On a class of pseudodifferential operators with mixed homogeneities Po-Lam Yung University of Oxford July 25, 2014
Introduction Joint work with E. Stein (and an outgrowth of work of Nagel-Ricci-Stein-Wainger, to appear) Motivating question: what happens when one compose two operators of two different homogeneities? e.g. On R N one can associate two different dilations: for x = (x, x N ) R N, λ > 0, one can define λ x := (λx, λx N ) λ x := (λx, λ 2 x N ) (isotropic) (non-isotropic) Associated to these are two norms, each homogeneous with respect to one of these dilations: x = x + x N x = x + x N 1/2
There are also the dual norms, on the cotangent space of R N, given by ξ = ξ + ξ N ξ = ξ + ξ N 1/2 One could look at multipliers e(ξ), with α ξ β ξ N e(ξ) ξ α β, and their associated multiplier operators T e f (ξ) = e(ξ) f (ξ). (These are just standard isotropic singular integral operators.) One could also look at multipliers h(ξ), with α ξ β ξ N h(ξ) ξ α 2 β, and their associated multiplier operators T h f (ξ) = h(ξ) f (ξ). (These are just non-isotropic singular integral operators, arising e.g. when one solves the heat equation.)
Some motivating Questions What happens when one compose T e with T h? What kind of operator do we get? What are the nature of the singularities of the multiplier, or the kernel, of T e T h? What mapping properties does T e T h satisfy? When is it (say) weak-type (1,1)? The question of composition is quite easy, since we are dealing with convolution operators on an abelian group R N. The question of mapping properties was already studied in a paper of Phong and Stein in 1982. We are interested in these questions, because they serve as toy model problems for what one needs to do in more general settings.
e.g. In several complex variables, in solving the -Neumann problem on a smooth strongly pseudoconvex domain in C n+1, one faces the problem of inverting the Calderon operator +. Roughly speaking, this amounts to composing an operator with isotropic homogeneity, with an operator with non-isotropic homogeneity: 1 + 1 b (at least when n > 1; c.f. Greiner-Stein 1977). Similar compositions arise in the study of the Hodge Laplacian on k-forms on the Heisenberg group H n (Müller-Peloso-Ricci 2012)
The role of flag kernels The flag kernels are integral kernels that are singular along certain subspaces on R N (e.g. Nagel-Ricci-Stein 2001, Nagel-Ricci-Stein-Wainger 2011, to appear). These are special cases of product kernels, which were studied by many authors (e.g. R. Fefferman-Stein 1982, Journe 1985, Nagel-Stein 2004). e.g. On the Heisenberg group H n, a flag kernel could be singular along the t-axis, and satisfies K(z, t) z 2n ( z 2 + t ) 1 along with some corresponding differential inequalities and cancellation conditions.
Simple examples of flag kernels include both singular integral kernels with isotropic homogeneities, and those with non-isotropic homogeneities. More sophisticated examples of flag kernels on H n are given by the joint spectral multipliers m(l 0, it ), where m is a Marcinkiewicz multiplier, and L 0 is the sub-laplacian (Müller-Ricci-Stein 1995, 1996). The singular integral operators with flag kernels map L p to L p, for 1 < p <, and form an algebra under composition. Thus if we want to compose a singular integral with isotropic homogeneity, with one that has non-isotropic homogeneity, we could have composed them in the class of all flag kernels. But then we get as a result a flag kernel, which is singular along some subspaces (whereas our original kernels are both singular only at one point).
It turns out one should consider the intersection of those flag kernels that are singular along the t-axis, with those that are singular along the z-axis, as in Nagel-Ricci-Stein-Wainger (to appear). This gives rise to operators with mixed homogeneities: K(z, t) ( z + t ) 2n ( z 2 + t ) 1 along with differential inequalities and cancellation conditions. Our first result will be a pseudodifferential realization of the above operators of with mixed homogeneities. The goal is to write them as pseudodifferential operators: ˆ T a f (x) = a(x, ξ)ˆf (ξ)e 2πix ξ dξ H n for some suitable symbols a(x, ξ).
