A NNALI DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze
M ARTIN S CHECHTER
On the dominance of partial differential operators II
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3
esérie, tome 18, n
o3 (1964), p. 255-282
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OPERATORS II*(1)
MARTIN SCHECHTER
1.
Introduction.
In
[19]
webegan
astudy
of certaintypes
ofinequalities
forpartial
differential
operators.
In thepresent
paper we continuealong
these lines butspecialize
toparticular
situations. Thisspecialization
allows us to re-move some of the restrictions of
[19]
and is directed toward certainappli-
cations.
Although
our methods are related to those of[19], they
are muchsimpler. Moreover,
thepresent
paper is self contained and iscompletely independent
of[19].
The fact that solutions of
elliptic boundary
valueproblems
are smoothup to the
boundary
is well known(cf.,
e. g.,[2, 3, 11, 14, 17, 18]).
Re-cently
similar results were obtained forhypoelliptic operators (for
defini-tions cf. Section
2).
Hormander[7]
considered the casewhere the
operators P(D)
and the have constant coefficients.(Here
x~, is the last coordinate in Euclidean
n-space.) Assuming
thatP (D)
ishypoelliptic,
he gave a necessary and sufficient condition that every solu- tion beinfinitely
differentiable inx.. > 0.
His method was tostudy
verycarefully
the solutions of theordinary
differentialequation
obtainedby
Pervenuto alla Redazione il 10 Dicembre 1963.
(1) The work presented in this paper is supported by the AEC
Computing
and Ap- plied MathematicsCenter,
Courant Institute of Mathematical Sciences, New York Uni-versity, under Contract AT (30-1)-1480 with the U. S. Atomic
Energy
Commission.1. Annali della Scuola Norm. Sup.. Pua.
taking
Fourier transforms withrespect
to theremaining
coordinates...,
Peetre
[13]
considered the Dirichletproblem
with
f infinitely
differentiableon x" >
0. He assumed that P(x, D)
is for-mally hypoelliptic,
i. e., that it is a variable coefficientioperator
which foreach x
equals
a constant coefficienthypoelliptic operator
and isequally strong
at eachpoint (for
aprecise
definitioncfr.,
e. g.,[8]).
He obtains asufficient condition that every solution v of
(1.3), (1.4)
should beinfinitely
differentiable in Xn > 0. This condition in also sufficient for the estimate
to hold for all
solutions,
whereR (D)
is anyoperator
weaker than P(x, D)
and the norm its that of
L" (xn ~ 0) (cf.
Section2).
In fact hisproof
ofregularity
wasaccomplished by
means of aninequality
similar to(1.5).
In this paper we are interested in
extending
the results of Peetre to theproblem
with
f
and thefj infinitely
differentiable on x. > 0and xn
=0,
respec-tively.
We do not assume that ishypoelliptic
but allow it tobelong
to a
slightly larger
class. We obtain sufficient conditions for the estimateto hold for all solutions v of
(1.6), (1.7),
where( - >
is anappropriate
normon x. = 0.
(Actually
we prove afamily
ofineqnalities stronger
than(1.8) (cf.
Theorem2.1).)
Thisinequality
enables us to prove that every solution of(1.6), (1.7)
isinfinitely
differentiable inx,, ~~ 0 (Theorem 5.1).
We mention two observations
concerning inequality (1.8).
1. When
P (D)
iselliptic,
everyoperator
of the same or lower orderis weaker than
P (D).
In this case(1.8)
becomes a coercivenessinequality
(cf.
Section 4 or[1, 2, 3, 14, 161).
In some instances the norm. )
may beslightly stronger
than the normusually employed. However,
and thisis
essential, inequality (1.8)
may hold eventhough
theQj (D)
do not coverP (D)
in the usual sense. The reason for this is that the lower order terms in theQj (D)
mayplay a
role indetermining
thevalidity
of(1.8). Examples
where this is the case are
given
in Section 2. Inbrief, (1.8)
is new evenfor
elliptic operators.
