Lectures on Pseudo-Differential Operators: Regularity Theorems and Applications to Non-Elliptic Problems. (MN-24)

by Alexander Nagel, Elias M. Stein

Overview

The theory of pseudo-differential operators (which originated as singular integral operators) was largely influenced by its application to function theory in one complex variable and regularity properties of solutions of elliptic partial differential equations. Given here is an exposition of some new classes of pseudo-differential operators relevant to several complex variables and certain non-elliptic problems.

Originally published in 1979.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Product Details

ISBN-13:	9780691630854
Publisher:	Princeton University Press
Publication date:	04/19/2016
Series:	Mathematical Notes , #24
Pages:	168
Product dimensions:	6.10(w) x 9.30(h) x 1.00(d)

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Lectures on Pseudo-Differential Operators

Regularity theorems and applications to non-elliptic problems

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By Alexander Nagel, E. M. Stein

PRINCETON UNIVERSITY PRESS

Copyright © 1979 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08247-9

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Contents

Preface,
Introduction, 1,
Chapter I Homogeneous Distributions, 7,
§1 Homogeneity and dilations in Rn, 7,
§2 Homogeneous groups, 14,
§3 Homogeneous distributions on the Heisenberg group, 21,
Chapter II Basic Estimates for Pseudo Differential Operators, 31,
§4 Examples of symbols, 31,
§5 The distance function p(x, [xi]), 35,
§6 LP estimates (p ≠ 2), 46,
§7 L2 estimates, 55,
Chapter III Further Regularity Theorems and Composition of Operators, 76,
§8 Sobolev and Lipschitz spaces, 76,
§9 Non-isotopic Sobolev and Lipschitz spaces, 82,
§10 Composition of operators, 89,
§11 A Fourier integral operator; change of variables, 98,
Chapter IV Applications, 104,
§12 Normal coordinates for pseudoconvex domains, 104,
§13 []b and the Cauchy-Szegö integral, 109,
§14 Operators of Hörmander and Grushin, 115,
§15 The oblique derivative problem, 122,
§16 Second-order operators of Kannai-type , 138,
Appendix, 144,
References, 156,


CHAPTER 1

Homogeneous distributions

Chapter I may be thought of as a review of some known facts which are basic in motivating our theory. Proofs are for the most part only sketched.

§1. Homogeneity and dilations in Rn

Denote x = (x1, ..., xn). Fix positive exponents a1, ..., an, and define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Observe that

(1) δs ο δt = δst

(2) δ1 = Id

(3) δt(x) -> 0 as t -> 0 for all x

(More generally, one might let δt be given by multiplication by the matrix eA log t where A is a real matrix whose eigenvalues all have positive real part.)

The change of variable formula for dilations δt is given by

d(δtx) = tadx

where a = (a1 ... + an). (In general, a = trace A.)

Denote Euclidean norm by x .

Proposition 1. There exists a norm function |x| satisfying

(a) |x| ≥ 0; |x| > 0 if and only if x ≠ 0

(b) x -> |x| is everywhere continuous and C∞ on Rn\{0}

(c) |δtx| = t|x| for all t > 0

(d) |x| = 1 iff x = 1. (This is just a normalization.)

Proof. Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. To obtain smoothness, apply the implicit function theorem. The rest of the proposition is obvious.

Define polar coordinates by x = (t, σ) where t = |x| and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i. e., σ = |σ| = 1).

Remark. |x| [equivalent] [summation] |xj|1/aj, since this holds when |x| = 1, and both sides transform the same way under the dilations δt.

Proposition 2. dx = ta-l ω(σ)dt dσ where dσ denotes the usual measure on the unit sphere and ω is a positive C∞ function on the sphere. The proof, a simple calculation of the Jacobian, is left to the reader. (See Fabes and Rivière, p.20.)

Corollary. There exists a constant c > 0 such that for all f: (0, ∞) -> [0, ∞) measurable,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark. The corollary implies that |x|α is locally integrable iff α > -a and |x|α is integrable at infinity iff α < -a.

Let λ [member of] C. We say that f is homogeneous of degree λ if f(δt(x)) = tλf(x), t > 0. Let K be a distribution. (By distribution we shall always mean tempered distribution.) If K were a function homogeneous of degree λ, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is therefore natural to call a distribution K homogeneous of degree λ if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every test function φ.

A distribution K is C∞ = in an open set Ω if there is a function f [member of] C∞(Ω) such that K(φ) = ∫fφ for all φ [member of] C∞0(Ω). We will call a distribution K of class λ if it is homogeneous of degree λ and C∞ on Rn\{0}.

