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#### Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability

**By Daryl Geller**

**PRINCETON UNIVERSITY PRESS**

**Copyright © 1990 Princeton University Press**

All rights reserved.

ISBN: 978-0-691-08564-7

All rights reserved.

ISBN: 978-0-691-08564-7

#### Contents

Introduction, 3,Chapter 1. Homogeneous Distributions, 69,

Chapter 2. The Space Zqq, j, 105,

Chapter 3. Homogeneous Partial Differential Equations, 146,

Chapter 4. Homogeneous Partial Differential Operators on the Heisenberg Group, 168,

Chapter 5. Homogeneous Singular Integral Operators on the Heisenberg Group, 202,

Chapter 6. An Analytic Weyl Calculus, 256,

Chapter 7. Analytic Pseudodifferential Operators on Hn : Basic Properties,

Chapter 8. Analytic Parametrices, 376,

Chapter 9. Applying the Calculus, 423,

Chapter 10. Analytic Pseudolocality of the Szegö Projection and Local Solvability, 453,

References, 489,

CHAPTER 1

Homogeneous Distributions

We assume given n-tuple of positive rationals [??] = (a1, ..., an). We put Q = [summation]a1. For x [member of] IRn we put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For f a function on Rn, r > 0, we define the functions Drf, Drf by Drf(x) = f(Drx), Drf = r-QDl/rf. For F [member of] S'(Rn) we define DrF,DrF [member of] S' by DrF|g) = (F|Drg), (DrF}g) = (F|Drg) for g [member of] S. (here and elsewhere) (F|g) denotes the sequilinear pairing, linear in g). For k [member of] C, we say that F is hcmogeneous of degree k if DrF = rkF for all r > 0. We let is Rhomk = Rhomk [??] = {K [member of] S'|K is homogeneous of degree k and is C∞ away from 0}. Such K are called regular homogeneous distributions. In studying there Rhomk [??] there is evidently no loss in assuming all a1 are positive integers, for we can always multiply them all by a common demoninator, changing k. For a while we make this assumption. For x [member of] IRn, we let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thens is homogeneous of degree one, satisfies for all |x| ≥ 0 for all x, |x| = 0 [??] x = 0. For these reasons is called a homogeneous norm function. Note is also (real) analytic away from 0.

Let G be a hcmogeneous function of degree j which is locally integrable away frcm 0; write G(x) = Ω(x)|x|j where Ω is homogeneous of degree 0. One then knows [53] that there exists a number M(Ω) such that for any 0 < A < B we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

We put M(G) = M(Ω). In particular, G [member of] L1loc if Rej > -Q, and G is in L1 at ∞ for Rej < -Q. It is well-known [53] that if Rek > -Q, K [member of] Rhomk and K = f away from 0, where f [member of] C∞(Rn\{0}), then K = f in the S' sense. (Indeed K - f must be supported at 0, and homogeneity considerations show it is zero.) Let ^ denote Fourier transform. The following proposition is known [66]; for convenience, we include the proof.

Proposition 1.1 (a) (Rhomk)^ = Rhom-Q-k

(b) Suppose Re k > -Q. There exists C1 > 0 such that whenever K [member of] Rhomk and |[xi]| = 1, |[??]([xi])| < C1 sup|[partial derivative]ΓK(x)| where the sup is taken over |γ| ≤ Re k + 2Q, 1 ≤ |x| ≤ 2. More precisely, given ε > 0 and 1 ≤ l ≤ n, there exists C2 > 0 such that whenever K [member of] Rhomk, |[xi]| = 1 and |[xi]1| > ε, we have |[??]([xi])| < C2sup|[partial derivative]r1K(x)| where the sup is taken over a1r ≤ Re k + Q + a1, l ≤ |x| ≤ 2. (Here [partial derivative]1 = [partial derivative]/[partial derivative][xi]1.)

Proof. (a) Suppose K [member of] Rhomk. Simple considerations show that [??] is homogeneous of degree -Q - k. Now select φ [member of] C∞c such that φ = 1 near 0. Put K1 = φK and K2 = K - K1. Since K1 is a distribution of compact support [??]1 [member of] C∞. (Indeed, let g[xi](x) = eix.[xi]. Then [??]1([xi]) = K1 (g[xi]) so that [??]1 is in fact the restriction to Rn of an entire function of exponential type.) Next, with α an n-tuple, 1 ≤ l ≤ n, put KNαl2(x) = [partial derivative]N1xαK2(x) for Na1 > Re k + |α| + Q. (Here and elsewhere, |α| = [??]·a.) Then, for large x, KNαl1 is homogeneous of degree m where Re m < - Q, and therefore KNαl2 [member of] Ll. Thus [xi]N1D'[??]2, is continuous on IRn if D' is the differential monomial D' = ([partial derivative]/ [partial derivative][xi])α and N is as before. If [xi] ≠ 0, same [xi]1 ≠ 0, so [??]2 is smooth away from 0. Thus [??] is as well.