2.
Inequality (1.8)
allows one to handleregularity
for variable coefficientoperators
as well. IfP (x, D)
isequally strong
at allpoints,
any errorobtained
by approximating
itby
a constant coefficientoperator
can be estimatedby (1.8).
This isessentially
the methodemployed
intreating formally hypoelliptic operators (cf. [8, 13]).
We shall carry out the details elsewhere.We also
give
anotherproof
of the usual coercivenessinequality
for(1.6), (1.7)
whenP(D)
iselliptic
and all theoperators
arehomogeneous.
This
proof
is muchsimpler
than anypresently
found in the literature.In Section 2 we
give pertinent
definitions and state our maininequal-
ities. Proofs are
given
in Section 3. In Section 5 we prove ourregularity
theorem
by
means of these results. Our shortproof
of the usual coerci-veness
inequality
isgiven
in Section 4.2. The Main
Results.
Let En denote n-dimensional Euclidean space with
generic points
... ,
zn).
We shall find it convenient to set x = ... , y = z,, and denotepoints by (x, y).
Thehalf spaces y > 0
andy > 0
aredenoted
by ~
andE+, respectively.
Let ~u = ... ,
¡.,tn}
be a multi-index ofnon-negative integers
withlength _
,ui+ ... +
Pn. LetDj
be theoperator BliBz; ,
1j
n, and setWe consider a
partial
differentialoperator
of the formwhere the coefficients a, are
complex
constants. Thepolynomial
correspon-ding
toP (Dae , Dy)
iswhere E -== (Et ,
... , We shall alsoemploy
the notationWe make the
following assumptions
on P(~, r~~.
Hypothesis
1. There are constantsK,
andK2
such thatwhenever E and I
are realand 1$1 ] where I
Hypothesis
2. Letm
q be thehighest
power of ’YJ in P(E,
and leta (~)
denote its coefficient. Then there is a constant such thatf’or all real
~.
We relnark that
Hypotheses
1 and 2 aresurely
satisfied if P ishypo- elliptic. By
definition P ishypoelliptic
ifas
(~, ’1) ~
oothrough
real valueswhenever [ p )
0. Thisimmediately implies Hypothesis
1.Hypothesis
2 follows from the well-known observationby
Hormander[6,
p.2391
that in ahypoelliptic operator
the coefficient of thehighest
powerof ’1
isindependent
ofE.
That there are
operators satisfying Hypotheses
1 and 2 which arenot
hypoelliptic
is seen fromsimple examples.
For instance theoperator corresponding
to thepolynomial
is not
hypoelliptic.
For each real
vector ~ let ’(1 (~),
... ,(E)
denote the roots ofWe claim that there is a
positive
constant c~,depending only
on m such thatfor
I ~ I ~ .g2 .
In fact let t7 be any real number and consider thepoly-
nomial 0
(t)
=+
in t. Thenby Hypothesis
1.Let t1,
... , denote the roots of 0(t).
Then there is apositive
constant cmdepending only
on ~n such that(cf.
Lemma 3.1 of[8]). But tk (ik (E)
-n). Setting n
=Re Tk (E)
weobtain
(2.1).
Since the
set ] > ~2
is connected for n> 2,
it follows that the number of roots Tk(~)
withpositive imaginary parts
is constant in this set for such n. Anoperator
with thisproperty
is said to beof
determinedtype (cf.
Hormander[7,
p.227]).
Ahypoelliptic operator
of determinedtype
iscalled
prope14ly hypoelliptic (cfr.
Peetre[13,
p.337]).
One sees from thereasoning
above that in dimensionshigher
than the second everyoperator satisfying Hypothesis
1 is of determinedtype
and hence everyhypoelliptic operator
isproperly hypoelliptic (cf.
also Hormander[7,
p.2271).
Forn = 2 we make the additional
Hypothesis
3. P(E, q)
is determinedtype.
