A new class of pseudodifferential operatorswith mixed homogenities

Po-Lam Yung

University of Oxford

Jan 20, 2014

Introduction

I Given a smooth distribution of hyperplanes on RN (or moregenerally on a smooth manifold), we find a class ofpseudodifferential operators adapted to that.

I Many situations of this kind: e.g. the heat distribution onRn × R+, or contact distribution on contact manifolds.

I Our motivation actually comes from several complex variables.

I On the boundaries of strongly pseudoconvex domains, there isoften a need to compose two kinds of pseudodifferentialoperators, one with Euclidean homogeneities, another withHeisenberg homogeneities.

I Our new class of pseudodifferential operators is an algebrathat is large enough to contain both of them, and yet smallenough to consists only of pseudolocal operators.

I This is joint work with E. Stein.

Outline

I Review pseudodifferential operators on RN

I Review singular integrals on the Heisenberg group Hn

I Our new class of pseudodifferential operators

I Geometric invariance, composition of operators, and regularityon various function spaces

I Applications to several complex variables

Pseudodifferential operators on RN

I A pseudodifferential operator is an operator of the form

Taf (x) =

ˆRN

a(x , ξ)f (ξ)e2πix ·ξdξ.

Here a(x , ξ) is a smooth function on the cotangent bundle,called the symbol of the operator.

I e.g. If a(x , ξ) = 2πiξj , then Ta = ∂∂x j

;

if a(x , ξ) = x j , then Ta is multiplication by x j .

I Studied in the early days by e.g. Kohn-Nirenberg andHormander.

I A standard (elliptic / Euclidean / isotropic) symbol class oforder m:

|a(x , ξ)| . (1 + |ξ|)m

|∂Jx ∂αξ a(x , ξ)| .J,α (1 + |ξ|)m−|α|.

I If a(x , ξ) is a standard symbol of order 0, then Ta admits thefollowing singular integral representation:

Taf (x) =

ˆRN

f (y)K (x , y)dy

where K (x , y) satisfies estimates

|K (x , y)| . 1

|x − y |N

|∂γx ∂δyK (x , y)| .γ,δ1

|x − y |N+|γ|+|δ|

and some suitable cancellation conditions.

The Heisenberg group Hn

I The Heisenberg group Hn is a non-abelian Lie groupdiffeomorphic to Cn × R.

I It arises in several complex variables as the boundary of theupper half space in Cn+1.

I We write a point x ∈ Hn as x = (z , t) ∈ Cn × R.

I The group law on Hn is then given by

(z1, t1) · (z2, t2) := (z1 + z2, t1 + t2 + 2Im(z1z2)).

I For y ∈ Hn, the group inverse of y will be denoted y−1.

I Haar measure is the Lebesgue measure dy on RN .

Two different homogeneities on Hn

I Automorphic dilation: δλ(z , t) := (λz , λ2t), λ > 0.Automorphic dimension Q := 2n + 2.Non-isotropic norm: ‖(z , t)‖ := |z |+ |t|1/2.

I Euclidean dilation: λ(z , t) := (λz , λt), λ > 0.Euclidean dimension N := 2n + 1.Euclidean norm: |(z , t)| := |z |+ |t|.

I We will write ξ = (η, τ) ∈ Cn × R for the dual variables tox = (z , t).

I We will study two classes of singular integral operators on Hn.

Two classes of singular integral operators on Hn

I First, identifying Hn with RN , we have the standardCalderon-Zygmund operators on RN . These are operators ofthe form

Tf (x) =

ˆRN

f (y)K (x − y)dy ,

where K (z , t) is a compactly supported kernel that satisfies

|K (z , t)| . 1

|(z , t)|N

|∂γz ∂δt K (z , t)| .γ,δ1

|(z , t)|N+|γ|+δ

and some suitable cancellation conditions.We call these singular integral operators with Euclidean (orisotropic) homogeneities.

I Second, we have the following automorphic class of singularintegral operators. These are operators of the form

Tf (x) =

ˆHn

f (y)K (y−1 · x)dy ,

where K (z , t) is a compactly supported kernel that satisfies

|K (z , t)| . 1

‖(z , t)‖Q

|∂γz ∂δt K (z , t)| .γ,δ1

‖(z , t)‖Q+|γ|+2δ

and some suitable cancellation conditions.(Note the distinction between the “good” derivatives ∂z andthe “bad” derivative ∂t .)We call these singular integral operators with Heisenberg (ornon-isotropic) homogeneities.

I Either class of singular integral operators form an algebraunder composition.

I However, there are situations when one wants to compose asingular integral operator with one homogeneity with anotheroperator with a different homogeneity. e.g. Solution of∂-Neumann problem.