Nagel-Stein (1979) and Beals-Greiner (1988) has realized those singular integrals with purely non-isotropic homogeneities as pseudodifferential operators. We will do so for singular integrals with mixed homogeneities; in doing so, we will also consider operators of all orders (not just order 0 ones, as singular integrals would be). Our results will actually hold in a more general setting, outside several complex variables; it will hold as long as a smooth distribution of tangent subspaces (of constant rank) is given on R N. We will also see some geometric invariance of our class of operators as we proceed. A very step-2 theory!
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 Aj 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!
An example: the contact distribution on H 1 (Picture courtesy of Assaf Naor)
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 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.
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 where 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 ξ = ξ + ξ N 1/2 if ξ = N ξ i dx i, i=1 then we say a 0 S m,n (D 0 ). (Note that in this situation, 1 + ξ 1 + ρ x (ξ) + ξ 1/2.)
More generally, suppose D is a distribution as before (in particular, we fix the frame X 1,..., X N, and its dual frame θ 1,..., θ N ). Given x R N and ξ = N ξ i dx i = i=1 N (M x ξ) i θ i, i=1 write N M x ξ = (M x ξ) i dx i. i=1 Then we say a S m,n (D), if and only if a(x, ξ) = a 0 (x, M x ξ) for some a 0 S m,n (D 0 ).
e.g. Suppose we are on R 3. Pick coordinates x = (x 1, x 2, t), and dual coordinates ξ = (ξ 1, ξ 2, τ). Define X 1 = and x 1 + 2x 2 t, X 2 = x 2 2x 1 t, X 3 = t, D := span{x 1, X 2 }. Then identifying R 3 with the Heisenberg group H 1, D is sometimes called the contact distribution. We then have a S m,n (D), if and only if a(x 1, x 2, t, ξ 1, ξ 2, τ) = a 0 (x 1, x 2, t, ξ 1 + 2x 2 τ, ξ 2 2x 1 τ, τ) for some a 0 S m,n (D 0 ). The condition a S m,n (D) can also be phrased in terms of suitable differential inequalities.
e.g. Still consider the contact distribution D on H 1. Recall coordinates x = (x 1, x 2, t) on H 1, and dual coordinates ξ = (ξ 1, ξ 2, τ). Let and D 1 = D τ = τ 2x 2 + 2x 1, ξ 1 ξ 2 x 1 + 2τ, D 2 = ξ 2 x 2 2τ, D 3 = ξ 1 t. Then a S m,n (D), if and only if a(x, ξ) C (H 1 R 3 ) satisfies a(x, ξ) (1 + ξ ) m (1 + ρ x (ξ) + ξ 1/2 ) n, α ξ Dβ ξ D Ja(x, ξ) (1 + ξ ) m β (1 + ρ x (ξ) + ξ 1/2 ) n α, where D J is composition of any number of the D 1, D 2, D 3.
Back to the general set up. We call elements of S m,n (D) a symbol of order (m, n). Our class of symbols S m,n is quite big. S m,0 contains every standard (isotropic) symbol of order m: a(x, ξ) (1 + ξ ) m α ξ J x a(x, ξ) α,j (1 + ξ ) m α 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 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. c.f. two-flag kernels of Nagel-Ricci-Stein-Wainger
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 2m + n 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 composing 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 isotropic) 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 interpolation between Ψ 0,0 and Ψ 1, (N 1). Proof of (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).
Epilogue On R, we know that if Tf := f y 1+α, 0 < α < 1, then whenever Tf L q C f L p, 1 q = 1 α, 1 < p < q <. p There are many known proofs; below we sketch one that is perhaps less commonly known, and that is reminiscent of our proof of the smoothing properties for our class Ψ m,n above.
The idea is to use complex interpolation: For s C with 0 Re s 1, let k s be the tempered distribution k s (y) = (1 + s) 1 2 y y 1+s, so that when 0 < Re s 1, we have k s (y) = s y 1+s. Let T s f = f k s for f S. Then when Re s = 1, we have T s : L 1 L, since it is then a convolution against a bounded function. Also when Re s = 0, we have T s : L r L r for all 1 < r <, since k s is then a Calderon-Zygmund kernel (one can check that k s L for such s, since k s, being a derivative, satisfies a cancellation condition). Interpolation then shows that T α : L p L q, when 1 q = 1 α, 1 < p < q <. p