Rend. Sem. Mat. Univ. Poi. Torino
Voi. 51, 1 (1993)
F. Serra Cassano
A COUNTEREXAMPLE ON THE WEIGHTED
POINCARÉ' INEQUALITY
Abstract. In this paper we exhibit a counterexample to some naturai conjectures by
constructing a weight function A for which a weighted Poincaré inequality does not
hold. Then we study the links between the Poincaré inequality with weight A and
some properties of A (such as, for instance, the summability of A and A - 1 , or the
regularity of A).
0. Introduction
Let p > 1 and A be a function in L11oc(Rn), positive a.e. in Rn.
We say that the weighted Poincaré inequality holds in X(Q), if there exists a positive
Constant e such that
P p (fì, A) : / \u\pXdx <c f \Du\p\dx
for every u e X(Q) ,
where £1 is an open set of R n and X(Q) is a suitable subset of W^(Cì). When n = 1 it
can be given a characterization in order that Vp(Ct, A) holds in X(Q); for instance, in [8]
(see, also, [3] § 1.3, Theorem 3) the following interesting result is given: Vp((0,R),X)
holds in X{(0,R)) = {u e C^lO.R]) : u{R) = 0} iff
(0.1)
sup [ ( / W ) ( / \~l/{v~~l)df^V~ ] < + o o .
In the general case n > 1, a characterization of the weighted Poincaré inequality in CQ°(Q)
is known (see [7]): let Q be a bounded open set of Rn, then VP{£1, A) holds in C§°(n)
iff there exists a positive Constant e such that, for every compact subset K of Q with C°°
boundary,
(0.2)
f \dx<cC%\K,\)
JK
where
(0.3)
C^\K, A) := inf { f \D<p\pXdx : tp e CJ°(Q) with cp > 1 in tfj
66
E Serra Cassano
denotes the p-capacity, with respect to À, of the compact K.
However let us observe that, except for the unidimensional case, it can be difficult
to compute the quantity in (0.3) for an arbitrary compact subset K of Q and so it can be
hard to verify (0.2).
Moreover there are also the following simple classical conditions in order the
VP(Q,\) holds in C^(fì):
(0.4) (see [4]) A e L^Q^X'1
e L*(fì) with s,t e [l,+oo] such that
- + 7 < ~ and (1 + 7 J <p<n( 1 + ì ) ;
s t
n
\
tJ
\
t )
(0.5) (see [2]) A satisfies the Muckenhoupt condition
(Ap) sup ( meas (Q)" 1 f Xdx\ ( meas (Q)" 1 f
X^lf^~l\h
p-i-
< + OO,
where the supremum is taken on ali cubes Q of R n with sides parallel to the coordinate
planes.
Unfortunately those simple conditions are not necessary, in fact, we have that
PP(Q, A) holds in C^(fì) also if (0.4) or Ap are not satisfied. For instance, if X(x) = \x\a
it can be proved that A e Av iff a e (-n, (p - l)n); so by (0.5) VP{£1, A) holds in C^(lì)
also if (0.4) is not verifled. On the other hand in [2], § 3 (see also [1]) it is proved Vp{Sl, X)
stili holds in CQ(Q) for every a > 0 even if X(x) = \x\a £ Ap for a: large.
Then a naturai problem is to study if it is possible, in some way, to weaken the
conditions (0.4) or Ap in order that VP(Q,X) holds in C£(Q). For instance, to see if it
holds under the only assumption that the two integrals in the Ap-condition are finite.
In this note we study the weighted Poincaré inequality when A is a p-weight on
n
R , i.e. A satisfies
(0.6)
A > 0 a.e. in R n , A and A" 1 /^" 1 ) are in Lloc(Rn).
We give for ali n > 2 and for every p > 1, an example of a p-weight Ao on Rw, for
which the weighted Poincaré inequality VV{B, A0) (where B denotes the unit open ball of
R n ) does not hold in C\(B) (see Proposition 1.3). Moreover by this weight Ao it can be
proved that the condition (0.4) cannot be improved (Remark 1.5).
