FUJIWARA, DAISUKE

Osaka City University Repository

Fujiwara, D.
Osaka J. Math.
11 (1974), 283-293
AN APPROXIMATE POSITIVE PART OF A SELF-ADJOINT
PSEUDO-DIFFERENTIAL OPERATOR II
DAISUKE FUJIWARA
(Received September 26, 1973)
1. Introduction
Let P=P(x, D) be a self-adjoint pseudo-differential operator with the
symbol p(x, ξ) in the class S\
 >0 of Hϋrmander. The positive part of P is
defined by
+ = [°χdE(X),
Jo
where dE(X) is the spectral measure of P. We shall be concerned with the
following question: To what extent the correspondence; u->P+u can be
localized? We shall prove a localization principle for the operator P+ which
is analogous to Theorem 6.3 of Hϋrmander [5], If we combine this with our
previous discussions in [2], we can explicitly construct an operator B such that
we have estimate
\((A+-B)u,v)\^C\\u\U\v\\l/t,
where u and v are arbitrary functions in £D{Rn) and C is a positive constant
independent of u and v.
2. Localized operators
Let us repeat our notations. p(x> ξ) is a function in the class S\t0 which
vanishes unless x lies in a compact set K in R". We treat pseudo-differential
operator P(xy D) defined as
(2.1) P(x, D)u(x) = (2*)
We assume that P=P(x, D) is self-adjoint in Hubert space L2(Rn).
Now we make use of the partition of unity of Hϋrmander [5], Let g
o
—O, gly
g29 ••• be the unit lattice points in R
n
. Then Rn is covered by open cubes of
side 2 with center at these points. Let θ(x) be a non-negative Co function
which equals 1 on \xj\ 5Π and 0 outside \xj\ ^3/2,y=l, 2, 3, •••, n. We set
284 D. FUJIWARA
(2.2)
 Ψk{x) = Θ(x-έ
and
{x) = ψk ( y {x-gk)+gk)
.2
Note that φ{x)=\ on supp <pk. We, by definition, have
(2.4)
and
(2.5)
for any multi-index « = ( « „ α2, ••• αw).
(2.6) I x—y I ^ 3\/ n ύ x and j are in supp φk.
We set
(2-7)
Then we have
(2.9) Σ
(2.10) l
and
(2.11) If-771 ^ C \ξ\p iίξ and ^ 7 are in supp fa .
Here and hereafter C stands for positive constants which are different from
time to time.
(2.12) Έ\Ψk(ξ)~Ψk(v)\2^ c (g-9) for any ξ,
Let δ y = I gj I p/(1-p\ Then g. δ y G supp i/ry. We shall denote by ψy(Z>) the
pseudo-differential operator corresponding to the symbol ψj(ξ). Then we have
(2.13)
J
The Sobolef norm \\u\\t of // is equivalent to
We put ψjh{x)=φβ\x) and φyjk(Λ?, ξ)=
where σ—(l—ρ)jp. It is obvious from definition that
SELF-ADJOINT PSEUDO-DIFFERENTIAL OPERATOR II 285
This means that the set {φJk}jk is bounded in the class SpA-p. We shall
frequently use the inequality
Choosing a point (xjΊs, ξk) in supp φJk, we set
(2.16) QJk(x, D) = ^
and PJk(x, D)=~(QJk(x, D)+Qjk{x, Z))*), where Q(x, Z>)* is the formal
adjoint of Qj&ix, D). We call these PJk(xy D) localized operators.
3. Statement of results
Theorem 1. For any given 7>—(1—p), there exists a constant C
v
>0 such
that inequality
(3.1) I (P+u, u ) - Σ (P+ΦJk(x, D)uy φJk(x, D)u) \ ^
holds for any uϊΞC*{Rn).
Theorem 2. Assume that the localized operators Pjk{xy D) are self-adjoint.