Let r be the number of roots with
positive imaginary parts (for
I ~ ~ > K2). By rearrangement
if necessary we assume thatSet
Let there be
given
rpolynomials Qi (E, ~~, ..., Qr (I, q)
ofdegree
r~zin fj. For
each j
we resolveQj (~, r~) ~ P (~, q)
intopartial
fractions :and set
We consider the r X r Hermitian matrices
and assume
Hypothesis
4. The set ofoperators (Qj
covers P(D).
This meansthat
( ) K2,
A is nonsingular
a. e. and there is a constantg4
such that
REMARKS. This definition of
covering
is ageneralization
of that for theelliptic
case. That A isnon-singular
isequivalent
tosaying
that theQj+ (~, .1)
arelinearly independent,
i. e., that theQj (~, ~)
arelinearly
inde.pendent
modutoP+(~,i7).
The estimate(2.4)
says, in a sense, that thisindependence
is uniform in>
In theelliptic
case(2.4)
holds auto-matically
when exists. A conditionequivalent
to(2.4)
isgiven by
Peetre
[13].
He assumed that every linear combinationQ
ofthe Qj
shouldsatisfy
Let denote the set of
complex
valued functions which areinfinitely
differentiable in.E+
and vanishfor sufficiently large.
For 8 real we
employ
thefamily
of normswhere u
(~, y)
is the Fourier transform of v(x, y)
withrespect
to the varia-bles xi ,
... ,(~x
=’1 31i +
...+
For 8 anon-negative integer
one seeseasily
that v 1.
isequivalent
to the sum of tlte L2(E+)
norms of all derivatives ofv (x, y)
withrespect
to xi 7 ... , X’l-l up to order 8. Inparticular v lo
isequivalent
to the L2(E+)
norm of v.V4’e shall also make use of the scalar
products
where zcc is the Fourier transform
of vi
withrespect
to x,i ---1, 2,
Thecorresponding
norms aregiven by
A
polynomial
R(~,
is said to be weaker than P(~,
iffor all real
E,
t7. Thecorresponding operator
R(D)
is said to be weaker than P(1)).
We can now state our main results.
2’HEOREM 2.1. Under
Hypotheses 1 4, for
eachoperator
weakerthan
P ( D)
and everypositive
nutnber b there is a constant C such thatfor
all v ECo (~E+)
all real 8, where tlceaii (~)
are the eloinent8of
COROLLARY 2.1. Under the gante
hypotheses
there is a constant d de-pending only
on P and theQj
such that(2.5)
bereplaced by
The
proofs
of these results aregiven
in the next section. Anappli-
cation is
given
in Section 5. In Section 4 we discuss theelliptic
case.We now consider. some illustrations. Assume that P is of second de- gree
in I
and that the coefficientof q2
is one. Thenand we take the case
for I E I > K2.
For theboundary
condition we takewhere p
(e)
is apolynomial
inE1, ... ,
Oneeasily
checks thatand hence
Condition
(2.4)
now reduces to1 sufficiently large.
If p is real andthe ri
are pureimaginary,
thisreduces further to
This in turn is valid if there is a constant g such that
for I E 1 sufficiently large
andfor such Otherwise there must be a
constant E > 1
_2 such that1 large,
where theright
handinequality
in(2.11)
needonly
be sa-tisfied when the last
expression
ispositive.
Peetre
[13]
observed thatinequality (2.9)
is necessary for aninequality slightly stronger
than(1.5)
to hold for Dirichletboundary
conditions. Anexample
which violates it iswhere
This
operator
ishypoelliptic
and hence satisfiesHypotheses
1 and 2.If p
(~)
is real and ofdegree 3,
then condition(2.11)
is satisfied and hencewe have
where the left hand side
represents operators
weaker thanP (D).
Another
example
shows how Theorem 2.1gives
new information forelliptic operators.
TakeThen
and
(2.9)
holds. If we takeone verifies
easily that Q (D)
does not cover J in the usual sense(cf.
Section4 or
[1, 2, 3, 14, 16]).