I In studying such compositions, flag kernels have beenintroduced in the 90’s; c.f. Muller-Ricci-Stein (1995-96),Nagel-Ricci-Stein (2001).

I These flag kernels form an algebra, in which the singularintegrals with different homogeneities can be composed, butthe class of flag kernels is too big - they fail to be pseudolocal.

I In another attempt, Nagel-Ricci-Stein-Wainger (2011)introduced the following class of pseudolocal singular integraloperators on Hn, where the kernels have mixed homogeneities:

A class of singular integral operators on Hn with mixedhomogeneities

I This is the class of operators of the form

Tf (x) =

ˆHn

f (y)K (y−1 · x)dy ,

where K (z , t) is a compactly supported kernel that satisfies

|K (z , t)| . 1

|(z , t)|2n‖(z , t)‖2

|∂γz ∂δt K (z , t)| .γ,δ1

|(z , t)|2n+|γ|‖(z , t)‖2+2δ

and some suitable cancellation conditions.

I It contains the two classes of singular integral operatorsintroduced earlier, and forms an algebra under composition.Furthermore, operators in this class are pseudolocal.

Two Questions

I Q1. For each of the above three algebras of singular integraloperators, is there a corresponding variable coefficient theory?

I 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), thatsatisfies the same kind of estimates.

I Certainly doable for the class of singular integral operatorswith Euclidean homogeneities.

I But being able to do so is important in applications since wewould like to adapt these singular integrals to boundaries ofstrongly pseudoconvex domains, and a variable coefficienttheory is necessary for such applications.

I Beals-Greiner (1988) answered this question for the class ofsingular integral operators with Heisenberg homogeneities.

I Q2. If one could find a variable coefficient theory for theseclasses of singular integral operators, can one find apseudodifferential realization of these operators?

I This means a theory where one rewrites such singular integraloperators as pseudodifferential operators, namely

Taf (x) =

ˆRN

a(x , ξ)f (ξ)e2πix·ξdξ,

and characterizes all the symbols a(x , ξ) that arises this way.

I Again well-known for the class of singular integral operatorswith Euclidean homogeneities; these are just the standardpseudodifferential operators on RN we recalled earlier.

I Beals-Greiner (1988) answered this question for the class ofsingular integral operators with Heisenberg homogeneities.

I Our goal: to answer both questions above for the class ofsingular integral operators with mixed homogeneities.

I In fact, our theory will allow treatment of pseudodifferentialoperators that are of different orders (not just 0th order ones)

I We will study these operators in a more general set-up, thatapplies not only to the Heisenberg group, but to a situationwhere a distribution of hyperplanes is given on RN .

I We will also see some geometric invariance of our class ofoperators as we proceed.

Our set-up

I Suppose on RN , we are given a (global) frame of tangentvectors, namely X1, . . . ,XN , with

Xi =N∑j=1

Aji (x)

∂

∂x j.

I We assume that all Aji (x) are C∞ functions, and that

∂Jx Aji (x) ∈ L∞(RN) for all multiindices J.

I We also assume that | det(Aji (x))| is uniformly bounded from

below on RN .

I Let D be the distribution of tangent subspaces on RN givenby the span of X1, . . . ,XN−1.

I Our constructions below seem to depend on the choice of theframe X1, . . . ,XN , but ultimately the class of operators weintroduce will only depend on D.

I No curvature assumption on D is necessary!

Geometry of the distribution

I We write θ1, . . . , θN for the frame of cotangent vectors dualto X1, . . . ,XN .

I We will need a variable seminorm ρx(ξ) on the cotangentbundle of RN , defined as follows.

I Given ξ =∑N

i=1 ξidx i , and a point x ∈ RN , we write

ξ =N∑i=1

(Mxξ)iθi .

((Mxξ)i =∑N

j=1 Aji (x)ξj will do.) Then

ρx(ξ) :=N−1∑i=1

|(Mxξ)i |.

I We also write |ξ| =∑N

i=1 |ξi | for the Euclidean norm of ξ.

I The variable seminorm ρx(ξ) induces a quasi-metric d(x , y)on RN .

I If x , y ∈ RN with |x − y | < 1, we write

d(x , y) := sup

1

ρx(ξ) + |ξ|1/2: (x − y) · ξ = 1

.

I We also write |x − y | for the Euclidean distance between xand y .

I Example: Xj = ∂∂x j

for j = 1, . . . ,N.

Then D is the distribution of horizontal hyperplanes on RN ,which we will denote by D0. In this case,

ρx(ξ) = |ξ′|+ |ξN |1/2

d(x , y) = |x − y |+ |xN − yN |1/2.