Let Pfk denote the non-negative part of PJk. Then, for any y>—(1—p), there
exists a constant C7!>0 such that we have estimate
(3.2) I (P+u, u)—2 (PjkφJk(x, D)u, φJk(x, D)u)\ ^C7(||w||v||z/||+||M||fCl_p))
for any u in Co(Rn).
REMARK 3.1. When p=2/3 and N=2y the assumption that PJk(x, D) is
self-adjoint is satisfied and P]k is easily constructed. See [2] for the details. We
can construct operator B for which the estimate | ((P+—B)u, v) | ^C||^ lIi/611^1 li/β
holds for any u and v in CJΓ.
4. Proofs
We begin our proof by the following lemma.
Lemma 4.1. Let Abe a self-adjoint operator in a Hίlbert space X. Let eisA
be the corresponding one-parameter group of unitary operators. Then the non-
negative part A+ of A is given by the formula
(A. 1 \ J+
 y __ (2τrY
286 D. FUJIWARA
for any x in D(A2). Here (s—iθ)~2 is the distribution lim (s—iε)~2. (cf. Gelfand-
Silov [3]) **°
Proof. Let λ+=max(λ, 0). Then we have
(4.2) J~ (s-iθ)-2eisλds= -2τrλ + .
If φ is in £B{Rn), then
(4.3) <(s-iO)-\ φ(s)> =
This and (4.2) mean that
(4.4) -2τrλ+ = [°(eisλ+e-isλ-2)ls2ds-π\ .
Jo
XdE(X) of A. Integrating
- o o
(4.4) with respect to λ by measure d
λ
E(X)x, we have
-2πA+x= [~(eisA+e-isA-2)/s2dsx-πAx = (°° eisAxl(s-i0)2ds .
Proof of Theorem 1. We have to deal with the difference
(4.5) (P+u, « ) - Σ (P+ Φik{x, D)u, φjk{x, D)u)
= Σ ([P+, ΦU*. D)]ΦM*> D)u> u)
Putting
φjfc x, D) = efWφjJx, Dyw and
(
 '
 j
 ΦUs; x, D) = e ^φ y f t(x, D)*e~^p,
we have
(4.7) [««", φjk{x, D)*]φ}.k(x, D)
= ^
p ( φ f t ( ί ; *, D)-φU-s',x, D))φJk(s; x, D)e^p.
Therefore by lemma 4.1,
(4.8) [P\ φjk{x, D)*]φJk(x, D)
The operator φ.k(s\xyD) is a pseudo-differential operator whose symbol is
given in the following manner; Let (y(t; x, ξ)y v(t; xy ξ)) be the solution of the
Hamilton-Jacobi equations
u 9) d v - dP(y>η) dy - _ dP(y>
V
 •
 ;
 A dy ' dt dη
SELF-ADJOINT PSEUDO-DIFFERENTIAL OPERATOR II 287
with initial conditions y (0 x, ξ) = x, and η(0; x, ξ) = ξ. The symbol of
φJk(s; xy D) is
(4.10) φjfa x, ξ) = φJk(y(s; x, ξ), v(s, x, ξ)).
(cf. Egoroίf [1], Hόrmander [6] and Nirenberg-Treves [7].). As a consequence,
the sequence φj/J(s; x, ξ) is bounded in *Spfl_p and the number of overlaps of
supp φjk(s; xy ξ) is bounded. Set
(4.11) ΦJk(s; x, D) = (φ%(s; x, D)-φ%(-s; xy D))φ.k(s; x, D).
Then we have
Lemma 4.2.
(4.12) 1° ΦJk(0;x,D) = 0,
(4.13) 2° fφJk(s;x,D) = \i{[P,Φ%{xy
as Z
(4.15) 3° \s\ -«{ |- ΦJk(s; x, D)-2i[P, φ%(x, D)]φjk(x, Z))}, , k=0, 1, 2 , - ,
is a bounded sequence in the space L^ι"^a~"\ if 0 < α < 1. Here we have used the
notation [P, φ%(x, D)]Cs>=e^p[P, φ%(x,
Proof.
