of the frame X 1,..., X N, but ultimately the class of operators we introduce will only depend on D. No curvature assumption on D is necessary!

Geometry of the distribution We write θ 1,..., θ N for the frame of cotangent vectors dual to X 1,..., X N. We will need a variable seminorm ρ x (ξ) on the cotangent bundle of R N, defined as follows. Given ξ = N i=1 ξ idx i, and a point x R N, we write ξ = N (M x ξ) i θ i. i=1 ((M x ξ) i = N j=1 Aj i (x)ξ j will do.) Then ρ x (ξ) := N 1 i=1 (M x ξ) i. We also write ξ = N i=1 ξ i for the Euclidean norm of ξ.

The variable seminorm ρ x (ξ) induces a quasi-metric d(x, y) on R N. If x, y R N with x y < 1, we write { } 1 d(x, y) := sup : (x y) ξ = 1. ρ x (ξ) + ξ 1/2 We also write x y for the Euclidean distance between x and y. Example: X j = x j for j = 1,..., N. Then D is the distribution of horizontal hyperplanes on R N, which we will denote by D 0. In this case, ρ x (ξ) = ξ + ξ N 1/2 d(x, y) = x y + x N y N 1/2.

A more interesting example: Suppose we are on R N, with N = 2n + 1. Define, for j = 1, 2,..., n, and X j = x j + 2x j+n x 2n+1, X j+n = j 2x x j+n X 2n+1 = x 2n+1. x 2n+1 Then identifying R N with the Heisenberg group H n, D is the distribution on H n given by the span of the left-invariant vector fields of degree 1 (sometimes also called the contact distribution). In this case, ρ x (ξ) = n ξ j + 2x j+n ξ 2n+1 + ξ j+n 2x j ξ 2n+1 j=1 d(x, y) = y 1 x if x, y are sufficiently close to each other.

Our class of symbols with mixed homogeneities The symbols we consider will be assigned two different orders, namely m and n, which we think of as the isotropic and non-isotropic orders of the symbol respectively. In the case of the constant distribution, it is quite easy to define the class of symbols we are interested in: given m, n R, if a 0 (x, ξ) C (T R N ) is such that a 0 (x, ξ) (1 + ξ ) m (1 + ξ ) n J x α ξ β ξ N a 0 (x, ξ) J,α,β (1 + ξ ) m β (1 + ξ ) n α where ξ = ξ + ξ N 1/2 if ξ = then we say a 0 S m,n (D 0 ). N ξ i dx i, i=1

More generally, suppose D is a distribution as before (in particular, we fix the coefficients A j i (x) of the frame X 1,..., X N ). Given x R N and ξ = N i=1 ξ idx i, write M x ξ = N (M x ξ) i dx i, where (M x ξ) i = i=1 N A j i (x)ξ j. j=1 Then we say a S m,n (D), if and only if for some a 0 S m,n (D 0 ). a(x, ξ) = a 0 (x, M x ξ) This condition can also be phrased in terms of suitable differential inequalities, as follows. Notation: let B i j (x) be the coefficients of the dual frame θ 1,..., θ N : i.e. θ i = N j=1 Bi j (x)dx j.

Define a derivative D ξ on the cotangent bundle by D ξ = N j=1 B N j (x) ξ j, where (ξ 1,..., ξ N ) are the coordinates of the cotangent space dual to x. We also define the following special derivatives on the cotangent bundle. For j = 1,..., N, define D j = x j + N N N k=1 p=1 l=1 B p k x j (x)al p(x)ξ l. ξ k We write D J for D j1 D j2... D jk, if J = (j 1,..., j k ), where each j i {1,..., N}. Note that in the special case when D = D 0, the above reduces to D ξ = ξ N, and D j =. x j

Back to a general distribution D, one can now show that a(x, ξ) S m,n (D), if and only if a(x, ξ) C (R N R N ), and satisfies a(x, ξ) (1 + ξ ) m (1 + ρ x (ξ) + ξ 1/2 ) n. α ξ Dβ ξ DJ a(x, ξ) α,β,j (1 + ξ ) m β (1 + ρ x (ξ) + ξ 1/2 ) n α. We call elements of S m,n (D) a symbol of order (m, n). When m = n = 0, this should be compared to the estimates satisfied by the Fourier transform of those singular integral kernels K on H n with mixed homogeneities: in fact then α η β τ K(η, τ) α,β ( η + τ ) β ( η + τ 1/2 ) α. Note that in our class of symbols S m,n (D), we allow non-trivial dependence of a(x, ξ) on x, and we control derivatives in x of a(x, ξ) via these special D j derivatives we introduced on the cotangent bundle.