Theorem 1. K is a distribution of class λ if and only if [??] is a distribution of class -a-λ.

Proof. Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it's easy to see that [??] is homogeneous of degree -a-λ.

Let K0 denote the C∞ function on Rn\{0} that agrees with K there. Choose ψ [member of] C∞0 such that ψ [equivalent] 1 in a neighborhood of 0.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ψ K has compact support, so its Fourier transform is C∞ (even analytic).

Denote K1 (1 - ψ)/K0 = K1 is C∞ = everywhere and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For sufficiently large M, using homogeneity, we see that ΔMK1 decreases quickly enough at infinity so that ΔMK1 [member of] L1. Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is continuous, and [??]1([xi]) is continuous except at the origin. Similarly, xαK1(x) is homogeneous (of degree λ + [summation]αj aj) for large x, so that any derivative of [??]1 is continuous outside the origin. qed.

Example 1. Suppose Re λ > -a. Let K(x) be a C∞ function away from 0 that is homogeneous of degree λ; then K [member of] L1loc (by Proposition 2). Therefore, K defines a distribution. Conversely, all distributions of class λ arise in this way. In fact, if K is such a distribution, let K0 as above Then K - K0 a distribution of class λ supported at the origin. Thus K - K0 is a sum of derivatives of the delta function at the origin. It is easy to check that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has homogeneity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore K - K0 = 0.

Example 2. Suppose λ = -a. In order that a function K(x), C∞ away from 0, of degree -a define a distribution, it must have mean value zero on the sphere. Conversely, each distribution of class -a is the sum of such a function and a constant multiple of the delta function at 0.

Proposition 3. More generally, suppose K0 [member of] C∞ (1Rn\{0}), K0 is homogeneous of degree λ. Then:

(a) There exists a distribution K that agrees with K0 in Rn\{0}.

(b) K can be chosen to be of class λ if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all multiindices α such that λ = -a - [summation]ajαj. (Notice that this condition is vacuous unless λ lies at certain exceptional points on the negative real axis λ ≤ -a.)

(c) The distribution K of class λ is unique up to linear combinations of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for those α for which λ = -a - [summation]ajαj.

Proof sketch. Write K0(x) = Ω(x)|x|λ, where Ω(x) is homogeneous of degree zero. Fix φ [member of] S.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

converges absolutely for Re λ > -a. It can be continued analytically to be meromorphic in C. It has at most simple poles at the points -a - [summation]αjaj, and these poles vanish under the compatibility conditions of b). In fact, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Clearly Iλ - I'λ is entire. The main part of I'λ is (by analytic continuation)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus poles arise only when λ = -a -[summation]αjaj, and in that case the pole vanishes if we impose the condition(s) c'α = 0, which are equivalent to those stated in b).

We carry out the argument in more detail for the case λ = -a.

Proposition 3'. Suppose K0(x) is homogeneous of degree -a and C∞ away from 0. K0(x) extends to a distribution of class - a if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Suppose K0 has mean value zero. We give an alternative 0 definition: We will define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Note that there exists b > 0 such that [parallel]x[parallel] ≤ c |x|b for [parallel]x[parallel] < 1. (In fact, b = l/maxaj. works because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].) It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

converges absolutely as ε -> 0. Thus K(ψ) is well defined. Its homogeneity is also obvious.

Now suppose K0 does not have mean value zero. There is a constant c ≠ 0 such that

K0(x) = K1(x) + c|x|-a where K1(x)

is homogeneous of degree -a and has mean value zero on the sphere. Since K1 defines a distribution of class -a we are reduced to showing that c|x|-a does not. Denote

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[??] is a distribution and agrees with c|x|-a away from 0. Assume there exists a distribution K of class -a that agrees with c|x|-a away from 0. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Choose φ so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all other α that occur in the sum. The formula implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By homogeneity, we also have K(φ ο δt) = K(φ) for all t > 0. Therefore [??](φ ο δt) = [??](φ) for all t > 0. But a change of variable in (4) shows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a contradiction.

Remark. Let K0 be homogeneous of degree -a. We have shown implicity that if K0 has mean value zero with respect to one homogeneous norm | |1, then it has mean value zero with respect to any homogeneous norm | |2 (satisfying Proposition 1 a), b), c) but not necessarily the normalization d) ). In fact, we can see this directly by observing that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The left-hand side is bounded, but the right-hand side would grow like -log b if K0 did not have mean value zero with respect to | |2.

Examples. (Recall the convention [??]([xi]) = ∫ e2πix[xi]f(x)dx.)