1° is obvious.
2° jj-φMs; x, D) = Ue**p[P, φfte-W = ±i[P, φ%(x,
3° £φ,k(s; x, D) =
= {iβf{[P, [P, φt*]]
ω
-[P, [P, Φ%(x, D)]]
c
_
s)}φjk(s , x, D)
+2(ί/2)2{[Λ φJk(x, D)*]^+[P, φ%(x, D)]C^}[P, φJk]Us)
+{il2Y(φ%(s; x, D)-φjk{-s; x, D))[P, [P, φJk]](s,.
— Φjk(s; x, D)\ is bounded in 5p,(ίrpp). Applying
as i jk
convexity argument, we can prove that the set | — Φ , - ^ ; x, D) — — Φy^ίOj x>
Ids ds
D)}\s\-« is bounded in 5p^ί_Λpκl-p) (Rn). This proves 3°.
Now we come back to the proof of Theorem 1. We divide integral (4.8)
into two parts;
288 D. FUJIWARA
(4.16) Ajk = ^s-2(e^sPΦJk(s; x, D)e^sP+e-^sPΦJk(-s; x,
and
(4.17) B)k = -2π [P, φ%(x, D)]φjh{x, D)+
"
1
'**/*; *» D)ei^p+e-iisPΦjk(-s; x,
We have to prove estimate
(4.18) \H{Ajkuy u)+Σ*(BJkuy it)I ^Cy|ii | | t | | tt | | .jk
Since {Φjk{*\ x, ζ)} jk is bounded in Sptl-P and the number of overlaps of supp
Φjk is bounded, the series Σ Φjk(s\ x, D) converges to an operator T(s; xy D)
in Lptl-P of Hϋrmander [5], Thus we have
(4.19) I Σ (AJkuy u) I = I {Y a {(2X* x, D)e^sPuy e~^pu)jk \ Jt
r, xy
We get estimate of 2 ( By*w> «) by virtue of lemma 4.2. The set
{I * I -°+-ϊ(φy»(ί *, D)-s-4-ΦjdP; x, D)\\ is bounded in S£ί_«p)α-p). If weI V as /)Jk
set Λ=(l—Δ)*and
Sjk(s; x, D) = A-^1+aχl-p's-σ+^(φJk(s; x, D)-s—ΦJk(0; x, Z))V"*α+β5C1"P5 *
V ds /
the sequence of their symbols iSiA.(ί; #, Z)) is bounded in S%Λ-P and the number
of overlaps of supports of them is also bounded. The series ^SJk(s; xy D)
thus converges to an operator S(s; x, D) in the space Lp,i_P. Hence we have
(4.20) Σ (*/*«,«) =
(-ί; x,
where A(s)=eitsPAe-itsP.
Since Λ(ί) and Λ(—s) are elliptic operators of order 1, we have
(4.21) I Σ (BJHU, U)\£C[5α-1ώ||«|l!c1+(»χ1-p)jk Jo
SELF-ADJOINT PSEUDO-DIFFERENTIAL OPERATOR II 289
Setting 7=— (l+α)(l—/>) and adding (4.19) and (4.21), we obtain
IΣί^yΛ u)+Έ(BJku, ιι)| ^ qί-llnllϊ+r^iill ).jk jk
Since £ was arbitrary positive number we take the minimum of the right side with
respect to t. This completes proof of Theorem I.
Proof of Theorem II.
This time we have to deal with
(4.22) I (P+u,
 M ) - Σ (Pί*Φ/ (*. D)u, φJh(x, D)u)\
ψ+-PΪ*)ΦMx, D)u, φjk(x, D)u)\.
Using Lemma 4.1 again, we have
(4.23) ((P+-P]k)φJk(X, D)u, φ^x, D)u)
= Γ {s-iϋ)-χ{ei°p-ei*p'>)φjk{x, D)u, φJk(x, D)u)ds .
