A Non-Inequality for Differential Operators
in the L1 Norm
DONALD O R N S T E I N
Comm. unicated by R. FINN
The starting point of this paper is the following question: Is there a universal
constant K such that
f f e,/<x,ay
a,y) ldxdy<----K(ff]a,/(.,
y)a.,
dxdy +
ff av(.,ay,Idxdy)
y)
is true for all coo,/ vanishing outside of the unit square9 We show t h a t this
inequality is not true and that no inequality at all of this sort is true9 This is
the content of Theorem 1.
T h e o r e m 1. Let B, D 1. . . . . DL be a set o/ linearly independent linear homogeneous di[[erential operators in n variables o/degree m. For any K > 0 there is
an / vanishing outside the unit cube and c~ in the whole n space such that f ]B /I > K
and f]Ddl<l, t<_i~L.
The first question is mitural since the answer is yes if the L 1 norm is replaced
by the Lp norm for t < p < oo and since this kind of inequality has proved useful
in studying differential equations. The question for the L1 norm was originally
raised b y HENRI MOREL, who showed that a negative answer would give a distribution, T, which is not a measure but whose first order derivatives are of order 1,
thus providing a counter-example to a question raised b y LAURENT SCHWARTZ
in his Theory of Distributions. This could be done b y constructing an / such that
ff
a*/(x'
ayY) ldxdy is
are finite.
butff
If we let T - - a*/(x' y)
ax ay
dxdy and f f a,t(x,
eV(*,
then
OT
ax
-
-
aY2
aIza*/(x'Y))\
Oy
and
dxdy
aT
ay
aY2
are of first order, but T is not a measure.
0x
Theorem t was suggested b y KAREL DE LEEUW,who, with HAZELTONMIRKEL,
proved the same theorem for the Loo norm 9
The paper will be divided into two parts. In the first part we shall give the
example for ],z, ]yy,/xy. This is simpler than the general theorem and already
contains some of the ideas used in the general theorem. Part 2 contains a proof
of Theorem 1 and can be read independently of Part t. Reading Part t, however,
should help motivate Part 2 and make it easier to read9
Non-inequality for differential operators
4t
Part 1
In this part we shall construct for any l < t a function gt(x, y) which is c~176
and vanishes outside of the square -- t ~ x < t, -- t < y < t, and such that
>
t
Ox ~y,>la x ay = -r ( f f
f f o.g,,;,
,, + o.g,,.,o,
9
ax dy)
"
We shall first introduce some notations. By "g" we will mean g (x, y). All
functions considered will vanish outside the square -- t =< x < t , - - t <: y ~ t .
dg
g * - dx'
etc.
X0
X
g(xo, Yo) = f g ( x , Yo}dx
[Note: ].,~ --/.]
-1
and
Yo
(Xo, Yo) = f g (Xo, y) dy.
-I
Varx g (x, Yo) = total variation of g (x, Yo) with Yo held constant (Var, g (x, y) is a
function of y), Vary g (Xo, y) = variation of g (xo, y), x o held constant. Note that
f Var,,h(x, y) dy = f f I/,,,,(x, Y}I dxdy.
~,
For each l < l we shall describe a sequence of functions p,,(x, y). ~.(x, y)
smoothed out for some (large) n will be gz" p,,(x, y) will have the following
properties:
(a) The y-axis between --1 and I is partitioned into K . non-overlapping
intervals Ji--- a i ~ y ~ ai+ t. The x-axis between -- 1 and l is similarly partitioned
into intervals L i = bi <=x ~ bi+1. p~, is defined and constant on each open rectangle
bi< x < bi+l, ai< y < ai+l and is not defined on the boundary.
(b) p, (x, y) -----0 outside the square -- t --< x ~ t, -- 1 < y < t.
1
1
(c) f p, (xo, y) dy = O, for all x o, f 13, (x, Yo) dx = O, for all Y0.
-1
-1
(d) fflP~[ >=88
T is a constant independent of n. (It depends on l.)