On the other hand we give also a simple characterization in order that VP(Q,X)
holds in CQ(Q) in the special case when A is of the kind
A(,)=/i(|*|),(£)
and Q, is an open ball of Rn centered in the origin of R n (Proposition 1.6).
A Counterexatnple on the Weighted Poicaré Inequality
67
1. The counterexample
Let ip : [0, +oo) —• [0, foo) be the function defined by
+°o
(1.1)
v(r):=r"+$>fc(r)
where tph '• [0, foo) —• [0, -f oo)(/i = 0,1,...) are the functions defined by
I
O
0 < r < 6 2 - h - 3 or r > 2~h
2—^ f 1 _ ^ - / X - V ) )
|r - 72- f c " 3 | < 2 - / l - 3
with o; and /? are positive numbers such that
(1.3)
0<a<(3
Then the following properties hold for the function (p.
LEMMA 1.1. Let a and fi be real numbersverifying (1.3), then (p : [0, +oo) —•
[0,-f-oo) is a continuous function, positive in (0,-foo) for which there exists a positive
Constant e > 1 such that
(1.4)
r@ < <p(r) < cra f r13
for every r e [0, -f-oo).
Moreover
(i)for every q > 0 and 7 > 0 verifying 0 < 7 < ag f 1, we have
f
r 1cpq(r)dr < f o o ;
'0
Jo
(ii)for every q > 0 and 7 > 0 verifying 7 > /?</ - 1, we flave
r1ip~q\r)dr
< foo .
Prao/ Since-iph(h = 0,1,...) are continuous and non negative in [0 -f-oo), by
(1.1) and (1.2) it follows at once that (p is continuous in [0, -foo), positive in (0, +00).
Moreover, since for every 7 > 0, VQ > 0,0 < d < ro
•r°(1~(!~T^))
~
r l
if\r-ro\<d,
from (1.2) and (1.3) it follows also that
(1.5)
'8
a
0 < iph(r) < ( - j ra
1
h
for every r e "2~h~i-h-1-n-h
, for every h .
ì2~
Therefore, by (1.5) and (1.1), we get (1.4).
Finally, by (1.4) we deduce at once (i) and (ii).
•
68
E Serra Cassano
Define now the function Ao : R n —• [0, +00) as
[O
x =Q
where <p : [0,-foo) —> [0, +00) is the function defined in (1.1).
Set B = Bi := {x e R n : \x\ < 1} and denote by wn the n-dimensional Lebesgue
measure of B.
Then A0 verifies the following summability and regularity properties.
1.2. Let n > 2, a and (3 be positive numbers verifying (1.3) and
let Ao be the function defined in (1.6).
Ttien A0 is a continuous positive function in R n \{0}, moreover
(i) ifO<a<n-l
then A0 G Lfoc(Rn) for every s e [1, n/(n - 1 - a)); ifa = n-l
then A0 G ££ c (R n ); if a > n / t o A0 G C^R").
(«j */0 < (3 < n-1 then A^1 e L™c(Rn); i / n - 1 < (3 < 2n - 1 tfzen A^1 G Lfoc(Rn)
for every £ G [1,ra/(/?- n + 1)).
PROPOSITION
Proof By (1.6) and Lemma 1.1 we deduce that Ao is a positive continuous function
in R \{0}.'
Let us prove (i).
By using polar coordinates, by (1.6) and (i) of Lemma 1.1, it follows that, if
0 < a < n - 1,
n
f Xs0dx = nujn f ^{ry^-^-^dr
< +00
Vs G [1, n/(n - 1 - a)).
Now, sinceAo('c) = \x\'3-n+1 for x e Rn\B, then A0 G Lfoc(Rn) Vs G [ l , n / ( n - l -a)).
If a > n — 1, by (1.4) and (1.6), we deduce that there exists a positive Constant ci such
that
0 < X0(x)
'.<ci\x\a-n+1
for every
xeB;
therefore A0 G Lg>c(Rn} if a = n- 1 and A0 G C°(R n ) if a > n - 1.
