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A new poincaré inequality and its application to the regularity of minimizers of integral functionals with nonstandard growth
Nonlinear Analysis, Theory. Methods & Applicahons, Vol. 17, No. 9, pp. 833-439, 1991. Printed in Great Britain.
0
0362-546X/91 WOO+ .OO 1991 Pergamon Press plc
A NEW POINCARii INEQUALITY AND ITS APPLICATION TO THE REGULARITY OF MINIMIZERS OF INTEGRAL FUNCTIONALS WITH NONSTANDARD GROWTH TEAK BHATTA~H~YA Department of Mathematics, Northwestern University, Evanston, IL 60208, U.S.A.
and FRANCESCOLEONETTI Dipartimento di Matematica Pura ed Applicata, Universit& degli Studi di L’Aquila, 67100 L’Aquila, Italy (Received 30 July 1990; received in revised form 23 October 1990; received for publication 20 February Key words and phrases:
Poincare inequality, integral functionals, minimizers, Morrey-regularity.
nonstandard
1991)
growth conditions,
INTRODUCTION LET s1 be
a bounded open subset of R”, n 1 2. Consider integral functionals of the type
mwx)l) dx
Z(u: i-2) = j
where u:a+RN, satisfy
Nr
(0.1)
61
1, is vector valued and F: [0, w) -+ [O,w) is continuous. atP - b (c F(t) I ctq + d
vtzo,
Let F also (0.2)
where a, b, c and d are positive constants. The regularity of minimizers of (0.1) in the case has been studied in great detail [2]. In recent years, there has been some attention focused on the case of p < 4; this is often referred to in the literature as a “nonstandard growth condition” [3, 81. In this work we present a result in the regularity theory of minimizers in this framework. We must point out that our result relies strongly on a new version of a PoincarC inequality on annular domains. This inequality may have some independent interest in itself. We have organized the work as follows: the statement of the results appear in Section 1; in Section 2 we prove the PoincarC inequaIity, and the proof of the regularity theorem appears in Section 3.
p = q
1. MAIN
RESULTS
Let x0 E R” and Q&O, R) = {x E R”: /Xi - x:1 < R/2, i = 1,2, . . . , n) the cube in R” of edge length R and centered at x0. We will often suppress x0 and write QR instead of Q&O, R) whenever there is no danger of confusion. By a minimizer of (0.1) we mean a function U: Q -+ RN, N 2 1, u E W’*‘(Q) with F(tDul) E J%V
i(u: Q) 5 Z(u + 56:Q)
and
for every cube Q contained in !A and every 4: IR -+ RN with 9 E W:*‘(Q). 833
(1.1)
T. BHATTACHARYA and F. LEONETTI
834
We study (0.1) for functions F that satisfy F: [0, co) + [0, a),
continuous,
convex, F f 0 and F(0) = 0;
there exists a number (T > 0 such that F(2t) I aF(t) v t L 0.
(1.2a) (1.2b)
The condition (1.2b) is referred to in the literature as the AZ condition [5]; it is not difficult to see that the aforementioned conditions imply that F is nondecreasing, o 1 2, v t, s 2 0,
F(t + s) 5 ; (F(t) + F(s))
(1.3)
and F(l)(t - 1) I F(t) 5 0F(l)(t’og”“og2
+ 1)
vtzo.
(1.4)
It is clear such an F satisfies nonstandard growth in (0.2) with 1 = p 5 q = log a/log 2. Some examples of F included in this class are (a) F(t) = tP, p L 1,
(b)
F(t)
=
tP+'
v t E [O, 1)
tP(log t + 1)
v t E [l,a),p
z 1;
and for p > 1, 0 < E I (p - 1)/5, take
(c>
F(t)
=
eP-‘t
v t E [0, e)
fp+esinloglogl
v t E [e, CD).