{e) f ['Var, i/~.(x, y) ldy < 2l f f [P.t"
(f) f ]Vary ~.(x, y) l a x - - - f [Vary :l(x, Y}I ax.
(g) : . (xo,, y) is a non-increasing step function of y except at the first and last
values of the ai:
I~.(Xo, y)l-~ a.
(h) "b,,(x, y) = - p , (x, We first define Pl:
pt=
t
for
P1=--t
for
Pt=--t
for
y).
(--t<x<O)
( 0 < x < 1)
(--t<x<O)
pl =
t
for
( 0 < x < 1)
Pl =
0
at all other points.
(--l<)J<-- 88
(--l< y<-- 88
(~l<y<l),
( 88
42
DONALD ORNSTEIN :
Given p., we define Pn+I. (We define it only for negative y and use (h) to
define it everywhere.)
Around each negative a i (except at ao=-- l) pick a small interval ~ of length
(a to be determined later). We redefine p. on each strip -- t < x < t, y E ~ as the
sum of two functions rl9 and r,i (defined on the strip - - t < x < t ,
yE~). Let
r{ (x0. y)y~= 89[p. (x0. y,) + p. (xo, y~-l)],
y~ E/~. y~-, E/~-1.
We define r ~* in the following way. Divide the strip -- t < x ~ t , _ y E ~ , into
rectangles K k of height a and base 7 " Divide each K k into four rectangles, each
of height !2 ~ and base 2t T " Choose/~ so that
[x~ is the x coordinate of the midpoint of Kh, YiEJi, Yi-lEf-1]. Define r~ to
equal ~ on the lower left and upper right rectangle of Kk, and let r{ equal -on the remainder of Kk.
We have thus defined P,+I, and we have to check that it satisfies properties
(a)--(h). The only ones that are not obvious are (d) and (e). These we shall
now show to be satisfied if ~ is small enough (r1 and r 2 were picked exactly to
make (g) [and therefore (f)] true, independently of ~).
First notice the following:
Yl
f Vat, Ir~l ay < t ff Ir~l,
0)
(2)
li~no: IVar- ~'~[ dy ~ o
(r, -- 27r{).
(1) and (2) and the fact that limoff I'll = 0 imply, for ~ small enough,
f.[Var. ~.+l(x, y)[dy < 2l ff.lP.+d d~dy.
(3)
yEXl~
y E,~/~
We have also
Y
(4)
Y
Var~ P.+I (x, y)y r zY~= Var~ p. (x, y)y r ~ ,
because for x, y~ 27f, p.+l(x,
~
y ) = ~ . ( x , y). Now (3) and (4) with (e) for 70.
imply (e) for P.+x. (d) follows from
(5)
lim f f I r { I = l f p.(x, y i - 1 ) - P . ( x , yi)dx,
Ct--+0
yI-xEJ~-I,
y,E~.
since
l~off I r d - ff IP.+II- ff IP.I
and since the right side of (5) summed over all i does not depend on n.
We must now make p. differentiable.
We first redefine p., call it ~b., on a small vertical strip around each line
x = b~ in such a way that (d) and (f) .remair~ true, and ~. (x, Y0) is a differentiable
function of x for all Y0=t= a~.
We' next redefine ~b. (call it ~.) on a small horizontal strip about each line
y = ai so that (d), (e) and (f) remain true and ~. is differentiable.
Non-inequality for differential operators
43
Part 2
I n this part we shall prove Theorem t. First, however,.we m a k e some motivating remarkS.
]~t will be-clear from what follows, since the construction that we shall perform
can be carried out in an arbitrarily small neighborhood of any point, that the
coefficients of B and D~ can be assumed constant.
I t is then easy to find a polynomial / such that B f = t and Di]---O for all
points in n space. We can make ] zero outside of the unit cube, and the proof
consists of rounding off the discontinuity at the boundary of the cube without
disturbing too much the inequality we want. We round off one face at a time.
