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Ar(λ)-Weighted Integral Inequalities for A-Harmonic Tensors
Journal of Mathematical Analysis and Applications 247, 466᎐477 Ž2000. doi:10.1006rjmaa.2000.6851, available online at http:rrwww.idealibrary.com on
A r Ž . -Weighted Integral Inequalities for A-Harmonic Tensors Gejun Bao Department of Mathematics, Harbin Institute of Technology, Harbin, People’s Republic of China Submitted by William F. Ames Received August 17, 1999 In this paper we prove the A r Ž .-weighted Caccioppoli-type inequality and weak reverse Holder inequality for A-harmonic tensors. We also obtain the A r Ž .¨ weighted Hardy᎐Littlewood inequality for conjugate A-harmonic tensors. These inequalities can be considered as extensions of the classical results. 䊚 2000 Academic Press
1. INTRODUCTION A-harmonic tensors are important extensions of p-harmonic tensors, p-harmonic functions, and harmonic functions, which have various applications in many fields, such as potential theory, quasiregular mappings, and the theory of elasticity. Many interesting results about A-harmonic tensors have been established recently; see w1᎐4, 6, 7, 9x. S. Ding and P. Shi introduce A r Ž ., a new class of weighted functions and obtain some basic properties of this class in w4x. In this paper we first obtain the A r Ž .weighted Caccioppoli-type estimate and the A r Ž .-weighted weak reverse Holder inequality for A-harmonic tensors. Then, we prove the A r Ž .¨ weighted Hardy᎐Littlewood inequality for conjugate A-harmonic tensors. These results can be used to study the integrability of A-harmonic tensors and estimate the integrals for A-harmonic tensors. Let ⍀ be a connected open subset of R n throughout this paper. Let e1 , e2 , . . . , e n be the standard unit basis of R n. For l s 0, 1, . . . , n, the linear space of l-vectors, spanned by the exterior products e I s e i1 n e i 2 n ⭈⭈⭈ e i l , corresponding to all ordered l-tuples I s Ž i1 , i 2 , . . . , i l ., 1 F i1 i 2 - ⭈⭈⭈ i l F n, is denoted by nl s nl ŽR n .. The Grassman algebra ns [n l is a graded algebra with respect to the exterior products. For ␣ s Ý ␣ I e I g n and  s Ý I e I g n, the inner product in n is given by 466 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
Ar Ž .-WEIGHTED
467
INTEGRAL INEQUALITIES
² ␣ ,  : s Ý ␣ I I with summation over all l-tuples I s Ž i1 , i 2 , . . . , i l . and all integers l s 0, 1, . . . , n. We define the Hodge star operator 夹: nª n by the rule 夹1 s e1 n e2 n ⭈⭈⭈ n e n and ␣ n 夹 s  n 夹␣ s ² ␣ ,  :Ž夹1. for all ␣ ,  g n. The norm of ␣ g n is given by the formula < ␣ < 2 s ² ␣ , ␣ : s 夹Ž ␣ n 夹␣ . g n 0 s R. The Hodge star is an isometric isomorphism on n with 夹: nl ª n ny l and 夹夹Žy1. lŽ nyl . : n l ª n l. Let 0 - p - ⬁; we denote the weighted L p-norm of a measurable function f over E by 5 f 5 p, E, s
1rp
p
žH
f Ž x . w Ž x . dx
E
/
.
A differential l-form on ⍀ is a de Rham current Žsee w7, Chap. IIIx. on ⍀ with values in n l ŽR n .. We denote the space of differential l-forms by D⬘Ž ⍀, n l .. We write L p Ž ⍀, n l . for the l-forms Ž x . s Ý I I Ž x . dx I s Ý i1 i 2 ⭈⭈⭈ i lŽ x . dx i1 n dx i 2 n ⭈⭈⭈ n dx i l with I g L p Ž ⍀, R. for all ordered l-tuples I. Thus L p Ž ⍀, n l . is a Banach space with norm 5 5 p, ⍀ s
žH
⍀
Ž x.
p
1rp
dx
/
s
žH ž Ý ⍀
I Ž x .