The last example may be found in [l, lo]. From here on we reserve G to stand for the constant in (1.2b). We recall that a function h belongs to the Morrey space L’,‘(a) if there exists k > 0 such that ]/r(x)\ dx I kRx s DnQR for every QR centered in 0, with 0 < R I 2 diam a. We now state the main results of this paper. THEOREM
1. (Regularity of minimizers.) Let fi be a bounded open subset of Rn, n 2 2, and F satisfy (1.2a) and (1.2b). Let U: n + RN, N 2 1, be a minimizer of (0. l), then (1.5)
F(\Du]) E L:;,x@); furthermore,
the following estimate holds h F((Du(x)()cLx
1QS
I C s (>i ’
mwx)O
dx
(1.6)
QI
for every pair of parallel concentric cubes Q, c Q, c i2 with s < r, and where C, a are positive constants depending only on n, N and o. The proof of this result employs the “hole filling technique” [ 11,2, p. 1631; in order to apply such a device we need a version of the Poincare inequality holding in the annular region where QR c !A.Writing uB = #u(x) dx = l//B1 Je U(X)dx, we have theorem 2. A = QR\&,,,
Poincarb inequality and nonstandard
835
growth
(Poincare inequality.) Let x0 E R”, n 2 2, A the annular region Q(x”, R)\Q(x’, R/3), and F be a function satisfying (1.2a) and (1.2b). Let U: Sz -+ RN, N 2 1, u E W’~‘(Q(x”, R)) with F(IDul) E L’(Q(x’, R)), then
THEOREM
2.
(1.7) where C is a positive constant depending only on n, N and 0. We furnish a proof in Section 2. It must be mentioned that a similar inequality for n = 2 was proven in [6] by using a different technique. Thus the generalization obtained here extends the regularity result in [6] to R”, n L 2. We also remark here that the higher integrability results in [l] and [9] are deduced under restrictions on the orders of nonstandard growth p and q, namely p I q < np/(n - p). To obtain our result we place no restrictions on p and q. Let us point out that the convexity of F plays an important role in our analysis. 2. A POINCARI?
INEQUALITY
Before proving theorem 2 we need the following. LEMMA 1. Let Sz be a convex bounded open subset of R”, n 2 1; F: [O,oo) + [O,a) be continuous and convex with F(0) = 0. Let U: a -+ RN, N 2 1, be in IV’*‘(n) such that F(IDul) E L’(a). Then
3’.F(‘(~),
“fl’) dx I ($$)‘-“‘1,
F(lDu(x)I)
dw,
(2.1)
where d is the diameter of fi and o,, is the volume of the unit ball in R”. Proof. The proof is a generalization of the one given in [4] for the case F(t) = tP and we give a brief sketch below.
Step 1. We first prove (2.1) for u E C’(0). We argue as in lemma 7.16 [4], using that F is nondecreasing, convex and F(0) = 0, to obtain
for every x E a. Now using the theory of Riesz potential [4, lemma 7.121 we deduce (2.1). Step 2. For u E W’*‘(sZ) with F(IDul) E L’(Sl) we consider the usual mollification since F is nondecreasing and convex, F(I%(x)l)
5 IF(IDul)],(x).
This inequality permits us to conclude that F(]%])
+ F(]Dul)
in L:,,(a),
as E + 0.
u,. Again,
836
T. BHATTACHARYA and F. LEONETTI
Step 3. Now consider
the family
(C&J of subset of fi where &2,,= (X E a: dist(x, afi) > h),
and h is small enough; thus as h -+ 0, S&, -+ Sz. Furthermore, each &2,,is convex. For every fixed h and E, we have (2.1) for u, and C& . Letting first E -+ 0 and then h + 0, we obtain (2.1). n We state a simple lemma LEMMA
2.
regarding
averages.
Let S c T, then
(2.2) Proof. Now lu(x) - !.d,ldxs
lu(x) - !+ldx+
T
Ius-
s T
=
iT The last term on the right-hand
u,ldx
s T
It.&) - t.+I dx + lTllus - UTl.
side is estimated
as follows
n
Thus (2.2) follows.
The basic idea in proving theorem 2 is the following. We subdivide the annular region A into equal cubes of edge length a third of that of the enclosing cube. Any two cubes in A can be joined by a connected chain consiting of cubes taken from the subdivision, such that the union of any two contiguous cubes is a convex set. As a consequence, we may apply lemma 1.
Proof of theorem 2. Subdivide excluding
Q(%, R/3),
Q(x”, R) into 3” equal parallel
subcubes
Sk of edge length R/3;
we write 3n-1 A
=
Q(x’,
R)\Q(x’,
R/3) = u k=l
Noting
xklSkl/lAI
By monotonicity
= 1, it
fdOWS
and convexity
of F,
Sk.