1. (See Stein, Chapter III.) Let a1 = a2 = ... = an = 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when α ≠ 0, -2, -4, ..., ≠ n, n+2, n+4, ..., taken in the sense of Proposition 3 b).

More generally, if Pk is a harmonic polynomial on Rn, homogeneous of degree k, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when α ≠ -k, -k-2, ..., ≠ k+n, k+n+2,....

2. a1 = a2 = ... = an = 1, an+1 = 2.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with x = (x1, ..., xn) t = xn+1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus K is the fundamental solution to the heat equation

([partial derivative]/[partial derivative]t - Δx) K(x, t) = δ

A fundamental fact connecting homogeneous distributions and operators is the following:

Theorem 2. Suppose K is homogeneous of degree -a (or more generally of degree λ with Reλ = -a) and is C∞ away from the origin. Then Tf = K * f is a bounded operator on L2.

The Proof of Theorem 2 is immediate from the fact that |[??]| is homogeneous of degree 0, and hence bounded.

Remark. We will see later that T extends to a bounded operator on Lp 1 < p < ∞. *

§2. Homogeneous groups

We shall now sketch the background for an important generalization of Theorem 2. Consider Rn with a group multiplication that makes it a Lie group. Assume (for simplicity) that x1</sub, ..., xn are canonical coordinates. (This means that they are the coordinates given by the exponential map. In particular, x-1 = -x and a straight line through the origin is a one-parameter subgroup.) Suppose we have a one-parameter set of dilations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that are automorphisms of the group. We will call a group with dilations a homogeneous group.

This situation arises often. If G is a semi-simple Lie group then G = KAN, where K is a maximal compact subgroup, A abelian, and N nilpotent. (This is the Iwasawa decomposition of G.) A acts on N by dilations, and L2(N) is of interest in representation theory. For example, let G = SU(n, 1), the biholomorphic self-mappings of the unit ball B in Cn. Then K is the stability group of the origin and B [equivalent] G/K. N is the Heisenberg group and can be identified with the boundary of (the unbounded realization of) B. The abelian group A is one-dimensional and acts on N as do our dilations. We will return to this example in the next section.

We add two remarks. First, all homogeneous groups are nilpotent. This is because the Lie algebra can be split into a sum of sub-spaces [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of vector fields X satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and because there are only finitely many indices aj, U is nilpotent. Second, dx (Lebesgue measure on Rn) is bivariant Haar measure. This is contained in a general theorem about the measure on the group inherited from the Lie algebra via the exponential mapping. In our case a simpler proof can be given. Suppose that a1 ≤ a2 ≤ ... ≤ an. If z = xy, then

zk = xk + yk + Rk (x,y), where

Rk is a polynomial in x and y with terms of (usual) degree at least two. But clearly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that Rk depends only on x1, ..., xk-1, y1, ..., yk-1. Thus the Jacobian matrix for x -> xy (and similarly for x -> yx) is lower triangular with 1 in each entry on the diagonal.

Let f be a function on a homogeneous group H. Lyf(x) = f(yx) and Ryf(x) = f(xy) define left and right translation, respectively. An operator T is left invariant if Ly(Tf) = T(Lyf), and similarly for right invariance. For example, right translation is left invariant. Define convolution by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since f -> f * g is just a "sum" of right translations, it is reasonable to expect (and easy to show) that it is left invariant. In general, it is helpful to think of any left invariant operator as being given (at least formally) by

f -> f * K, where K is some distribution.

Let K be a distribution which is a C∞ function away from the origin and is homogeneous of degree λ, Reλ = -a. Define Tf = f * K.

Theorem 2'. T extends to a bounded operator on L2.

Proof. We need the following lemma

Lemma. Suppose {Tj} is a family of bounded operators on Hilbert space, a(j) ≥ 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Assume that

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then [summation]Tj [??] A.

Proof of the lemma. Suppose that there are N operators. Denote T = [N.summation over (j = 1) Tj. T T* is self-adjoint, so we can compute its spectral norm:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where each ik runs from l to N.

We can estimate each term of the sum in two ways. First,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Second,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and similarly for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].)

(Continues...)

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Excerpted from Lectures on Pseudo-Differential Operators by Alexander Nagel, E. M. Stein. Copyright © 1979 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents

• Frontmatter, pg. i
• Table of Contents, pg. iii
• Introduction, pg. 1
• Chapter I. Homogeneous Distributions, pg. 7
• Chapter II. Basic Estimates for Pseudo Differential Operators, pg. 31
• Chapter III. Further Regularity Theorems and Composition of Operators, pg. 76
• Chapter IV. Applications, pg. 104
• Appendix, pg. 144
• References, pg. 156