(In the process we add some planes of discontinuity, so that actually we get rid
of discontinuities on m a n y (n -- t )-dimensional planes perpendicular to a fixed
coordinate axis, and we do this for one coordinate axis at a time.) Getting rid
of the discontinuities on n - - 1 planes perpendicular to a fixed axis, x,, also is
done in stages. We first make / continuous on these planes, then make all partial
derivatives involving only one differentiation in the direction x, continuous, then
all partial derivatives involving two d~fferentiations in the x, direction. (To be a
little more precise, we do this process on each closed cube into which the unit
cube is divided b y the (n--1)-dimensional planes of discontinuity'in directions
not yet fixed up and different from r. / need not match on these boundaries.)
This is the content of L e m m a 2; i.e., we can add one degree of differentiability
in the r direction. In doing this we must maintain our inequality (condition (4)
of L e m m a 2), and this can be done if certain partial derivatives of degree m - - t
have small variation (compared to J [B/[) in the r :direction (condition (5) of
L e m m a 2). In L e m m a I we modify / to make this variation small. L e m m a t
makes the variation small for one operator at a time, and we must therefore use
L e m m a t m a n y times before we can use L e m m a 2. I t is the construction in
L e m m a t that is very similar to th~ construction in Part f of this paper. I n
Part I the construction was the end resuit; here we use it for technical assistance.
The idea that we carry over from Part 1 is roughly this: We have a function ]
of n variables and homogeneous operations B of degree m, D, (i = 1: .... k) of
degree m, Ci (i : 1 . . . . . l) of degree m - - 1 and E of degree m - - t. E is an operator
of the form
~
and is discontinuous on a certain set of parallel n - - t planes.
f
ID,/I <
IB/I
(a condition similar to (d)), and some inequality involving the variation of C J in
the r direction and f ]B/] (similar to (e)) holds. Then we can modify / b y adding
small copies of ] around the planes of discontinuity so that the variation Of E
(similar to f ]Vary p,,(x, y)] dx) across the new plane of di*continuity has not
increased, a fixed amount has been added to f ]B/I and the copies of / are placed
in a set of such small measure that B / a n d B/i (]i are the copies of/) do not interfere and therefore our inequalities, since they hold for / and copies of ], are.not
disturbed.
Proof of T h e o r e m 1. We first establish some notations: x denotes a point im
n space, x~ the/th variable. We write ] : / ( x ) = / ( x l . . . . . x,,). \Ve call an operator
44
DONALD ORNSTEIN :
tgk
of the form ex i ..... x~ elementary. With f there will be associated a finite n u m b e r
of planes 1 p~, where p~ is perpendicular to x i. (] will be c~ except on the p~.)
If x is a point in p~ and ~9 a small sphere about x and A an operator such t h a t
A (]) is defined and continuous on both closed hemispheres into which p~ divides •,
then we denote the absolute value of the jump discontinuity at x b y Vp~t(x).fVp~ l(x)
means integration over p~ with respect to (n--1)-measure. [All other integrals,
such as f ]D(])I, mean integration over n-space. 1 We shall sometimes write
V '~ ! (x) ~ Vp~! (x) if there is no possible confusion.
L e m m a 1. Let ] (xl, ...., x~) have the ]ollowing properties."
(1) ] = 0 on the exterior o] the unit cube.
(2) There are a finite number o/ hyperplanes p~ and a fixed r such that p~ is
perpendicular to xi, 1 < j < = r < n and ] is c~ on each closed set into which the R~ is
thus divided. (p~ =.lower boundary o / I and pM = upper boundary.)
(3) There is an s such that all elementary operators involving less than s x/s
exist and are continuous on the closed sets into which the unit cube is divided by
p~ (1 < i < r - - I ) . [Note that when s=O condition 3 is vacuous. When s = t , (3) is
taken to mean that ~ (as well as all partial derivatives not involving x,) is continuous.]
(4) ~f l D- d l < e (t < i ~ L) (here B~ and D / are not necessarily defined on the p~).
(5) There is a set So] operators o] the ]orm
~Ya-- I
a C,.)" .~ .....
,~
(i.e., o] degree m-- 1 and involving exactly s di]]erentiations with respect to x,) such
that/or A in S,
M
y. fvp~!(,)
fiB(t)]
[summation is taken over all planes o[ discontinuity pe@endicular to x,].