Moreover, if n < a < (3, by (1.1) and by (1.2), it results at once that ip G
C^pj-f-oo)) and there exists a positive Constant c-2 such that
(1.7)
b'(r)| < c2ra_1
for every rG [0,1].
On the other hand, by (1.6) we have that
(1.8) A„ e CHR-UO}) and !*,(,) = ^ ( M ) N ^ - ( ^ - I M W ) ^ - ^ , ^ 0 .
A Counterexample on the Weighted Poicaré Inequality
69
by (1.8), (1.7) and (1.4) mere exists a positive Constant c% such that
(1.9)
| m o W | < J ^ ^ +(nH-l)^^<c3N—
V,efl\{0},
so, by (1.9) it follows that A0 G C^R"') if a > n.
Finally, by using similar arguments as in (ì), it can be proved also (ii).
•
We can now give the counterexample.
PROPOSITION
(1.10)
1.3. Let n > 2,p > 1, a and (3 be positive numbers verifying
a + p<f3
<pn-ì,
and let X0:M.n—> [0,+oo) be the function defìneùUn (1.6).
Then Ao is a p-weight on R n such that VP(B,XQ) does not hold in
D:=UeC^(B):u{x)
= v(\x\)
with v e C 1 ^), 1])} .
Proof. By (1.10) and Lemma 1.1 (ii), it follows that A0 and Aó 1 / ( p _ 1 ) belong to Lf oc (R n );
so, by (0.1), Ao is a p-weight on R n .
Now we will prove that VP(B, Ao) does not hold in D.
Let vh; [0,1].—> R(h == 1,2,...) be C1 functions such that
vh = lm[0,2-(h+1)
/h],vh=()m
f
2~(A+1)/'\1
l\
K|< C l 2Mn[0,l],
where c\ is a positive Constant independent on fi.
Consider the function uh : B —->• E (h = 1,2,...)
uh(x) = vh(\x\(h+^h)
(1.12)
.
Clearly we have that un € D for every h. Moreover by (1.11), (1.12), (1.1) and
(1.2) we deduce that
(1.13)
[ \uh\nQdx>
JB
[
h
•\vh(\x\^h+1Vh)\p>^(x)dx='f
J\x\<1- -i
r\ — h~ 1
c
r
= nujn I
Jo
2
= c 2 2~ (Q+1)/l
n—h — 1
2
r
(p(r)dr > nu)n I
<p(r)dr =
h 4
J7 2- for every h > 1,
where c2 = mjn2~(a+4) $(1 - z2)2dz .
J\x\<2-h-x
X0(x)dx =
70
E Serra Cassano
On the other hand, by (1.6), (1.12), (1.1) aiid (1.2) we have that
(1.14)
/ \Duh\pX0dx = (^±)P[
\v\X\x\^^\yAv,hM^)^
for every h > 1, where c^ is a positive Constant independent on h.
Then by (1.13) and (1.14) it follows that
,.*,'', "
JB\Duh\PX0dx
T
> — 2^-a-p)h
for every h ,
C2
and so, by (1.10), VP(B, XQ) does not hold in D. •
1.4. Let us explicitly observe that, by Proposition 1.2 and 1.3, if
n > 2, p > n/{n — 1) and
REMARK
n
1 < a < p(n — 1)
,
a-\-p<(3<pn
— 1,
the function Ào turns out to be a continiious p-weight on Rn such that Vp(B,Xo) does not
hold in C^{B).
On the other hand, if n > 2,p > n + l / ( n - 1) and
n < a <;?(n — 1) — 1 ,
a + p < /? < pn — 1 ,
A0 turns out to be a C1 weight on E n .
1.5. By Proposition 1.2 and 1.3, we deduce that the summability
condition (0.4) cannot be improved for p e [2,2n - 1).
In fact, when p e [2,2n — 1], by Proposition 1.2 and 1.3, if n > 2, then, for every
s e [n/(n - l),+oo), t e (1,-t-oo) with f + j > f-, it is possible to find, by a suitable
choice of a and /3, a p-weight Ao such that VP{B, Ao) does not hold in C\[B).