(b) We keep the notation of the proof of (a). Suppose φ = 1 for |x| < 1 and φ = 0 for |x| > 2. If Re k > -Q, K1 [member of] L1. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (Here and elsewhere, C denotes a constant which may be different in different appearances.) Also, if 1 ≤ l ≤ n, pick N = N1 with Re k + Q < Na1 ≤ Re k + Q + a1. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Adding these estimates for [??]1 and [??]2 we find the "more precise" form of (b); the "less precise" form follows, since for some ε > 0, |[xi]| = 1 implies |[xi]1| > ε for some l.

If Re j ≤ -Q, we can obtain more insight into Rhomj by considering the distributions which follow. For G a homogeneous function of degree j [member of] C which is smooth away from 0, we define [??]G [member of] S' by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

where N is chosen arbitrarily with N ≥ - Q - Re j - 1. By (1.1), it is easy to see that the choice of N is immaterial. In particular, if Re j > -Q, we can choose thus N = -1, thus [??]G = G. Suppose instead that Re j ≤ - Q. Say -j - Q [??] Z+ = {0, 1, 2, ...}. It is then easy to compute, using (1.1), that [??]G [member of] Rhomj. Further, if K [member of Rhomj, say K = G away from 0; then K - [??]G is supported at 0 and homogeneous of degree j; it follows easily that K - [??]G = 0. Thus, if -j - Q [??] Z+, it follows that Rhomj = {[??]G|G [member of] C∞(Rn\{0}), G homogeneous of degree j}. We discuss the case -j - Q [member of] Z+ presently.

First, however, we introduce a space of distributions which includes Rhomk when k [member of] Z+. If k [??] Z+, we let Kk = Rhomk. If k [member of] Z+, we let Kk = Rhomk + {p(x) log|x|:p a homogeneous polynomial of degree k}. We determine [??]k. For this, suppose j [member of] C. If -j - Q [??] Z+, we let Jj = Rhomj. If -j - Q[member of]Z+, we let Jj = {[??]G|G [member of] C∞Rn\{0}), G homogeneous of degree j} + {P(δ)δ|P a homogeneous polynomial of degree -Q - j}. Here, of course, if P(x) = [summation]cαxα, then P([partial derivative])δ = [summation]cα[partial derivative]αδ.

Proposition 1.2 (a) [??]k = Jj for k = -Q - j.

(b) Proposition 1.1 (b) holds for "Kk" in place of "Rhomk".

Proof. We may assume k [member of] Z+ + and set j = -Q - k. Let K [member of] Kk, J [member of] Jj, K = F + p log|x| (F [member of] Rhomk), J = [??]G + P([partial derivative])δ. The same proof as that of Proposition 1.1(a) shows that [??] and [??] are each smooth away from 0. Next note that

DrK = rkK = rk(log r)p (1.3)

while

DrJ - rkJ = rk(log r)q([partial derivative])δ (1.4)

by a computation using (1.1), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Put [??] = J'; suppose J' = G' away from O. By (1.3), DrJ' rkJ' = rk(log r)p(-i[partial derivative])δ. Thus if φ [member of] S is supported away from 0, (DrJ'-rkJ')(φ) = 0. It follows easily that G' is homogeneous of degree j. Put J" = J' - [??]G,. By (1.4) for [??]G in place of J, DrJ" - rkJ" = rklog r q" ([partial derivative])δ for some homogeneous polynomial q" of degree k. However, J" is supported at the origin; say J" = [summation]cα[partial derivative]αδ. Then DrJ" - rkJ" = [summation]cα(r|α|-rk) [partial derivative]αδ. Putting r = 2, we find that cα = 0 unless |α| = k. Thus J' = [??]G, + J" [member of Jj.

Conversely, put [??] = K'. By (1.4), DrK' - rkK' = rk(log r)p' for some homogeneous polynomial p' of degree k. Put K" = K' - p'(x)log|x|. Then K" [member of] Rhomk, so K' [member of] Kk as desired. This proves (a).

For (b), let S = sup|K(x)| for 1 ≤ |x| ≤ 2. Choose [??] as in Proposition 1.1(b), and let K1 = [??]K. Imitating the proof of Proposition 1.1(b), we see that it suffices to show that K1 1< CS for some C > 0, depending only on k. For this, write K = K' + p(x)log|x|. If |x| = 1, K(x) = K'(x). Thus |K'(x)| ≤ |x|kS for all x. If |x| = 2, p(x) = [K(x) - K'(x)]/log 2. So |p(x)| ≤ C1|x|kS for all x, where C1 depends only on k. Finally, then,

|K(x)| ≤ C2|x|k(1+|log|x )S

for all x, C2 depending only on k, and (b) follows.

Suppose now that j [member of] C, - j - Q [member of] Z+. It follows at once from Proposition 1.2 that Rhomj [subset] Jj. This enables us to clearly understand Rhomj if - j - Q [member of] Z+. Using (1.4) and the explicit formula for q, we see that, in this case, Rhomj = {[??]G|G [member of] C∞(Rn\{0}), G homogeneous of degree j, μ([xi]αG) = 0 whenever |α| = -j - Q} + {P([partial derivative])δ|P a hemogeneous polynomial of degree - j - Q}.