J - o o
We put
L(s) = {{eisp-eispi>)φjk{x, D)u, φJk(x, D)u) and
divide the integral in (4.23) into two parts;
5 is ιp"1* s-%L(s)+L(-s))ds and0
(4.25) Njh = πi
The latter is easily majorized. In fact, unitarity of operators eisP and eisPjk
imply that
(4.26)
while
(4.27) I L'(0) I = I {{P-Pjk)φjk{x, D)u, φJk(x, D)u) \
And we have
(4.28) N).k^C\ξk\1-'\\φjk(x, D)u\\2.
L(s) can be written in the form
290 D. FUJIWARA
(4.29) L(s) = \'9£(<[e»pe-«-*>pi')φJk(xt D)u, φjh{x, D)u)dt
= Γ\e"p{P-P J k)e^-'^φ j h{x, D)u, φJk(x, D)u)dt.
JO
The integrand can be divided into two parts
(4.30) J{t) = eitpφ%(2t *, D)(P-P
shγ<-»
p
»
and
(4.31) K(t) = e>'<p(I-φ%(2t; x,
Here φ%(2t; x, D) = e-itPφjk(x, D)*eitP. The symbol φJk(2t; x, ξ)* of it is
obtained from φjk(xy | ) * in exactly the same manner as φjk(t\ x, ?)* is obtained
from φ%(x, I). A consequence of this is that there exists constant C > 0
such that \x-x^\^C\ξk\
p
~
1
 and \ξ-ξk\^C \ξk\
p
 hold if (*, ξ) is in
supp φ%(2t; x, ξ) and | ί | ^\ξk\p~\ This fact together with definition of Pjk
imply that {φ%(2t; x, ξ)(P—PJk)}Jk is bounded in *?*,!% and at most bounded
number of them have non-empty intersection.
Lemma 4.3. We have the following estimates-,
(4.32) (1) I (J(t)φJk(xy D)uy Φjk{x, D)u)\^C\ ξk\ ^Wφ^x, D)u\\2,
(4.33) (2) ||*I -«((J(t)φJk(x, D)u, φjk(xy D)u)-(J(0)φJk(x, D)uy φJk(xy D)u))\
Proof.
o
(1) Since {φfk(2t; x, D)(P—Pjk)}Jk is a bounded set in L\~λ%, we have
\{J{t)φjk{x, D)u, φjk(x,D)u)\
t; x, D){P-Pjk)e^-*^φjh{x, D)u, Kι-\-2t)φjk{x, D)u)\
*, D)u\\
(2) Differentiating (4.30), we have
j-tJ{f) = e"pφ\{2t; x,
= e
itp
φ%{2t; x, D){(P-Pjkγ+[P, P-Pjk]}e^
We know, just as above, that
φ%(2t; x, D){(P-PJkγ+[P, P-
is bounded. This fact implies that
SELF-ADJOINT PSEUDO-DIFFERENTIAL OPERATOR II 291
(x, D)u} φjk(x, D)u) ^C \ξk\2σ~?Λ\Φjk{x> D)u\\2.
Convexity argument again proves
(x, D)u, φJk(x, D)u)-{J(ϋ)φjk{x, D)u, φJk(x9 D)u)}\
n\\φJk(x, D)u\\ \\u\\
Lemma 4.4.
(4.34) I (K(t)φJk(x, D)u, φJk(x, D)u) \^
and
(4.35) x, D)u, φ)k (x, D)u
Proof. By definition (4.31) we have
φ%(x, D)K(t) = e»pφ%(2t; x, D)(ί-φ%(2t; x,
o
Lemma 4.4 is a consequence of this and the fact that φ%(2t; x, D)(ί—φfk(2t;
xy D)) belongs to L~°°.
Now we are able to manage (4.23). L(s) turns out to be
(4.36) L(s) = Γ((/(0-/(0))<M*, D)u, φJk(x, D)u)dt
Jo
+s(J(0)φJh(x, D)u, φJk(x, D)u)
+ \\s-t)(£κ(t)φJk(x, D)u, φjfa D)u)dt
+s(K(0)φ}.k(x, D)u, φjk(x, D)u).