Then, given an elementary operator E involving s x,'s and of degree m - - t ,
there is an ~ satisfying (t) ; (2) with more p~ but the same n u m b e r of directions, r;
(3) with the same r and s, but more p~; (4) with more p~, and (5) with more p~
and larger M and the set S enlarged by E (i.e., we can enlarge S by E without
disturbing things too much.)
Before proceding to the proof of L e m m a t note t h a t in this argument r, s, m
and n are constant integers, i, ?', k variable integers, ['s are functions, p's and g's
are hyperplanes and the a's are thickened hyperplanes.
P r o o f of L e m m a 1. [We are going to describe a process which will have to
be repeated m a n y times. Hence for the sake of clarity we assume t h a t we have
two functions / and 0[ satisfying the conditions of L e m m a t. 0f m a y have different hyperplanes of discontinuity oPt, but the number of directions, r, and the
degree of discontinuity s on planes perpendicular to x, will be the same. Also
the set S will be the same for / and o/.]
Planes will mean planes (or hyperplanes) of dimension n-- t.
Non-inequality for differential operators
45
We also pick our o[ in such a way that the sum of the jump discontinuities
of E along any line in the x, direction will add up to O. If 0[ does n6t a!ready have
this property (which we shall refer, to as (6)), we take the sum of o] and -- 0[
translated b y an amount in the x, direction such that its support is disjoint from
that of 0]. This new function could then be shrunk down (replace ](x) by [(cox))
to have support in the unit cube.
Pick a strip, a~, of width y about p', (for all t . < i ~ M ) .
(y will be a small
number to be determined later.)
Step 1. Redefine
8'1
in each a i in the following way: Let g~'i('t _~/'~M)
be a hyperplane in a, and parallel to p~ and such that d(g~", g~,k) =yd(0p~, opk)
where d (p, g) means distance between p and g; a i is thus divided into M0 substrips
which we call hi. [For simplicity we assume that the boundaries of a, are included
in the g~'J and that o] vanishes outside of a strip bounded by its two extreme opt-!
We redefine ~ f on* a line l Il~p~ parallel to x,. Let a, and fl, be ~(x,)s
~t
evaluated at the right and left end-point of l,",a,, respectively. Let ],..s =
~i + ~ - (fli--ai) on a{~l (we assume then a! are numbered consecutively, so that
~ / on the rest of l. Define
/,,s will be monotone), and/,,s = ~(x,)s
G
(a)
.....
'
......
f
~1
x,)' - '
"
.....
x ......
,,,)dx,
-i
Note that ] , , , - e(x,)s.
Step 2. Divide each a i into disjoint cubes Ci,,; the length of an edge will be 7.
Let y,,i be the center of C~,i. Let o[],, J.x
to Yi.i. Define [ a s
be 0[
with the origin translated
j, *
(summation is taken over all Ci, i) , where
K1 i = J
(b)
'
4
v E [ (y],i) . ~_o_ . 7 m-1 ~
,
max
i
{ I "E (o])(x)}.~ C:oplr"(l_~k =<.llo)
(m-- t = degree of E).
We now demonstrate some properties of 7. Conditions (2) and (~) of this lemma
imply for t --<b _ s
-G
Oi[ (~1 . . . . . r . . . . . . ~,,)
-
l
f
~
.
8j+b [ (~i. . . . . x, :r+x . . . . . ~J') d x ,
.
.
.
.
--1
(the line x,. is assumed not to be in any of the p~). (c) remains true if we"replace !
by ]. From this it is easy to check that for any elementary operator A,
limAf(x)=A/(x)
(d) 3
y---~ 0
for
lim f lA [t = O.
(e)
~,---~0
2
~]
0(.,)o
Uaj
w i l l b e t a k e n t o m e a n [.