Our class of symbols S m,n is quite big. S m,0 contains every standard (Euclidean) symbol of order m. Also, S 0,n contains the following class of symbols, which we think of as non-isotropic symbols of order n: a(x, ξ) (1 + ρ x (ξ) + ξ 1/2 ) n. α ξ Dβ ξ DJ a(x, ξ) α,β,j (1 + ρ x (ξ) + ξ 1/2 ) n α 2β. When D is the contact distribution on the Heisenberg group, pseudodifferential operators with these non-isotropic symbols give rise to the non-isotropic singular integrals on H n we mentioned before. Many of the results below for S m,n has an (easier) counter-part for these non-isotropic symbols.

Our class of pseudodifferential operators with mixed homogeneities To each symbol a S m,n (D), we associate a pseudodifferential operator ˆ T a f (x) = a(x, ξ) f (ξ)e 2πix ξ dξ. R N We denote the set of all such operators Ψ m,n (D). We remark that Ψ m,n (D) depends only on the distribution D, and not on the choice of the vector fields X 1,..., X N (nor on the choice of a coordinate system on R N ). (Geometric invariance!) One sees that Ψ m,n (D) is the correct class of operators to study, via the following kernel estimates:

Theorem (Stein-Y. 2013) If T Ψ m,n (D) with m > 1 and n > (N 1), then one can write ˆ Tf (x) = f (y)k(x, y)dy, R N where the kernel K(x, y) satisfies K(x, y) x y (N 1+n) d(x, y) 2(1+m), (X ) γ x,y δ x,y K(x, y) x y (N 1+n+ γ ) d(x, y) 2(1+m+ δ ). Here X refers to any of the good vector fields X 1,..., X N 1 that are tangent to D, and the subscripts x, y indicates that the derivatives can act on either the x or y variables.

Main theorems Theorem (Stein-Y. 2013) If T 1 Ψ m,n (D) and T 2 Ψ m,n (D), then T 1 T 2 Ψ m+m,n+n (D). Furthermore, if T 1 is the adjoint of T 1 with respect to the standard L 2 inner product on R N, then we also have T 1 Ψ m,n (D). In particular, the class of operators Ψ 0,0 form an algebra under composition, and is closed under taking adjoints. Proof unified by introducing some suitable compound symbols.

Theorem (Stein-Y. 2013) If T Ψ 0,0 (D), then T maps L p (R N ) into itself for all 1 < p <. Operators in Ψ 0,0 are operators of type (1/2, 1/2). As such they are bounded on L 2. But operators in Ψ 0,0 may not be of weak-type (1,1); this forbids one to run the Calderon-Zygmund paradigm in proving L p boundedness use Littlewood-Paley projections instead, and need to introduce some new strong maximal functions. Nonetheless, there are two very special ideals of operators inside Ψ 0,0, namely Ψ ε, 2ε and Ψ ε,ε for ε > 0. (The fact that these are ideals of the algebra Ψ 0,0 follows from the earlier theorem). They satisfy:

Theorem (Stein-Y. 2013) If T Ψ ε, 2ε or Ψ ε,ε for some ε > 0, then (a) T is of weak-type (1,1), and (b) T maps the Hölder space Λ α (R N ) into itself for all α > 0. An analogous theorem holds for some non-isotropic Hölder spaces Γ α (R N ). Proof of (a) by kernel estimates Proof of (b) by Littlewood-Paley characterization of the Hölder spaces Λ α.

Theorem (Stein-Y. 2013) Suppose T Ψ m,n with m > 1, n > (N 1), m + n 0, and m + 2n 0. For p 1, define an exponent p by 1 p := 1 { } m + n p γ, 2m + n γ := min, N N + 1 if 1/p > γ. Then: (i) T : L p L p (ii) If m+n N for 1 < p p < ; and 2m+n N+1, then T is weak-type (1, 1 ). These estimates for S m,n are better than those obtained by interpolating between the optimal results for S 0,n and S m,0.

For example, take the example of the 1-dimensional Heisenberg group H 1 (so N = 3). If T 1 is a standard (or Euclidean) pseudodifferential operator of order 1/2 (so T 1 Ψ 1/2,0 ), and T 2 is a non-isotropic pseudodifferential operator of order 1 (so T 2 Ψ 0, 1 ), then the best we could say about the operators individually are just T 2 : L 4/3 L 2, T 1 : L 2 L 3. But according to the previous theorem, which is better. T 1 T 2 : L 4/3 L 4, Proof of (i) by complex interpolation; (ii) by kernel estimates.

Some applications One can localize the above theory of operators of class Ψ m,n to compact manifolds without boundary. Let M = Ω be the boundary of a strongly pseudoconvex domain Ω in C n+1, n 1. Then the Szegö projection S is an operator in Ψ ε,2ε for all ε > 0; c.f. Phong-Stein (1977). Furthermore, the Dirichlet-to-Neumann operator +, which one needs to invert in solving the -Neumann problem on Ω, has a parametrix in Ψ 1, 2 ; c.f. Greiner-Stein (1977), Chang-Nagel-Stein (1992).

Thank you very much!