The reason is that thehighest
order termof Q (D)
is
tangential
and hence the characteristicpolynomial
ofQ (D)
vanishes for0
(in
fact whenever$,
=0).
Butnevertheless,
Theorem 2.1applies.
Inequality (2.10)
holdstrivially
and hence we haveOne
might explain
this situationby saying
that theboundary
condition(2.12)
is notelliptic
withrespect
to theLaplacian
but ishypoelliptic
withrespect
to it(H6rmander [7,
p.245]).
3.
Fourier Transform Methods.
We now
give
theproof
of Theorem 2.1..Assume thatP($, q)
and the(~1 (~, ’1),
...Q,. (~, ’1) satisfy Hypotheses
1-4 and let anypolyno-
mial weaker than
P ($, q). For $ ~ g~
itclearly
satisfies(3.1 ) 1
R(~, r~) ~ ($, r~) ~ I const. 1) 1
.We consider functions u
($, y)
as functions of ywith I
as a(vector) parameter.
We let Hm denote thecompletion
of(E1)
witbrespect
to the norm
For a
particular u (~, y)
E0000
setand let D’ be the column vector with
components
We aregoing
toshow that
for all and
real E
suchthat I E I > K2 .
a
simpler inequality
holds. In fact we havewhere C is an upper bound for the coefficients of
P ($, Dy)
on the set1 $ K2 (recall
thatK3
is an upper boundfor a ()’-i,
wherea ($)
is thecoefficient of
’1}m
in P(I, ’1})).
ThusEmploying
the well knowninequality (cfr. [4, 11])
and
taking a 1
we have 2Thus
co - 1
for all
00 (.E+) I K2
where K"depends only
on boundsfor I a () 1-1
and the coefficients of andI K2.
Once
(3.2)
isproved
we see, in view of(3.3),
thatThis
immediately implies
Theorem 2.1. For letv (x, y)
be any function inCo
and let u(~, y)
he its Fourier transform withrespect
to xl , ...... , x"_i .
Substituting
into(3.4)
we obtain the desired result.The
proof
of Theorem 2.1 can therefore be made todepend
on(3.2).
We now prove
(3.2) by
a method due to Peetre[ 13 J.
Let
u(~,y)
be any function in The trick is to find an exten- sionu1 (~, y)
of u to such that] ~~ ~
where C isindependent
of u and~.
For thenby (3.1)
andParseval’s
identity
where
denotes the Fourier transform with
respect
to y. Thisgives (3.2)
when com-bined with
(3.5).
Let
u((E, y)
be an extension of u(~, y)
to Hm(E1),
i. e. a function in whichequals u
on Defineand set
We have
and hence
by
Parseval’s relationThe
problem
is now reduced toproving
where the constant C is not
permitted
todepend
upon u, U1 ore.
In
proving (3.6)
we shallemploy
the two identitiesholding
for anypolynomial Q (E, ’1)
ofdegree
in q, whereWe note that the second relation
(3.8)
follows from the first and the inversion formula for Fourier transforms:In fact
since Q = Q+ -~- Q-.
In order to prove
(3.7)
we define u~~, y)
to be zero for y 0 and ob-serve that
which is
easily
obtainedby integration by parts. Secondly,
we note thatwhen h
(~, y)
isdiscontinuous,
the left hand side of(3.9)
should bereplaced
~y "~ ~ (~ ~ +) + ~ (~ ~ 2013)]’ 2
mBy
andy = 0
we havewhere the
integral
is taken in theCauchy principal
value sense.Set
Then
by (3.10)
Expanding
intopartial
fractions we have(2)
(2)
For simplicity we assume that the roots’rk (~)
are simple. The reader will haveno difficult,y filling in the details for the general case.
and note that
Hence
Substituting
the aboveexpressions
in the left hand side of(3.7)
weobtain
where we have
employed (3.11).
Thisgives (3.7).