REMARK
We give now a simple characterization of the weighted Poincaré inequality in a
special case.
Let BR := {x e Mn : \x\ < R], B = £i,Boo = E n and denote by Hn-i the
(n — l)-Hausdorff measure on R n .
lì
A Counterexample on the Weighted Poicaré Inequalìty
1.6. Lei R e (0,oo],p > 1 and lei \i : (0,JR) —• [0,oo) and
v : dB —> [0,oo) be two measurable functions with v e L1(dBìHn-i{dd)).
Suppose
that A is of the kind
PROPOSITION
(1.15)
X{x) = p{\x\)v (•£-) x e BR\{0} .
\ \x /
Then Vp(BRìX) holds in C^(BR) iff
1 /- (l /p( P_ -1l ) j ^ P " "
, nn- l J ^ ( f - / . . ^ N _. n - l V
sup
[
(
/
^
)p
-Up){j
ow
T
W)
P
0
0<r<fi
(ne)
R
r
1
' < +OO
Proo/ We treat only the case when i? is finite, the case R = 00 being similar.
Set y(r)..:=7z.(r)r n ~ 1 and suppose that PP(BR, A) holds iirC^I?/?). Then, by
a density arguyment, we can get that VP(BR, A) stili holds in Lipo{B) := {w Lipschitz
continuous in B, supp(w) C f?} and so, by considering the radiai functions u(x) = v(\x\),
it follows that Pp((0,i?),.y>) holds in X((0,R)) = {v e ^ f M ] ) : v(i?) = 0}.
Conversely, if (1.16) holds, by the result in [8] (see also [3], § 1.3 Theorem 3) we
get that 7V((0, R), tp) holds in X(0, flj.
On the other hand, for a given « G CQ(BR), if we defìne
v$(r) := w(a;) = w(r#)
( r = |x| and ti = —• ) ,
^ e X(0,.i2) for every ti e dB. In particular, there exists a positive Constant e = c(p)
(depending only on p,) such that, for every u e CQ(BR), for every & e dB
nR
\
pR
pR
pR
p
p
\%t{rd)\ (p(r)dr = / \v*\ (pdr <c
\v^\ ipdr <c
\Du(rd)\p ip(r)d,r .
p
JoJo
Jo
Jo
Then, integrating on dB and using polar coordinates, we nave that
\u\pXdx=f
/
JBR
<cl
JdB
for every u e
JdB
v(i9)Hn-i(dd)
i/(#)?tn-i(d'd)
CQ(BR)
[
Jo
\u{rd)\pip{r)dr
f
Jo
\Du{rti)\pLp(r)dr = e f
and so the thesis follows.
\Du\pXdx,
JBR
•
1.7. By arguing as in the proof of Proposition 1.6, we deduce that, if
A is a measurable fiinction, positive a.e. in BR, a necessary condition on A in order that
Tp(BRì X) holds in C^(BR) is
REMARK
sup
0<r<R
Xdx
{(L )(l
R
/
f
x\ - 1l //((7p3 - 1D)
(K [ X{Pd)Hn-_!(dl9)J
JdB
\ JP) - l li
N
dp)
< +OO .
72
E Serra Cassano
1.8.
Let A be a radiai function on BR, i.e. \(x) = p{\'A) a n d
suppose that A e L}oc(BR), /z" 1 /^" 1 ) e L1((<5,JR)) for every 6 e (0,1?) and there
exists So € (0, R) such that
REMARK
r — • ^(r)rn~x
is strictly increasing in (0, So).
Then, by using Proposition 1.6, it follows that VP(BR,X)
holds in
CQ(BR).
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Francesco SERRA CASSANO
Dipartimento di Matematica, Università di Bologna
P.zza di Porta San Donato, 5, 40127 Bologna, Italy.
Lavoro pervenuto in redazione il 29.10.1992 e, informa definitiva, il 1.3.1993.