I
2
1rp
pr2
/
dx
/
.
Similarly, Wp1 Ž ⍀, n l . are those differential l-forms on ⍀ whose coefficients are in Wp1 Ž ⍀, R.. The notations Wp,1 loc Ž ⍀, R. and Wp,1 loc Ž ⍀, n l . are self-explanatory. We denote the exterior derivative by d : D⬘Ž ⍀, n l . ª D⬘Ž ⍀ , n lq 1 . for l s 0, 1, . . . , n. Its formal adjoint operator d 夹 : DŽ ⍀, n lq1 . ª D⬘Ž ⍀, n l . is given by d夹 s Žy1. nlq1夹 d夹 on D⬘Ž ⍀, n lq1 ., l s 0, 1, . . . , n. In recent years there have been remarkable advances made in the study of the A-harmonic equation d 夹A Ž x, d . s 0
Ž 1.1.
for differential forms, where A : ⍀ = n l ŽR n . ª n l ŽR n . satisfies the conditions A Ž x, . F a < < py 1
and
² A Ž x, . , : G < < p
Ž 1.2.
for almost every x g ⍀ and all g n l ŽR n .. Here a ) 0 is a constant and 1 - p - ⬁ is a fixed exponent associated with Ž1.1.. A solution to Ž1.1. is an element of the Sobolev space Wp,1 loc Ž ⍀, n ly1 . such that
H⍀² AŽ x, d . , d : s 0 for all g Wp1 Ž ⍀, n ly1 . with compact support.
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GEJUN BAO
DEFINITION 1.3. We call u an A-harmonic tensor in ⍀ if u satisfies the A-harmonic equation Ž1.1. in ⍀. A differential l-form u g D⬘Ž ⍀, n l . is called a closed form if du s 0 in ⍀. A differential form u is called a p-harmonic tensor if d 夹 Ž < du < py 2 du . s 0
d 夹 u s 0,
and
where 1 - p - ⬁. The equation A Ž x, du . s d 夹 ¨
Ž 1.4.
is called the conjugate A-harmonic equation. For example, du s d 夹 ¨ is an analogue of a Cauchy᎐Riemann system in R n. Clearly, the A-harmonic equation is not affected by adding a closed form to u and coclosed form to ¨ . Therefore, any type of estimates between u and ¨ must be modulo such forms. Suppose that u is a solution to Ž1.1. in ⍀. Then, at least locally in a ball B, there exists a form ¨ g Wq1 Ž B, n lq1 ., 1rp q 1rq s 1, such that Ž1.4. holds. DEFINITION 1.5. When u and ¨ satisfy Ž1.4. in ⍀, and Ay1 exists in ⍀, we call u and ¨ conjugate A-harmonic tensors in ⍀. DEFINITION 1.6. We call u a p-harmonic function if u satisfies the p-harmonic equation div Ž ⵜu < ⵜu < py 2 . s 0 with p ) 1. Its conjugate in the plane is a q-harmonic function ¨ , py1 q qy1 s 1, which satisfies ⵜu < ⵜu < py 2 s
ž
⭸¨ ⭸y
,y
⭸¨ ⭸x
/
.