-
Once
(3.7)
and(3.8)
areknown,
weproceed
as follows.Write g
in theform
where the
coefficients Aj depend on ~
while 1p satisfiesWe are
going
to show that we mayalways
choose ul sothat
= 0. As-suming
this for themoment,
weproceed. By (3.8)
But
by (3.12)
and hence
where is the inverse of A. This
gives
where
U+
is the column vector withcomponents Uj+. (We
haveemployed
the fact that A is
Hermitian).
This is almost what we want. In fact itgives (3.2)
with Ureplaced by U +.
Of course theQj+ automatically satisfy Hy- pothesis
4 if exists.A
simple argument
nowgives
us the form we desire. Set U- = U- U +.Then
The idea is to estimate the unwanted
expression
U-* A-’ IT- in terms ofN ~
f.
This may be done as follows.Write f
in the formwhere
and set
f, = f -
0. Thenwhere y
is the column vector withcomponents
y; . Now in view ofHypothesis
4We now
merely
observe thatBy
= 1I- sinceby (3.7).
HenceThis,
combined with(3.14) gives
the final result.It therefore remains
only
to show that we maychoose u1
in such away
that
= 0. Webegin by taking
ei inCo (E1).
In this case g is in-finitely
difierentiablein y
0 and vanishes for - ylarge.
One then veri- fieseasily
that9
is bounded in1m r¡ > 0
andThe same is therefore true for the
corresponding ~.
Asimple
contour inte-gration
shows thatBut every
polynomial Q
ofdegree in q
can beexpressed
in the formwhere the Cj
depend only
on~.
Inparticular
this is true forQ
=1 lc r. Hence
by (3.13)
2. Anriadi della Scuola Norm. Pisa.
Thus y (E, «k)
= 0 for each of the roots withpositive imaginary parts.
This shows thatis bounded in
Im q >
0.Moreover, w
is in sinceand
(1 + ~2)m ~ P y2
is bounded(cf. [20 J).
Henceby
thePaley-Wiener
theo-rem
(cf. [12,
p. vanishes for y > 0. SinceP
(~, Dy) w equals Setting 2ci
= ul - w, we see thatu’
is an extensionof it to Hm
(E1).
In additiong’
= P(~, Dy) u’
-f
= g - 11’ and we see thatul
has the desiredproperties.
Thiscompletes
theproof
of Theorem 2.1.PROOF OF COROLLARY 2.1. We
merely
let d be anyexponent
such thatwhere I is the
identity
matrix. Thisimmediately gives (2.6).
4.
The Elli ptic Case.
Coerciveness
inequalities
forelliptic operators
haveproved
to be very useful tools in thestudy
ofboundary
valueproblems
and otherinvestiga-
tions
(cf.,
e. g.,[2, 3, 9, 14, 17, 18]).’
Suchinequalities
for various situations have beenproved by
several authors(cf.,
e. g.,[1, 2, 3, 14, 16]).
In thecase of
L2
estimates for oneoperator,
the usual method is to reduce theproblem
toshowing
thatholds for all v E
Co (E+)
under thefollowing assumptions :
a)
P(~, q)
and theQj (~, q)
arehomogeneous polynomials
ofdegree
m
and mj
1n,respectively.
b) P (~,
iselliptic,
i. e., there are no realsatisfying ] $ ]2 + q2 ~
0and P
(~, q)
= 0.c)
P(E, q)
isproperly elliptic,
i. e., is of determinedtype.
Thus m iseven and r ~
w/2.
d)
TheQj (D)
coverP (D),
i.e., they
arelinearly independent
mo-dulo
P+ .
e)
R(~, q)
is anyhomogeneous polynomial
ofdegree
m.Actually, assumption c)
is neededonly
for n - 2. Asimple argument
shows that every
elliptic operator
isproperly elliptic
in dimensionshigher
than the second
(cf. [10, 2]).
The reduction of the
problem
to(4.1)
is standard andeasily
carriedout;
themajor difficulty
is inproving (4.1).