3 W h e n s----O, (c) is v a c u o u s a n d (d) is o b v i o u s .
x~Ua,,
46
D O N A L D ORNSTEIN :
Condition (t) implies that (d).can be modified to
if)
lim A/(x) = A / (x)
x~U a/;
for
~'--+0
another property is
(g)
~ f v~.'(x) = ~, f v:r(x),
P
g
for y small enotigh. [The left side is s u m m e d over all p~ and the right side over
all
g~,i. r is fixed;
i and ~"vary.~ g is true because
(~) the planes of discontinuity perpendicular to
the g~'k,
(/3) if y is small enough
x, of di,~(--~
t •
with
21< VE/(Yi'i)VE/(x)
for xCCi,~,
is t,,e, e eoo,
L
/
--J
(~) VF'7(x) > ~t ~ t Vm(y/,~)
for x in
Ci,~:~g~,'k.
(~) (fl) (y) (~) together with (b) imply that VEThas the same sign as VET for x
in some g~'i, and, since o/also satisfies (6) (see p. 45), (g) is proved.
xf
(h)
c:j,,:,
[o:;.,
where
C=
-"
fIB(d)l
4 Mo max { VE (d) (x)}.E op~(~.<k<M,) "
[ ,')]
(h) follows from (r (b), and the fact that B [ d (x)l = 7.~ B 0/(-~ x . Because of (e),
9 (f) and (h), we have
(i)
lim B / =
~--*0
B / + B(.~. K,,j[o[Li(~x)] )
"
To finish the proof we first apply Step t and Step 2 to the pair ] = d .
then apply Steps t and 2 to the pair ], d instead of/,
d, getting
We
a new function/.
We now apply Steps I and 2 t o / , d, etc. Eventually we shall get a n / , which we
call ~
Let us check the properties of ]. Conditions (2) and (3) of the lemma are
easy to verify with the aid" of (c). (4) is satisfied because of (i) and the fact that it.
is true of K,,,Io//,i( ~ x)l" (5)stillholds for all theA in S because for A in Swe have
x s v:, : x
i
,
*,7
/
1
and (5) are true at each step and hence are true of ~ Now because of (g),
Non-inequality for differential operators
47
[where the summation is taken over all planes of discontinuity perpendicular to x,
of f and f respectively]. (h) and (e) imply that each time we repeat Steps t and 2
we add a fixed amount to f IB]I. This shows that if we repeat t and 2 long enough
will have (5) satisfied for E also.
To make ] satisfy t we note that if each y was small enough, ] and all partial
derivatives up to order m can be made arbitrarily small on any fixed cube containing the unit cube. We could therefore modify f to be 0 outside this fixed
cube containing the unit cube (just multiply by a fixed coo function that is t on
the unit cube and 0 outside the other cube). We now have a new ] that satisfies
our lemma.
L e m m a 2. Let ] (Xl... x,) have the ]ollowing properties:
(t) ] = 0 on the exterior o/the unit cube.
(2) There are a [inite number o/hyperplanes p~ and a number r suck that p~ is
perpendicular to x i, t < j < r < n and ] is c~ on each closed set into which the R ,
is thus divided.
"
(3) There is a number s such that all elementary operators involving < s x,'s
exist and are continuous on the closed sets into which the unit cube is divided by
Pj0 < i < r - t). [(3) is vacuous/or s = 0 . ]
(4) f [h~ II < e (t ~ i ~ Z).
f lB tl
-_
(5) I/A is an elementary operator o/degree m-- t involving exactly s differentiations with respect to x,, then
M
<e.
flB
/I
(There are M planes o/discontinuity perpendicular to x,.)
Conclusion. We can find an ] satislying (t) ; (2) with dillerent p~, pZ~ but the
same number o/directions r; (4); and (3) with all dementary operators involving
< s (instead oI < s). [When s = 0 we conclude that [, and A I, where A is an
dementary operator not involving x,, is continuous on the closed sets into which the
unit cube is divided by p~(t < ~<r-- t).]
Proof of L e m m a 2. About each p; pick a strip a~ of width }, (the bounding
planes of these strips will be the ~,~). Along each line l [l d~p}] parallel to x,
0,f on lc~a~ to be linear on aic~l and continuous on the whole line and
redefine 0-ff(~-,)*
to agree with ~ 0,t
outside a,c~l. Call the new ~ 0't
(a) Define
~ (~ ..... ~, ..... r
, ],,,.
r
-
( ~ _,~ ) ! . [
( ~ ' , - x,)'- ~ l,., (r ....
