 # Estimates of the Neumann Problem. (MN-19), Volume 19

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ISBN-13: 9780691616575 Princeton University Press 03/08/2015 Mathematical Notes , #19 202 6.00(w) x 9.10(h) x 0.60(d)

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#### Estimates for the [??]-Neumann Problem

By P. C. Greiner, E. M. Stein

#### PRINCETON UNIVERSITY PRESS

ISBN: 978-0-691-08013-0

CHAPTER 1

Symbols on the Heisenberg groups

Let

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the usual left invariant vector fields on the Heisenberg group Hn. Following the notation and terminology of Folland-Stein  we define

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The purpose of this chapter is to compute the symbol of the fundamental solution of [??]α. Let Gα denote a (presumptive) fundamental solution and set

(1.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking [??]α under the Fourier transform we obtain the operator

(1.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Assume [xi]0 > 0. (The case [xi]0< 0 will follow by replacing α by -α.) Since we want Gα to act on the Heisenberg group by convolution we shall try a kernel of the following form

(1.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we used the notation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The choice (1.5) is dictated by the following considerations. First by (0.3) it suffices to consider the special case when y = 0 (when w= 0). Next [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is invariant under unitary linear transformation of the z-variables, and so one may look for a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which depends only on |z|. Also observe that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] annihalates any function of |z|. Thus we are led to the form (1. 5) (when w = 0). For general w we then use the group law (0.2) to reduce matters to the case w = 0. Next we shall solve

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if z ≠ w. A bit of algebra yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We set [xi]0|z-w|2 Hence we need to solve

(1.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if x ≠ 0. This is a confluent hypergeometric differential equation. We set

W (x) = e-x y(2x)

which reduces the equation (1.6) to the following better known form

(1.7) uy"(u) + (n-u)y'(u) – n-α/2 y(u) = 0

if u > 0. The identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a solution of

uy"(u) + (c-u) y'(u) – ay(u) = 0

if u > 0 and Re a > 0. Therefore we assume that

(1.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We still need to determine the unknown function a([xi]0). To this end we note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as long as and n-α/2 > 0 and n > 1. In particular

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if [xi]0 > 0, n > 1 and |z-w| is small. Thus to have the "correct" fundamental singularity we set

(1.9) a([xi]0) = cα [xi]0n-1.

We would like to point out that we are still in the process of trying to find the symbol σ(Gα) by heuristic considerations. Once found, we shall prove its correctness for all n ≥ 1.

We continue by applying the operator induced by the kernel

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

After interchanging the order of integration, we obtain the kernel

(1.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where wj = y2j-1 + iy2j, j=1, ..., n and we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

j=1, ..., n. Thus (1.10) becomes

(1.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally, to obtain the symbol, we multiply (1.11) by e-1, set xj – yj = σj, j=1, ..., 2n and note that

<2ω, x' > = < [xi]', x'>.

This yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The interchange of the order of integration is justified by Fubini's theorem because the second iterated integral is easily seen to be convergent. Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which yields

(1. 12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Led by our heuristics, we state our f i r s t result as follows:

1.13. Proposition. If Re(n-α/2) > 0, then

(1.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

induces the operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which has the property that

[??]α Gα f(x) = f(x)

if f is in the Schwartz space and supp [??] is contained in {[xi] [member of] lR2n+1; [xi]0 > 0}

Proof. We replace 1/1 + 2s by s in (1.12) and obtain the form (1.14). We will show that if cα = 1, then Gα is a fundamental solution of [??]α, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

if f belongs to the Schwartz space of functions, such that supp [??] [subset] {[xi] [member of lR2n+1; [xi]0 > 0}. This requires

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where we set

ωj = ω2j-1 + i ω2j, j=1, ..., n.

To simplify matters we note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A simple computation yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every function h. Next we reduce the problem to solving an inhomogeneous ordinary differential equation as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We set |ω|2/[xi]0 = x and require that the following differential equation

is satisfied

(1.15) cα (xg"(x) + ng'(x) + (α – x) g (x)) = -1.