It is often useful, by the way, to know the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any k, l. To see this, we may assume k [member of] Z+; say K [member of] Kk. Select φ [member of] C∞(Rn) such that φ(x) = 0 for |x| ≤ 1 and φ(x) = 1 for |x| ≥ 2. Let φε = D1/εφ. Away from 0, [partial derivative]1K equals a smooth function K'; it suffices to show that [partial derivative]1K = K' in the S' sense. Simply let ε -> 0 (K'|φεg = -(K|[partial derivative]1(φεg)) and use the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We turn now to analyticity. Let AKk = {K [member of] Kk|K is real analytic away from 0}, and AJj = (AK-Q-j)[conjunction]. We seek to characterize AJj. Here is the main idea. For a α multi-index, we write α = α1+ ... +αn. Suppose K [member of] AKk; we shall examine Kαβ = [partial derivative]βxαK for multi-indices α,β with |α| = |β|. Note all Kαα [member of] AKk. We have that for some R ≥ 0, |xα[partial derivative]βK(x)| < CR β β! whenever |x| = 1. It is easy to believe the variant |Kαβ(x)| < CR β β! (|x| = 1), and that for any fixed N, sup|[partial derivative]ΓKαβ(x)| < CR β β! (sup over |γ| ≤ N, |x| = 1. By Proposition 1.1(b), then, we have for |[xi]| = 1 that |[??]αβ([xi])| < CR β β!. That is, if [??] = J,

|[xi]β[partial derivative]αJ([xi])| < CR β β! for |[xi]| = 1, |α| = |β|. (1.5)

The analogue of Proposition 1.1(b) for the inverse Fourier transform similarly shows that (1.5) is not only necessary for J to be in AJj but is also sufficient, in that one can derive |Kαβ(x)| < CR β β! (for |x| = 1) from (1.5). We shall prove everything rigorously in a moment.

Actually, we shall be using not (1.5) but rather several equivalent variants of it. These we prove below in Theorem 1.3, and discuss briefly now; particularly we wish to indicate the significance of these conditions.

First, however, here are a few simple inequalities that we shall frequently use. If |α| = |β|, we have

α ≤ |α| = |β| ≤ Q β , β ≤ |β| = |α| ≤ Q α . (1.6)

Also, by the multinomial theorem, for any multi-index β, we have

β! ≤ β ! ≤ n β β!. (1.7)

Again by the multinomial theorem, if m, r [member of] N, then

(r!)m ≤ (mr)! ≤ mmr(r!)m. (1.8)

Returning now to (1.5), we examine a special case, when [xi]β = [xi]r1 for some l, l ≤ l ≤ n. In that case β = r = |α|/a1. Consequently, using (1.6) and (1.8) we have that for some C,R,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This argument works for all α, provided |α|/a1 is an integer. Even if it is not, it is not hard to prove (1.9), and we do so below. Further, we shall show that the conditions (1.9) for all α are sufficient to imply J [member of] AJj. Using (1.7), (1.8), we conclude

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

Assuming a1 ≥ a2 ≥ ... ≥ an, ak/a1 ≤ 1 and it follows readily that J is real analytic away from [xi]1 = 0. Even more, suppose a1 > an and for fixed [xi]' = ([xi]1, ..., [xi]n-1) with [xi]1 ≠ 0, set F([xi]n) = J([xi]', [xi]n). Then F is better than real analytic — it is the restriction to R on an entire function on C. Along {[xi]1 = 0}, J may well be worse than real analytic. This agrees with the statements in the introduction about the case [??] = (p, 1, ..., 1). We study that case in detail after Theorem 1.3.

With this motivation in mind, we turn to the details. (b) below is a restatement of (1.10), while (c) and (d) are variants.

Theorem 1.3. For J [member of] Jj, the following are equivalient:

(a) J [member of] AJj

(b) For some C,R,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(c) For some [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(d) For some [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all m,s, |[xi]| = 1,

and all n-tuples n of nonnegative real numbers with [??]·n = am.

(We use the convention 00 = 1 if some [xi]1 = n1 = 0 in (d); similarly in (b), (c).) Further, with k = -Q - j, we have these uniformities:

(1) Suppose Re k > -Q. Then, for any R1 > 0 there exist C,R,Co,Ro > 0 so that the inequalities of (b), (c) and (d) hold whenever J = [??] for a K [member of] AKk which satisfies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(2) Suppose Re j > -Q. Then for any C,R,Co,Ro > 0 there exist C1, R1 > 0 so that if the inequalities of (b), (c) or (d) hold and if J = [??] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

*(Continues...)*

Excerpted fromAnalytic Pseudodifferential Operators for the Heisenberg Group and Local SolvabilitybyDaryl Geller. Copyright © 1990 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.

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