, x , ,. ....
. ~,)
d x, .
--1
Note IT,s-- O~-~tis 9 W e n o w show t h a t ] has t h e properties t h a t we w an t .
9(b) It is easy to see that 0 (x,)*,O,+k
T x~ "when restricted to a line] parallel to xr,
z[ .....
is linear in each lr~ai and continuous on the whole line.
48
DONALD ORNSTEIN :
(c) We now have
~r
ei/(Ct ..... r ...... C.) _
l
,9 ( .. ) s - ~,, .,:, .....#
x,)b_~ ~/+bf(r
f (r
(b--l)! .
. . . . . x, . . . . . G) d x ,
--1
(t < b < _ s ) .
(d) I~ follows from (b) and (c) that [ is c~ on the closed sets into which the
unit cube is divided by the p~ (1 < i < r) and the ~ .
From (c) it follows that for an elementary operator A,
(e)
(f)
limAf(x)=A[(x),
for
7-.+0
~kf~x)
~ (~,)'+~,
~ .....
=0,'for
4,
xffUa~;
xEOai(t~b).
-
This follows from (b):
(g)
~lim
oj
f] a(x,),-b,,!
O*'l(x).....
x~ I = 0
(O~_b<=s)
13 at
(integration is taken over the a,).
This follows from (c):
(hi
lim f
7~oj
am[(x)
a(~,)'+'-, xh . . . . .
4
= V~t(X),
al
where the integral on the left is integrated over a i and where
~rt$--1
A=
(*,)', *,{ ..... 4 "
(f),.(g), (h) and hypothesis (5) imply
,, f l -a/I
,..~a~
(i)
--fl/~?l-
~ ~,
where A is any operator of degree m.
(i) and (e) imply that [ satisfies condition (4). (2) is the same as (d) ~and (3)
is clearly satisfied with =<s instead of < s . Because of ( e ) / c a n be easily modified
'to satisfy (1) as at the end of the first lemma.
Proof of the Theorem. Let ]0 be a polynomial such that B/o(x ) ~ t and
Let / 1 = / 0 Orl a closed cube I ' interior to the
unit cube and/1---- 0 on the exterior of I'. ]1 satisfies the conditions of Lemma t
with r = n, s-~ 0 and S empty. We apply Lemma t t o / 1 several times, each time
increasing S by one until S Contains all elementary operators of degree m-- t not
involving x,,. Call this function/2. /~ satisfies the conditions of Lemma 2 with
r -----n, s = 0. Applying Lemma 2 t o / 2 gives us ]2 which again satisfies Lemma t
with r = n and s = t and S empty. Repeated application of Lemma I will fill S,
DJo(X) = 0 for all x and t < _ i ~ L .
Non-inequality for differential operators
49
giving us /a- Lemma 2 applied t o / 3 gives us/a, which satisfies the condition of
Lemma t with r = n, s = 2. Repeating this process m times, we get a function/1
such that for all elementary operators A involving ~ m x,'s A # is defined and
continuous on the closed sets into which the unit cube is divided by the
p~.4 ( t ~ j ~ n - - t ) .
Repeated application of the process of Lemma 2 will give ]]
A
which is coo on these sets. Also f l Di t I [ < e 1 ~ i--< L.
tB31
We can now apply Lemma t to/1 (r = n - - l, s = 0). After repeated application
Lemma 2 applies, etc. At the end of all this we get a function ]2 which is c~
except on p] 1 ~ ~"=<n - - 2 and such that f ]Di 721 < e (t ~ i ~ L). Repetition of this
&
f l B f~l
-- _
process proves the theorem, with e = ~ - .
4 p~ denotes the planes of discontinuity of [1.
Stanford University
California
(Received M a y 5, 1962)
Arch, R a t i o n a l Mech. Anal., Vol. t t
4