Following previous calculations we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then the left-hand side of (1.15) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which yields cα = 1. This proves Proposition 1.13.

Next we continue σ(Gα)(x, [xi]) analytically on the complex α-plane. Let D denote the contour

In other words D starts at -1, encircles the origin once counterclockwise and returns to -1. Consider

(1.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Again we apply [??]α to ψα as we did in (1.15), we obtain

(1.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

On D we set

sa = ea log s

where log s is the principal branch of the logarithm, i. e., log s is real if s is on the positive part of the real axis Therefore

(1.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1.18. Proposition. If Re (n-α) > 0, n-α/2 ≠ 0, [+ or -], [+ or -] 2, ... then for [xi]0 > 0

(1.19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

σ(Gα)(x, [xi]) can be continued analytically on the complex α-plane to all α, such that

n-α/2 ≠ 0, -1, -2, ...

Proof. In (1. 19) we deform the path of integration, D, into integrating along the real axis from –1 to –δ, 0 < δ < 1, then integrating on the circle of radius δ around the origin and, finally returning from -δ to –1. On the first part

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where log (-s) is real, on the last part

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If Re n-α/2 > 0 we let δ [right arrow] 0 and the contribution of the integral on the circle vanishes. Thus we are left with

(1.20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We set -t = l-s and (1.20) combined with (1.19) yields (1.14). This proves the proposition.

Analogous calculations yield the symbol σ(Gα)(x, [xi]) for [xi]0< 0, as we have already remarked.

We collect the results of this chapter in the following form.

1.21 Theorem. The symbol σ(Gα)(x, [xi]) of a fundamental solution of [??]α is given by

(1.22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(1.23) [xi]0 ≠ 0,

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for j=1, ..., n and [??]α is given by

(1.24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here σ = σ0, σ'). Moreover, [??]α(σ) can be continued analytically in the complex α plane to n-αsignσ0/2 = 1,2,3,. according to the formula

(1.25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is also valid for all α with n > Re α sign σ0.

Observe that when σ0 (=[xi]0) ≠ 0, then kα is defined by either (1.24) or (1.25) as long as n [+ or -]/2 ≠ 0, -1, -2,....

1.26 Corollary. [??]α (σ0, σ') can be extended by continuity to σ0 = 0 as follows

(1.27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus defined [??]α(σ) is C∞ outside of the origin.

Proof. This follows from a simple integration by parts. For example, if Re n-α/2 > 1, then

(1.28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This proves Corollary 1.26.

CHAPTER 2

A comparison

Set

(2.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In (2. 1) we assume n[+ or -]α/2 ≠ 0, -1, -2,. ... According to Proposition 7.1 of  the operator Kα, defined by

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

f [member of C∞0 (Hn) is inverse to [??]α as long as α is admissible, i.e., n[+ or -]α/2 ≠ 0, -1, -2,....

2.4 Theorem. Let α be admissible. Then

σ(Kα)(x, [xi]) = σ(Gα)(x, [xi]),

[xi] ≠ 0, where σ(Gα)(x, [xi]) is given by (1.22). In other words,

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever f [member of][??],

Proof. Starting with

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we shall compute

(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We shall do the computation only if [xi]0 > 0. It is similar if [xi]0< 0. We use the notation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and recall that

dHn(x) = 2-n dx.

First we compute

(2.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence we need to evaluate

(2.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we set z = 2u. If –n+α/2 > 0 we can change the path of integration from (-i∞, i∞) to (1/2 - i∞, 1/2 + i∞), so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where L denotes the following path

This deformation is justified by noting that for |v| = R, R large, the integrand is bounded by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since n [??] 1. Finally (-v)γ = eγlog(-v), where log denotes the principal branch of the logarithm, i.e., log(-v) > 0 if v < 0. This easily yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, we have derived

2.10 Lemma. Let x > 0 and n+α/2 < 0. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The next result concerns changing the parameters n and α.