Note that if p s q s 2, we get the usual conjugate harmonic functions. We write R s R1. Balls are denoted by B and B is the ball with the same center as B and with diamŽ B . s diamŽ B .. The n-dimensional Lebesgue measure of a set E : R n is denoted by < E <. We call w a weight if w g L1loc ŽR n . and w ) 0 a.e. Also in general d s w dx where w is a weight. The following result appears in w6x. Let Q ; R n be a cube or a ball. To each y g Q there corresponds a linear operator K y : C⬁Ž Q, n l . ª C⬁Ž Q, n ly1 . defined by 1
Ž K y . Ž x ; 1 , . . . , l . s H t ly1 Ž tx q y y ty ; x y y, 1 , . . . , ly1 . dt 0
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INTEGRAL INEQUALITIES
and the decomposition
s dŽ K y . q K y Ž d . . We define another linear operator TQ C⬁Ž Q, n l . ª C⬁Ž Q, n ly1 . by averaging K y over all points y in Q TQ s
HQ Ž y . K
y
dy,
where g C0⬁Ž Q . is normalized by HQ Ž y . dy s 1. We define the l-form Q g D⬘Ž Q, n l . by
Q s < Q
HQ Ž y . dy, l s 0,
Q s d Ž TQ . , l s 1, 2, . . . , n,
and
for all g L p Ž Q, n l .,k 1 F p - ⬁. 2. THE A r Ž .-WEIGHTED CACCIOPPOLI-TYPE ESTIMATE DEFINITION 2.1. We say the weight w Ž x . satisfies the A r Ž . condition for r ) 1 and 1 - - ⬁, write w g A r Ž ., if w Ž x . ) 0 a.e., and sup B
ž
1 < B<
HBw
dx
1
1
/ž Hž / < B<
B
ry1
1r Ž ry1 .
w
dx
/
-⬁
Ž 2.2.
for any ball B ; R n. The above A r Ž .-weights are introduced in w4x. See w4x for the properties of A r Ž .-weights. We will also need the following generalized Holder’s inequality. ¨ LEMMA 2.3. Let 0 - ␣ - ⬁, 0 -  - ⬁, and sy1 s ␣y1 q y1 . If f and g are measurable functions on R n, then 5 fg 5 s, ⍀ F 5 f 5 ␣ , ⍀ ⭈ 5 g 5  , ⍀
Ž 2.4.
for any ⍀ g R . n
In w7x, C. A. Nolder obtains the following Caccioppoli-type estimate. THEOREM A. Let u be an A-harmonic tensor in ⍀ and let ) 1. Then there exists a constant C, independent of u, such that 5 du 5 s, B F C diam Ž B . y1 5 u y c 5 s, B for all balls or cubes B with B ; ⍀ and all closed forms c. Here 1 - s - ⬁.
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GEJUN BAO
The following weak reverse Holder inequality appears in w7x. ¨ THEOREM B. Let u be an A-harmonic tensor in ⍀, ) 1, and 0 - s, t - ⬁. Then there exists a constant C, independent of u, such that 5 u 5 s, B F C < B < Ž tys.r st 5 u 5 t , B for all balls or cubes B with B ; ⍀. We prove the following A r Ž .-weighted Caccioppoli-type estimate for A-harmonic tensors. THEOREM 2.5. Let u g D⬘Ž ⍀, n l ., l s 0, 1, . . . , n, be an A-harmonic tensor in a domain ⍀ ; R n and ) 1. Assume that 1 - s - ⬁ is a fixed exponent associated with the A-harmonic equation and w g A r Ž . for some r ) 1 and ) 1. Then there exists a constant C, independent of u, such that 1rs
žH
< du < s w 1r dx
B
/
F
1rs
C
2
žH
diam Ž B .
< u y c < s w 1r dx
B
/
Ž 2.6.
for all balls B with B ; ⍀ and all closed forms c. Proof. Choose t s s 2rŽ 2 y 1.; then 1 - s - t. Since 1rs s 1rt qŽ t y s .rst, by Holder’s inequality and Theorem A, we have ¨ 1rs
ž
HB< du< w s
1r
dx
/
s F
HB Ž < du< w
ž žH
1r s
1rs
s
. dx
/ Ž tys .rst
1rt
< du < t dx
B
F 5 du 5 t , B ⭈
/ ž
žH
y1
1r s str Ž tys .
.
dx
5 u y c5 t, B
w dx
w dx
B
s C1 diam Ž B .