I A more interesting example: Suppose we are on RN , withN = 2n + 1. Define, for j = 1, 2, . . . , n,

Xj =∂

∂x j+ 2x j+n ∂

∂x2n+1, Xj+n =

∂

∂x j+n− 2x j ∂

∂x2n+1

and

X2n+1 =∂

∂x2n+1.

Then identifying RN with the Heisenberg group Hn, D is thedistribution on Hn given by the span of the left-invariantvector fields of degree 1 (sometimes also called the contactdistribution). In this case,

ρx(ξ) =n∑

j=1

|ξj + 2x j+nξ2n+1|+ |ξj+n − 2x jξ2n+1|

d(x , y) = ‖y−1 · x‖

if x , y are sufficiently close to each other.

Our class of symbols with mixed homogeneities

I 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.

I In the case of the constant distribution, it is quite easy todefine the class of symbols we are interested in: givenm, n ∈ R, if a0(x , ξ) ∈ C∞(T ∗RN) is such that

|a0(x , ξ)| . (1 + |ξ|)m(1 + ‖ξ‖)n

|∂Jx ∂αξ′∂βξN

a0(x , ξ)| .J,α,β (1 + |ξ|)m−β(1 + ‖ξ‖)n−|α|

where

‖ξ‖ = |ξ′|+ |ξN |1/2 if ξ =N∑i=1

ξidx i ,

then we say a0 ∈ Sm,n(D0).

I More generally, suppose D is a distribution as before (inparticular, we fix the coefficients Aj

i (x) of the frame

X1, . . . ,XN). Given x ∈ RN and ξ =∑N

i=1 ξidx i , write

Mxξ =N∑i=1

(Mxξ)idx i , where (Mxξ)i =N∑j=1

Aji (x)ξj .

Then we say a ∈ Sm,n(D), if and only if

a(x , ξ) = a0(x ,Mxξ)

for some a0 ∈ Sm,n(D0).

I This condition can also be phrased in terms of suitabledifferential inequalities, as follows.

I Notation: let B ij (x) be the coefficients of the dual frame

θ1, . . . , θN : i.e. θi =∑N

j=1 B ij (x)dx j .

I Define a derivative Dξ on the cotangent bundle by

Dξ =N∑j=1

BNj (x)

∂

∂ξj,

where (ξ1, . . . , ξN) are the coordinates of the cotangent spacedual to x .

I We also define the following special derivatives on thecotangent bundle. For j = 1, . . . ,N, define

Dj =∂

∂x j+

N∑k=1

N∑p=1

N∑`=1

∂Bpk

∂x j(x)A`p(x)ξ`

∂

∂ξk.

I We write DJ for Dj1Dj2 . . .Djk , if J = (j1, . . . , jk), where eachji ∈ 1, . . . ,N.

I Note that in the special case when D = D0, the abovereduces to Dξ = ∂

∂ξN, and Dj = ∂

∂x j.

I Back to a general distribution D, one can now show thata(x , ξ) ∈ Sm,n(D), if and only if a(x , ξ) ∈ C∞(RN ×RN), andsatisfies

|a(x , ξ)| . (1 + |ξ|)m(1 + ρx(ξ) + |ξ|1/2)n.

|∂αξ Dβξ DJa(x , ξ)| .α,β,J (1 + |ξ|)m−β(1 + ρx(ξ) + |ξ|1/2)n−|α|.

We call elements of Sm,n(D) a symbol of order (m, n).

I When m = n = 0, this should be compared to the estimatessatisfied by the Fourier transform of those singular integralkernels K on Hn with mixed homogeneities: in fact then

|∂αη ∂βτ K (η, τ)| .α,β (|η|+ |τ |)−β(|η|+ |τ |1/2)−|α|.

I Note that in our class of symbols Sm,n(D), we allownon-trivial dependence of a(x , ξ) on x , and we controlderivatives in x of a(x , ξ) via these special Dj derivatives weintroduced on the cotangent bundle.

I Our class of symbols Sm,n is quite big.

I Sm,0 contains every standard (Euclidean) symbol of order m.

I Also, S0,n contains the following class of symbols, which wethink of as non-isotropic symbols of order n:

|a(x , ξ)| . (1 + ρx(ξ) + |ξ|1/2)n.

|∂αξ Dβξ DJa(x , ξ)| .α,β,J (1 + ρx(ξ) + |ξ|1/2)n−|α|−2β.

I When D is the contact distribution on the Heisenberg group,pseudodifferential operators with these non-isotropic symbolsgive rise to the non-isotropic singular integrals on Hn wementioned before.