2.11 Lemma. Let a,c be real numbers, a > 0, and a – c > 0.

Suppose x > 0. Then

(2.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Erdelyi: , v. 1. p. 256 proves the following formula

(2.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where –a < γ < min (0 , -c). We replace a by a-c+1, and c by c-2 in

(2.13), we obtain

(2.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as long as –a+c-1 < ω < min (0, c-2). Actually, since a > 0 and a > c we may assume that ω satisfies the more restrictive condition

-a+c-1 < ω < min (-1, c-1).

Setting t=s+1-c in the right-hand side of (2. 14) we obtain that the left-hand side of (2. 14) is equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which gives the right-hand side of (2.12), since –a < γ+ 1 –c < min (-c, 0).

This proves Lemma 2.11.

Next we return to the computation of σ(Kα)(x, [xi]). Set a = 1 – n+α/2 and c = 2-n. Then the hypotheses of Lemma 2.11 can be put in the form

1 – n+α/2 > 0, n-α/2 – 1 > 0,

or , equivalently,

(2.15) α < min (-1, 2-n).

According to Lemma 2.11 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus Lemma 2.10 yields

2.16 Lemma. Let α < min (-1,2-n). (Then

(2.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we are ready to complete the proof of Theorem 2.4. First

(2. 7) and (2.17) yield

(2.18) σ(Kα)(x, [xi])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next the analysis that yields (1.14) from (1.11) applies and we obtain

(2. 19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as long as [xi]0 > 0 and

α < min (-1, 2-n).

To remove the restriction on α, we note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and by introducing parabolic coordinates with v = (|z|4+t2)1/4, φα is integrable at the origin and entire in α. Therefore

(2.20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is holomorphic in α whenever α is admissible.

Next we note that σ(Kα)(x, [xi]) given by (2.19) if [xi]0 > 0 and α < min (-1, 2-n) can be extended to all admissible α holomorphically, the extension given in Theorem 1.22, also for [xi]0< 0. From the formula (1.23) it follows immediately that if satisfies the hypothesis of Theorem 2.4 then (Kα f)(x) given by

(2.21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is holomorphic in α when α is admissible. Since (2.20) and (2.21) agree when α is in an open interval of the real axis, they must agree for all admissible α. This proves Theorem 2.4.

2.22 Corollary. Set x = (x0, x') = (x0, x1, ..., x2n) and zj = x2j-1 + ix2j, j=1, ..., n. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [??]α(σ) is defined in (1.24).

As a by-product of these considerations we found a solution of an inhomogeneous confluent hypergeometric differential equation. We recall that in Whittaker's standard form the confluent hypergeometric differential equation is given by

(2.23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let us consider the differential equation (1.15)

xh" (x) + nh'(x) + (α-x)h(x) = -1.

We substitute

h(x) = x-n/2 y(2x).

Then an elementary calculation yields the following result.

2.24 Proposition. Let x > 0. Then

(2.25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for Re n-α/2 > 0 and its analytic continuation for all other α, α ≠ n, n+2, n+4, ... yields a solution of the following inhomogeneous Whittaker differential equation

(2.26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The analytic continuation is given by

(2.27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if n-α/2 ≠ 0, [+ or -] 1, [+ or -] 2, ..., where D is the Hankel contour appearing in(1.16).

CHAPTER 3

[]b on functions and the solvability of the Lewy equation

A q-form f on Hn is a sum

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where fJ are complex-valued functions on Hn indexed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The [??]b operator (mapping q - forms to q+1-forms) is then defined by

(3.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] see (1.1), and the formal adjoint [??]b is given by

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The interior product [??] is defined after formula (6. 10). One then defines the Laplacian corresponding to this complex; it is

(3.4) []b = [??]b [??]b + [??]b [??]b.

(Continues...)

Excerpted from Estimates for the [??]-Neumann Problem by P. C. Greiner, E. M. Stein. Copyright © 1977 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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• Frontmatter, pg. i
• Preface, pg. iii
• Introduction, pg. 1
• Part I. Analysis on the Heisenberg group, pg. 10
• Part II. Parametrix for the ∂̄ -Neumann problem, pg. 44
• Part III. The Estimates, pg. 130
• Principal notations, pg. 190
• References, pg. 192

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