The first term is estimated as a consequence of Lemma 4.3.
(4.37) I J[((/(*)-/(O))Φ,*(*, D)u, φJk(x, D)u)dt
= \St«r»(J(t)-J(0))(φJk(x, D)u, φjk{x, D)u)dt
Jo
1l£*lα+"χl-ΊI<M*, D)u\\\ a>0.
Estimate of the third term follows from Lemma 4.4;
x, D)u, φ,k(x, D)u(4.38)
Thus we have proved that L(s)=sW(s)-i~R(s)9 where
292 D. FUJIWARA
(4.39) W(s) = ((P-Pjk)e"pnφ.k(χ, D)u, φίk(x, D)u)
and
(4.40) \R(s)\ ^ C(s^\ξk\^^-^\\φjk(x, D)u\\*+s*\ξk\-in\\φjk{x, D)u\\ \\u\\)
Now we majorize Mjk. First we have
*'" s-%R(s)+R(-s))ds
'**'
The remainder is
ί-
ι(βin{sPjk)φjk{x, D)u, {P-P}k)*φ}k{xy D)u)ds .
Therefore we have proved estimate
(4.41) \M}k\ ^C{\ξkV->\\φjk{x, Z>)M||2+|fft!-4"+1-p||φyΛK D)u\\\\u\\)
if we admit the following lemma that will be proved later.
Lemma 4.5. Let Abe a self-adjoint operator in a Hilbert space X, then
It follows from (4.23), (4.24) and (4.26) that we must prove estimate
This is proved in the following manner: Taking summation of (4.41) with
respect to j and k, we have
Σ
On the other hand
Σ I ^ ΣlξkVUΦM, W+ξT^'-'WΦjt*, D)u\\\\u\\)
jk jk
This is because the number of those/s for which supp φJkf]KxRn, k being
fixed, is of order | ^ | α " P : > Λ χ ( t h e volume of the set K). Theorem II is now
proved up to Lemma 4.5.
Proof of Lemma 4.5. Let A=\ \dE(X)be the spectral representation of
J —oo
A. Then we have
SELF-ADJOINT PSEUDO-DIFFERENTIAL OPERATOR II 293
s-
1(sin(sA)x9 y)ds = \ ds\ s'1 sin (Xs)d(E(X)x, y)0 Jo J-oo
= Γ d(E(X)x,y) ^s'1 sin (\s)ds
J-oo JO
ί
rκ\
d(E(\)xfy) \ s'1 sin sds .
- o o JO
Therefore,
K
s
 1
 sin sA ds
THE UNIVERSITY OF TOKYO
Sup I ( V 1 si
T I Jo
1
 sin s ds
References
[1] Yu. V. Egoroff: On canonical transformations of pseudo-differential operators,
Uspehi Mat. Nauk. 24 (1969), 235-236.
[2] D. Fujiwara: Approximate positive part of a self-adjoint pseudo-differential operator.
I. Osaka J. Math. 11 (1974), 265-281.
[3] Gelfand-Silov: Generalised Functions.
[4] L. Hormander: Pseudo-differential operators and non elliptic boundary value
problems, Ann. of Math. 83 (1966), 129-209.
[5] : Pseudo-differential operators and hypoelliptic equations, Proc. Symp in
pure Math. A.M.S., Vol. X, 138-183.
[6] : Fourier integral operators I, Acta Math. 127 (1971), 79-183.
[7] L. Nirenberg and F. Treves: On local solvability of linear partial differential
equations Part I I , sufficient conditions, Comm. Pure Appl. Math. 23 (1970), 459-510.



AN APPROXIMATE POSITIVE PART OF A SELF-ADJOINT PSEUDO-DIFFERENTIAL OPERATOR II

Fujiwara, Daisuke

Osaka University Knowledge Archive