Of course the
inequality (4.1)
ismerely
aspecial
case of(2.6)
and fol-lows
immediately
from Theorem 2.1.However,
when one is concernedonly
with
proving (4.1),
theargument
can be made evensimpler
than that of, the last section. The
proof given
below is muchsimpler
than anypresently
found in the literature. The
only
result of Section 3 which we shallemploy
is formula
(3.7) (actually
even this is notneeded).
We first note that the function R
(~, .1) / P ($, il)
is continuous on thesurface j (2 + q2
= 1 in B". Hence there is a constant.R5
such thatfor all
such ~,
-1.By homogeneity
this extends to such that~~~~+~l2$0.
Secondly
we observe thatby multiplying each Qj ($, ~) by
an appro-priate
power , we may assume thateach Qj (~, q)
ishomogeneous (1)
of
degree 1n
- 1.Resolving
intopartial
fractions we have(4)
where
(3)
TheQj
may no longer be polynomials in the~y,
but this does not affeet the ar-gnment.
(4) Cf. footnote 2.
One
easily
verifies from thehomogeneity
of P(~, q)
that each root(~)
ishomogeneous in $
ofdegree
one. Itfollows, therefore,
that the sameis true of
each ek ($).
Likewise each(E)
ishomogeneous in 8
ofdegree
zero. In
particular,
it follows that there are constantsK.
and87
such thatLet u
(~, y)
be the Fourier transform of v(x, y)
withrespect
to the va-riables ..., and define it to be zero for y 0. As before we con-
sider u
(, y)
as a function of ywith E
aparameter.
We definef
as inSection 3. A
simple integration by parts gives
where
(Recall
thatf
is the Fourier transform off
withrespect
to y andPk
==
P/ (q
-Tk)).
Sincewe have
by (4.5)
and hence
Thus
by
Parseval’s formula and(4.4)
Since Im Tk 0 for r
k m
we haveby (3.7)
Hence
by (4.4)
and Schwarz’sinequality
Therefore
(4.6)
becomesNext we observe that there is no
complex
vector w =(Wi’
... ,ror) =t=
0such that
For otherwise there would be a
complex
vector A =(Ål , .. , ~
0 such thatand hence
showing
is amultiple
ofP+, contradicting assumption d).
Thusthe
expression
r
is
positive
on thecompact set I I
=1. · Henceby
the ho-1
mogeneity properties
of theqjk
we haveior and
E.
Sinceive have
by (4.7)
and(4.9)
Nov
by
anotherapplication
of(3.7)
and hence
One
easily
checks that theexpression
is
homogeneous in ~
ofdegree
zero and hence is boundedby
a constantfor all real
E.
ThusCombining (4.10)
and(4.11)
andemploying
thetriangle inequality
wefinally
obtain
If we now
integrate
withrespect to E
we obtain(4.1).
Thiscompletes
theproof.
5.
Regularity.
The power of Theorem 2.1 is in its
ability
to proveregularity
up to theboundary
of solutions ofequations
with variable coefficients. We shallsave a detailed discussion of this for a future
publication.
At the momentvie shall content ourselves with
proving regularity
of solutions of constant coefficientequations.
This will illustrate the ideas withoutgetting
involvedin technical difficulties.
Let
P (D)
andQj (D),
1j
r, begiven
constant coefficientoperators satisfying Hypotheses
1.4 of Section 2. For the sake ofsimplicity
we shalleven assume that
Hypothesis
2 isreplaced by
Hypothesis
2’. The coefficient of thehighest
pover of ’YJ in P(~, ’1)
is aconstant.
We shall consider solutions v
(x, y)
ofin
on
where is considered as the
hyperplane y
= 0 inEU.
For eachinteger
k
>
0 letC~’ (.E+)
denote the set of functionshaving
derivatives up to order lc continuous in For each real s we let denote thecompletion C§° (E+)
withrespect
to thenorm [
.Is. Clearly
eachT~ (E+)
is a Hilbertspace.