HB Ž w
/
/ žH
B
1r 2 s
/
Ž 2.7.
for all balls B with B ; ⍀ and all closed forms c. Since c is a closed form and u is an A-harmonic tensor, then u y c is still an A-harmonic tensor. Taking m s 2 srŽ 2 q r y 1., we find that m - s - t. Applying Theorem B yields 5 u y c 5 t , B F C2 < B < Ž myt .r m t 5 u y c 5 m , 2 B s C2 < B < Ž myt .r m t 5 u y c 5 m , B ,
Ž 2.8.
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INTEGRAL INEQUALITIES
where s 2 . Substituting Ž2.8. in Ž2.7., we have 1rs
ž
HB< du< w s
1r
dx
F C3 diam Ž B .
/
y1
< B < Ž myt .r m t
= 5 u y c5 m, B
žH
w dx
B
1r 2 s
/
.
Ž 2.9.
Now 1rm s 1rs q Ž s y m.rsm, by Holder’s inequality again, we obtain ¨ 1rm
5 u y c5 m, B s s
F
< u y c < m dx
žH žH Ž žH B
B
B
/
< u y c < w 1r s wy1r 2
2
s
m
1rm
/ / žH ž / .
dx
1rs
< u y c < w 1r dx 2
s
1
dx
w
B
Ž ry1 .r 2 s
1r Ž ry1 .
/ Ž 2.10.
for all balls B with B ; ⍀ and all closed forms c. Combining Ž2.9. and Ž2.10., we obtain 1rs
žH
< du < s w 1r dx
B
/
F C3 diam Ž B .
y1
< B < Ž myt .r m t 1rs
s 1r s = 5 w 5 1r , B 51rw 5 1rŽ ry1., B 2
žH
B
< u y c < s w 1r dx 2
/
. Ž 2.11.
Since w g A r Ž ., then we have s 1r s 5 w 5 1r , B ⭈ 51rw 5 1rŽ ry1., B 2
2 ry1 1r s
F
ž žH / žH B
ž
w dx
s < B< r
ž
1 < B<
F C4 < B < r r s . 2
B
/ / /ž H ž /
Ž 1rw .
H Bw
dx
1r Ž ry1 .
dx
1
< B<
1
B
w
2 ry1 1r s
1r Ž ry1 .
dx
/ / Ž 2.12.
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GEJUN BAO
Substituting Ž2.12. in Ž2.11., we find that 1rs
žH
< du < s w 1r dx
B
F
/
1rs
C diam Ž B .
< u y c < s w 1r dx 2
žH
B
/
for all balls B with B ; ⍀ and all closed forms c. This ends the proof of Theorem 2.5. 3. THE A r Ž .-WEIGHTED WEAK REVERSE ¨ HOLDER INEQUALITY We prove the following A r Ž .-weighted weak reverse Holder inequality. ¨ THEOREM 3.1. Let u g D⬘Ž ⍀, n l ., l s 0, 1, . . . , n, be an A-harmonic tensor in a domain ⍀ ; R n, ) 1. Assume that 0 - s, t - ⬁ and w g A r Ž . for some r ) 1 and ) 1. Then there exists a constant C, independent of u, such that 1rs
žH
< u < s w r2 dx
B
1rt
F C < B < Ž tys.r st
/
žH
B
< u < t w t r2 s dx
/
,
Ž 3.2.
for all balls B with B ; ⍀. Note that Ž3.2. can be written as
ž
1rs
1
< u< w < B < HB s
r2
dx
FC
/
1rt
1
< u< w < B < H B
ž
t
t r2 s
dx
/
.
Ž 3.2. ⬘
The proof of Theorem 3.1 is similar to that of Theorem 2.5. For completion of the paper, we prove Theorem 3.1 as follows. Proof. Applying Holder’s inequality, we have ¨ 1rs
ž
HB< u < w s
r2
dx
/
s F
HB Ž < u < w
ž žH
r2 s
s
1rs
. dx
/
1r2 s
< u < 2 s dx
B
/ žH žH /
Ž w r2 s .