(2.3)
PoincarC
uk = I&l/ IA 1 and integrating
Setting
Using lemma
inequality
and nonstandard
growth
837
over A, we obtain
1 for a = S,, monotonicity
and the A2 property,
where C, depends only on n, N and o. We now estimate the last integral on the right-hand side as follows. Observe that every pair of cubes Sj and S, may be joined by a connected chain of such cubes contained in A so that any two contiguous cubes of the chain have an (n - 1) hyperface in common. Such a requirement renders the union of two such continuous cubes a convex set. Let us remark that we can choose chains of this type consisting of no more than 6n cubes. For the pair Sj and Sk let Qr , Q2, . . . , Q, be an enumeration of the cubes in such a chain so that Sj = Qr , S, = Q,, Qi and Qi+l are contiguous in the sense described above and r I 6n. Setting Ui = uQ,, i = 1, 2, . . . , r, we have r-1
l”Cx)- url 5 l”Cx)- ull + 1 l”i - Ui+ll; i=l
applying
(1.3) r times,
where C, = (~r/2)~~-l.
Again
by lemma
we have
Thus
1,
(2.6) Foreveryi=
1,2,...,r-
I
(2.7) Now
838
setting
T. BHATTACHARYA and F. LEONETTI
Ei = LI~,~~,+, and using lemma
2, we have
1+
IQ;,"";"/
[u(x) - lil dx QiUQ,+l
I+1
56
lu(x) f Q,UQ,+I
noting
4 d-c
/Qi U Qi+ll = 2lQi I. Thus from (2.7)
I,,F(lU’ +;;+I’) dx
5
c3
l”;l+ll I’Q,uQL+, F(l”yy“‘>dx
,Qi
where we have used (1.2a), (1.2b) and IQr 1 = IQi 1, and C, depends Applying lemma 1 on the convex set Qi U Qi+l
only on n, N and 0.
where C, = C&n, N, a). Using (2.6) and (2.8) in (2.3,
The statement
of the theorem
now follows
from (2.4).
n
Remark. Theorem 2 continues to hold for any function G such that there is a function F satisfying (1.2a, b) with the ratio G/F bounded above and below by positive constants. This observation may be useful in studying nonconvex minimization problems. 3. PROOF
OF
THEOREM
1
Let Q(x’, r) be a cube in a and r: R” + R belong to C’,‘(a), Q(x’, r/3), q = 0 outside Q(x’, r) and l&l I c/r, where 4 = -V(U - <) where r = fQ,\Q,,, U. Then
0 5 q 5 1 such that q = 1 on C depends only on n. Set
n
F(bWI)
.i Q,
dx 5
0F(ID(u + d~)(x)I)
dx
Q,
522 s-
F((1 - rl(x))lWx)l)
+ F(IMx)l
I+)
- (1)
a 2
F(kWx)l)
do +
i,,,,,,,
+
‘“‘“‘,-
“)
q
9
Poincart
where we have used the properties integral
inequality
and nonstandard
of q, (1.2a)
F(IDu(x)))
Applying
NMx)l)
i.e. add (k - 1) SQr,SF(IDu(x)l)
2 on the last
d-C
dx to both sides resulting ~(D4x)l)
i Qr/3 Iterating
theorem
s Qr\Qr//,
Q,
now we “fill the hole”,
and (1.2b).
d.x I e
839
growth
in
d.x.
Q,
m times,
dx I
~mwl)
If s E (0, r], there is an integer
m such that r/3”+’
< s 5 r/3”’ and hence
where ,l = log(&(&
3. This estimate
F(IDu(x)()
d-.X.
Qr/3m
- l))/log
implies
F((Dul)
E L:;:(Q).
n
Acknowledgements-The
authors thank the referee for useful suggestions and remarks. This work was written while the second author was visiting Northwestern University on a C.N.R. grant. He wishes to thank the Department of Mathematics for its hospitality.
REFERENCES 1. Fusco N. & SBORDONE C., Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions, Communspure uppl. Math. XLIII, 673-683 (1990). 2. GIAQUINTA M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton (1983). 3. GIAQUINTA M., Growth conditions and regularity, a counterexample, Manuscripta math. 59, 245-248 (1987). 4. GILBARG D. & TRUDINGER N. S., Elliptic Partial Differential Equations of Second Order, 2nd edition. Springer, New York (1983). 5. KRASNOSEL’KIIM. A. & RUTICKII Y. B., Convex Functions and Orlicz Spaces. Noordhoff, Groningen (1961). 6. LEONETTI F., Two-dimensional regularity of minima of variational functionals without standard growth conditions, Rc. Mat. 38, 41-50 (1989). 1. LIEBERMAN G. M., The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Communs partial diff. Eqns (to appear). 8. MARCELLINI P., Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Archs ration. Mech. Analysis 105, 267-284 (1989). 9. SBORDONE C., On some inequalities and their applications to the calculus of variations, Boll. (In. mat. ital. An.