I Many of the results below for Sm,n has an (easier)counter-part for these non-isotropic symbols.

Our class of pseudodifferential operators with mixedhomogeneities

I To each symbol a ∈ Sm,n(D), we associate apseudodifferential operator

Taf (x) =

ˆRN

a(x , ξ)f (ξ)e2πix ·ξdξ.

We denote the set of all such operators Ψm,n(D).

I We remark that Ψm,n(D) depends only on the distribution D,and not on the choice of the vector fields X1, . . . ,XN (nor onthe choice of a coordinate system on RN).(Geometric invariance!)

I One sees that Ψm,n(D) is the correct class of operators tostudy, via the following kernel estimates:

Theorem (Stein-Y. 2013)

If T ∈ Ψm,n(D) with m > −1 and n > −(N − 1), then one canwrite

Tf (x) =

ˆRN

f (y)K (x , y)dy ,

where the kernel K (x , y) satisfies

|K (x , y)| . |x − y |−(N−1+n)d(x , y)−2(1+m),

|(X ′)γx ,y∂δx ,yK (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 X1, . . . ,XN−1 thatare tangent to D, and the subscripts x , y indicates that thederivatives can act on either the x or y variables.

Main theorems

Theorem (Stein-Y. 2013)

If T1 ∈ Ψm,n(D) and T2 ∈ Ψm′,n′(D), then

T1 T2 ∈ Ψm+m′,n+n′(D).

Furthermore, if T ∗1 is the adjoint of T1 with respect to thestandard L2 inner product on RN , then we also have

T ∗1 ∈ Ψm,n(D).

In particular, the class of operators Ψ0,0 form an algebra undercomposition, and is closed under taking adjoints.

I Proof unified by introducing some suitable compound symbols.

Theorem (Stein-Y. 2013)

If T ∈ Ψ0,0(D), then T maps Lp(RN) into itself for all 1 < p <∞.

I Operators in Ψ0,0 are operators of type (1/2, 1/2). As suchthey are bounded on L2.

I But operators in Ψ0,0 may not be of weak-type (1,1); thisforbids one to run the Calderon-Zygmund paradigm in provingLp boundedness → use Littlewood-Paley projections instead,and need to introduce some new strong maximal functions.

I Nonetheless, there are two very special ideals of operatorsinside Ψ0,0, namely Ψε,−2ε and Ψ−ε,ε for ε > 0.(The fact that these are ideals of the algebra Ψ0,0 followsfrom the earlier theorem). They satisfy:

Theorem (Stein-Y. 2013)

If T ∈ Ψε,−2ε or Ψ−ε,ε for some ε > 0, then

(a) T is of weak-type (1,1), and

(b) T maps the Holder space Λα(RN) into itself for all α > 0.

I An analogous theorem holds for some non-isotropic Holderspaces Γα(RN).

I Proof of (a) by kernel estimates

I Proof of (b) by Littlewood-Paley characterization of theHolder spaces Λα.

Theorem (Stein-Y. 2013)

Suppose T ∈ Ψm,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

p− γ, γ := min

|m + n|

N,|2m + n|

N + 1

if 1/p > γ. Then:

(i) T : Lp → Lp∗ for 1 < p ≤ p∗ <∞; and

(ii) If m+nN 6= 2m+n

N+1 , then T is weak-type (1, 1∗).

I These estimates for Sm,n are better than those obtained byinterpolating between the optimal results for S0,n and Sm,0.

I For example, take the example of the 1-dimensionalHeisenberg group H1 (so N = 3).

I If T1 is a standard (or Euclidean) pseudodifferential operatorof order −1/2 (so T1 ∈ Ψ−1/2,0), and T2 is a non-isotropicpseudodifferential operator of order −1 (so T2 ∈ Ψ0,−1), thenthe best we could say about the operators individually are just

T2 : L4/3 → L2, T1 : L2 → L3.

But according to the previous theorem,

T1 T2 : L4/3 → L4,

which is better.

I Proof of (i) by complex interpolation; (ii) by kernel estimates.

Some applications

I One can localize the above theory of operators of class Ψm,n

to compact manifolds without boundary.

I Let M = ∂Ω be the boundary of a strongly pseudoconvexdomain Ω in Cn+1, n ≥ 1.

I Then the Szego projection S is an operator in Ψ−ε,2ε for allε > 0; c.f. Phong-Stein (1977).

I Furthermore, the Dirichlet-to-Neumann operator +, whichone needs to invert in solving the ∂-Neumann problem on Ω,has a parametrix in Ψ1,−2; c.f. Greiner-Stein (1977),Chang-Nagel-Stein (1992).

Thank you very much!