Let q
be thehighest
order of theoperators P (D)
and the Wehave
THEOREM 5.1. A88u.me
that f E C~ ( li’+~
and thatIf
V E
Cq (EI{.)
f1T ° (E+) is a
solutionof (5.1), (5.2),
then v isinfinitely
E+
..In
proving
Theorem 5.1 we shall find it convenient toemploy
Frie-drichs’ mollifters
[5]
withrespect
to the variables X1, ... , 31n-l.Let j (x)
beany
non-negative
function in which vanishesfor I x I ~
1 andsuch that
Set
and
for any function w E
L2 (E’~-~).
Sincewe see that
J, w approaches w
as e -~ 0uniformly
on anycompact
set wherew is continuous.
We next note that for v E T8
(E+)
This follows from the fact that the Fourier transform of
J, v
istimes the Fourier transform of v and
The eame
reasoning
shows that for v E T °(Efl)
and each fixed e>
0 the functionJ,
v is in T’ for every B.+
Our main
step
inproving
Theorem 5.1 is to establish5.1. Tlnder the
hypotheses of
Theorem 5.1for
each s is aconstant K such that
for
all -> 0,
where m is the orderof
P(D)
with’respect
to y.We assume Lemma 5.1 for the moment and show how it
implies
The-orem 5.1. Note that the constant .~
depends
on v and 8 but not on e.We first make use of the Sobolev
type inequality
holding
for all v E 01 fl Tt(Ell)
whenever t> (n - 1)/2.
This follows from the fact thatwhere u
(~, y)
is the Fourier transform of v(x, y)
withrespect
to x. Thusand
Since
and
(5.5)
followsimmediately.
Applying (5.5)
to(5.4)
forlarge s
we haveI + (n - 1)/2 s,
where.v:
denotes differentiation withrespect
tothe variables x
only.
We shall refer to it as an x-derivative. We see that thefamily
of functions( Ja v )
has bounded x-derivatives of orders8
--
(n -1)/2
inE+n.
Hence for each fixed y and eachcompact subset 0
ofEn-1 ,
the x-derivatives of orders8
-(n -1)/2
-1 areequicontinuous.
Thus there is a
subfamily
such that all x-derivatives convergeuniformly
on0. Since
T, v
convergesuniformly
to v on0,
it follows that v has x-deri- vatives of orders8 - (n -1)/2
-1 on 45. Since s and ø werearbitrary,
we see that v is
infinitely
differentiable withrespect
to x in2~ .
It remains to consider derivatives with
respect
to y.By applying
thesame
reasoning
to derivativesD~ v with k m,
we see thatthey
too areinfinitely
differentiable vithrespect
to x.Moreover, (5.1)
can be written in the formwhere c is the constant coefficient mentioned in
Hypothesis
2’. From(5.6)
we see that is
infinitely
differentiable withrespect
to the x variables.Differentiating (5.6)
withrespect to y,
we see that the same is true of.D~’~
v.Continuing
in this way we see that each derivative exists and isinfinitely
differentiable withrespect
to x. Thiscompletes
theproof
ofTheorem 5.1.
It now remains
only
to prove Lemma 5.1.By (2.6)
and(5.3)
we have(1-lere
we have taken 1 = q and made use of the f’act that J, comnintes withdiSerentiation).
We note that is veaker thanq)
and hence we may take
R (E, q)
= 1 in(5.7).
Thisgives
Since v E TO
(B 11),
we haveIf we now
reapply (5.8)
for the values s =2q, 3q,...,
we obtainfor each real 8. We note next that P
(~,
is weaker than P(~, ~)
and is of the form
where Pt
(~)
is apolyuomial in ~ only. By (5.7)
and(5.9)
we see thatfor each real s. But
and hence another
application
of(5.9) gives
for each real s.
Repeating
the process m times weeventually
come to(5.4).
This
completes
theproof.
Added in
proof.
The methods of this paper can beapplied
tosystems
of
equations
as well.Moreover, Hypothesis
1 can be relaxedconsiderably.
Details will be
given
in a futurepublication.
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