Fumz. Appl. 5, 73-94 (1986). 10. TALENTI G., Boundedness of minimizers, Hokkaido math. J. 19, 259-279 (1990). 11. WIDMAN K. O., Holder continuity of solutions of elliptic systems, Manuscripta math. 5, 299-308 (1971).
0
0362-546X/91 WOO+ .OO 1991 Pergamon Press plc
A NEW POINCARii INEQUALITY AND ITS APPLICATION TO THE REGULARITY OF MINIMIZERS OF INTEGRAL FUNCTIONALS WITH NONSTANDARD GROWTH TEAK BHATTA~H~YA Department of Mathematics, Northwestern University, Evanston, IL 60208, U.S.A.
and FRANCESCOLEONETTI Dipartimento di Matematica Pura ed Applicata, Universit& degli Studi di L’Aquila, 67100 L’Aquila, Italy (Received 30 July 1990; received in revised form 23 October 1990; received for publication 20 February Key words and phrases:
Poincare inequality, integral functionals, minimizers, Morrey-regularity.
nonstandard
1991)
growth conditions,
INTRODUCTION LET s1 be
a bounded open subset of R”, n 1 2. Consider integral functionals of the type
mwx)l) dx
Z(u: i-2) = j
where u:a+RN, satisfy
Nr
(0.1)
61
1, is vector valued and F: [0, w) -+ [O,w) is continuous. atP - b (c F(t) I ctq + d
vtzo,
Let F also (0.2)
where a, b, c and d are positive constants. The regularity of minimizers of (0.1) in the case has been studied in great detail [2]. In recent years, there has been some attention focused on the case of p < 4; this is often referred to in the literature as a “nonstandard growth condition” [3, 81. In this work we present a result in the regularity theory of minimizers in this framework. We must point out that our result relies strongly on a new version of a PoincarC inequality on annular domains. This inequality may have some independent interest in itself. We have organized the work as follows: the statement of the results appear in Section 1; in Section 2 we prove the PoincarC inequaIity, and the proof of the regularity theorem appears in Section 3.
p = q
1. MAIN
RESULTS
Let x0 E R” and Q&O, R) = {x E R”: /Xi - x:1 < R/2, i = 1,2, . . . , n) the cube in R” of edge length R and centered at x0. We will often suppress x0 and write QR instead of Q&O, R) whenever there is no danger of confusion. By a minimizer of (0.1) we mean a function U: Q -+ RN, N 2 1, u E W’*‘(Q) with F(tDul) E J%V
i(u: Q) 5 Z(u + 56:Q)
and
for every cube Q contained in !A and every 4: IR -+ RN with 9 E W:*‘(Q). 833
(1.1)
T. BHATTACHARYA and F. LEONETTI
834
We study (0.1) for functions F that satisfy F: [0, co) + [0, a),
continuous,
convex, F f 0 and F(0) = 0;
there exists a number (T > 0 such that F(2t) I aF(t) v t L 0.
(1.2a) (1.2b)
The condition (1.2b) is referred to in the literature as the AZ condition [5]; it is not difficult to see that the aforementioned conditions imply that F is nondecreasing, o 1 2, v t, s 2 0,
F(t + s) 5 ; (F(t) + F(s))
(1.3)
and F(l)(t - 1) I F(t) 5 0F(l)(t’og”“og2
+ 1)
vtzo.
(1.4)
It is clear such an F satisfies nonstandard growth in (0.2) with 1 = p 5 q = log a/log 2. Some examples of F included in this class are (a) F(t) = tP, p L 1,
(b)
F(t)
=
tP+'
v t E [O, 1)
tP(log t + 1)
v t E [l,a),p
z 1;
and for p > 1, 0 < E I (p - 1)/5, take
(c>
F(t)
=
eP-‘t
v t E [0, e)
fp+esinloglogl
v t E [e, CD).
The last example may be found in [l, lo]. From here on we reserve G to stand for the constant in (1.2b). We recall that a function h belongs to the Morrey space L’,‘(a) if there exists k > 0 such that ]/r(x)\ dx I kRx s DnQR for every QR centered in 0, with 0 < R I 2 diam a. We now state the main results of this paper. THEOREM
1. (Regularity of minimizers.) Let fi be a bounded open subset of Rn, n 2 2, and F satisfy (1.2a) and (1.2b). Let U: n + RN, N 2 1, be a minimizer of (0. l), then (1.5)
F(\Du]) E L:;,x@); furthermore,
the following estimate holds h F((Du(x)()cLx
1QS
I C s (>i ’
mwx)O
dx
(1.6)
QI
for every pair of parallel concentric cubes Q, c Q, c i2 with s < r, and where C, a are positive constants depending only on n, N and o. The proof of this result employs the “hole filling technique” [ 11,2, p. 1631; in order to apply such a device we need a version of the Poincare inequality holding in the annular region where QR c !A.Writing uB = #u(x) dx = l//B1 Je U(X)dx, we have theorem 2. A = QR\&,,,
Poincarb inequality and nonstandard
835
growth
(Poincare inequality.) Let x0 E R”, n 2 2, A the annular region Q(x”, R)\Q(x’, R/3), and F be a function satisfying (1.2a) and (1.2b). Let U: Sz -+ RN, N 2 1, u E W’~‘(Q(x”, R)) with F(IDul) E L’(Q(x’, R)), then
THEOREM
2.
(1.7) where C is a positive constant depending only on n, N and 0. We furnish a proof in Section 2. It must be mentioned that a similar inequality for n = 2 was proven in [6] by using a different technique. Thus the generalization obtained here extends the regularity result in [6] to R”, n L 2. We also remark here that the higher integrability results in [l] and [9] are deduced under restrictions on the orders of nonstandard growth p and q, namely p I q < np/(n - p). To obtain our result we place no restrictions on p and q. Let us point out that the convexity of F plays an important role in our analysis. 2. A POINCARI?
INEQUALITY
Before proving theorem 2 we need the following. LEMMA 1. Let Sz be a convex bounded open subset of R”, n 2 1; F: [O,oo) + [O,a) be continuous and convex with F(0) = 0. Let U: a -+ RN, N 2 1, be in IV’*‘(n) such that F(IDul) E L’(a). Then
3’.F(‘(~),
“fl’) dx I ($$)‘-“‘1,
F(lDu(x)I)
dw,
(2.1)
where d is the diameter of fi and o,, is the volume of the unit ball in R”. Proof. The proof is a generalization of the one given in [4] for the case F(t) = tP and we give a brief sketch below.
Step 1. We first prove (2.1) for u E C’(0). We argue as in lemma 7.16 [4], using that F is nondecreasing, convex and F(0) = 0, to obtain
for every x E a. Now using the theory of Riesz potential [4, lemma 7.121 we deduce (2.1). Step 2. For u E W’*‘(sZ) with F(IDul) E L’(Sl) we consider the usual mollification since F is nondecreasing and convex, F(I%(x)l)
5 IF(IDul)],(x).
This inequality permits us to conclude that F(]%])
+ F(]Dul)
in L:,,(a),
as E + 0.
u,. Again,
836
T. BHATTACHARYA and F. LEONETTI
Step 3. Now consider
the family
(C&J of subset of fi where &2,,= (X E a: dist(x, afi) > h),
and h is small enough; thus as h -+ 0, S&, -+ Sz. Furthermore, each &2,,is convex. For every fixed h and E, we have (2.1) for u, and C& . Letting first E -+ 0 and then h + 0, we obtain (2.1). n We state a simple lemma LEMMA
2.
regarding
averages.
Let S c T, then
(2.2) Proof. Now lu(x) - !.d,ldxs
lu(x) - !+ldx+
T
Ius-
s T
=
iT The last term on the right-hand
u,ldx
s T
It.&) - t.+I dx + lTllus - UTl.
side is estimated
as follows
n
Thus (2.2) follows.
The basic idea in proving theorem 2 is the following. We subdivide the annular region A into equal cubes of edge length a third of that of the enclosing cube. Any two cubes in A can be joined by a connected chain consiting of cubes taken from the subdivision, such that the union of any two contiguous cubes is a convex set. As a consequence, we may apply lemma 1.
Proof of theorem 2. Subdivide excluding
Q(%, R/3),
Q(x”, R) into 3” equal parallel
subcubes
Sk of edge length R/3;
we write 3n-1 A
=
Q(x’,
R)\Q(x’,
R/3) = u k=l
Noting
xklSkl/lAI
By monotonicity
= 1, it
fdOWS
and convexity
of F,
Sk.
(2.3)
PoincarC
uk = I&l/ IA 1 and integrating
Setting
Using lemma
inequality
and nonstandard
growth
837
over A, we obtain
1 for a = S,, monotonicity
and the A2 property,
where C, depends only on n, N and o. We now estimate the last integral on the right-hand side as follows. Observe that every pair of cubes Sj and S, may be joined by a connected chain of such cubes contained in A so that any two contiguous cubes of the chain have an (n - 1) hyperface in common. Such a requirement renders the union of two such continuous cubes a convex set. Let us remark that we can choose chains of this type consisting of no more than 6n cubes. For the pair Sj and Sk let Qr , Q2, . . . , Q, be an enumeration of the cubes in such a chain so that Sj = Qr , S, = Q,, Qi and Qi+l are contiguous in the sense described above and r I 6n. Setting Ui = uQ,, i = 1, 2, . . . , r, we have r-1
l”Cx)- url 5 l”Cx)- ull + 1 l”i - Ui+ll; i=l
applying
(1.3) r times,
where C, = (~r/2)~~-l.
Again
by lemma
we have
Thus
1,
(2.6) Foreveryi=
1,2,...,r-
I
(2.7) Now
838
setting
T. BHATTACHARYA and F. LEONETTI
Ei = LI~,~~,+, and using lemma
2, we have
1+
IQ;,"";"/
[u(x) - lil dx QiUQ,+l
I+1
56
lu(x) f Q,UQ,+I
noting
4 d-c
/Qi U Qi+ll = 2lQi I. Thus from (2.7)
I,,F(lU’ +;;+I’) dx
5
c3
l”;l+ll I’Q,uQL+, F(l”yy“‘>dx
,Qi
where we have used (1.2a), (1.2b) and IQr 1 = IQi 1, and C, depends Applying lemma 1 on the convex set Qi U Qi+l
only on n, N and 0.
where C, = C&n, N, a). Using (2.6) and (2.8) in (2.3,
The statement
of the theorem
now follows
from (2.4).
n
Remark. Theorem 2 continues to hold for any function G such that there is a function F satisfying (1.2a, b) with the ratio G/F bounded above and below by positive constants. This observation may be useful in studying nonconvex minimization problems. 3. PROOF
OF
THEOREM
1
Let Q(x’, r) be a cube in a and r: R” + R belong to C’,‘(a), Q(x’, r/3), q = 0 outside Q(x’, r) and l&l I c/r, where 4 = -V(U - <) where r = fQ,\Q,,, U. Then
0 5 q 5 1 such that q = 1 on C depends only on n. Set
n
F(bWI)
.i Q,
dx 5
0F(ID(u + d~)(x)I)
dx
Q,
522 s-
F((1 - rl(x))lWx)l)
+ F(IMx)l
I+)
- (1)
a 2
F(kWx)l)
do +
i,,,,,,,
+
‘“‘“‘,-
“)
q
9
Poincart
where we have used the properties integral
inequality
and nonstandard
of q, (1.2a)
F(IDu(x)))
Applying
NMx)l)
i.e. add (k - 1) SQr,SF(IDu(x)l)
2 on the last
d-C
dx to both sides resulting ~(D4x)l)
i Qr/3 Iterating
theorem
s Qr\Qr//,
Q,
now we “fill the hole”,
and (1.2b).
d.x I e
839
growth
in
d.x.
Q,
m times,
dx I
~mwl)
If s E (0, r], there is an integer
m such that r/3”+’
< s 5 r/3”’ and hence
where ,l = log(&(&
3. This estimate
F(IDu(x)()
d-.X.
Qr/3m
- l))/log
implies
F((Dul)
E L:;:(Q).
n
Acknowledgements-The
authors thank the referee for useful suggestions and remarks. This work was written while the second author was visiting Northwestern University on a C.N.R. grant. He wishes to thank the Department of Mathematics for its hospitality.
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