B
2s
1r2 s
dx
/
1r2 s
s 5 u 5 2 s, B
w dx
Ž 3.3.
B
for all balls B with B ; ⍀. Choosing m s 2 strŽ2 s q t Ž r y 1.., by Theorem B we obtain 5 u 5 2 s, B F C3 < B < Ž my2 s.r2 m s 5 u 5 m , B .
Ž 3.4.
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INTEGRAL INEQUALITIES
Since 1rm s 1rt q Ž t y m.rmt, by the Holder’s inequality again, we have ¨ 5 u5 m , B s F s
H B Ž < u < w
ž žH žH
1r2 s
wy1r2 s .
1rm
m
dx
/ Ž tym .rm t
1rt
B
< u < t w t r2 s dx
/ žH Ž / žH Ž B
1rw .
m tr2 s Ž tym .
1rw .
1r Ž ry1 .
B
B
/
Ž ry1 .r2 s
1rt
< u < t w t r2 s dx
dx
dx
/
.
Ž 3.5.
Combining Ž3.3., Ž3.4., and Ž3.5. yields 1rs
ž
HB< u < w s
r2
dx
/ 1r2 s
F C1 < B < Ž my2 s.r2 m s
ž
HBw
dx
/
Ž ry1 .r2 s
=
žH
B
Ž 1rw . 1r ry1 dx Ž
.
/
1r6
žH
B
< u < t w t r2 s dx
/
. Ž 3.6.
Since w g A r Ž ., then we have Ž ry1 .r2 s
1r2 s
žH
w dx
B
/ žH
B
F < B < r r2 s
žž
Ž 1rw . 1r ry1 dx Ž
1 < B<
.
w dx
HB
/ž
/ 1r2 s
1 < B<
H B
Ž 1rw . 1r ry1 dx Ž
.
//
F C2 < B < r r2 s s C3 < B < r r2 s .
Ž 3.7.
Substituting Ž3.7. into Ž3.6. yields 1rs
žH
< u < s w r2 dx
B
/
1rt
F C < B < Ž tys.r st
žH
B
We have completed the proof of Theorem 3.1.
< u < t w t r2 s dx
/
.
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GEJUN BAO
4. THE A r Ž .-WEIGHTED HARDY᎐LITTLEWOOD INEQUALITY Hardy and Littlewood in w5x proved the following result. THEOREM C.
For each p ) 0, there is a constant C such that
HD u y u Ž 0.
p
dx dy F C
HD ¨ y ¨ Ž 0.
p
dx dy
for all analytic functions f s u q i¨ in the unit disk D. C. A. Nolder proves the following Theorem in w7x. THEOREM D. Let u and ¨ be conjugate A-harmonic tensors in ⍀ ; R n, ) 1, and 0 - s, t - ⬁. Then there exists a constant C, independent of u and ¨ , such that 5 u y u B 5 s, B F C < B <  5 ¨ y c 5 tq,rpB for all balls B and B ; ⍀. Here c is any form in Wp,1 loc Ž ⍀, ⌳ . with d 夹 c s 0 and  s 1rs q 1rn y Ž1rt q 1rn. qrp. Now we prove the following A r Ž .-weighted Hardy᎐Littlewood inequality. THEOREM 4.1. Let u and ¨ be conjugate A-harmonic tensors in a domain ⍀ ; R n and w g A r Ž . for some r ) 1. Let 0 - s, t - ⬁. Then there exists a constant C, independent of u and ¨ , such that 1rs
žH
< u y u B < s w r p dx
B
/
F C < B<␥
qrp t
žH
B
< ¨ y c < t w t r q s dx
/
Ž 4.2.
for all balls B with B ; ⍀ ; R n and ) 1. Here c is any form in Wq,1 loc Ž ⍀, ⌳ . with d 夹 c s 0 and ␥ s 1rs q 1rn y Ž1rt q 1rn. qrp. Note that Ž4.2. can be written as the symmetric form
ž
1 < B<
1rq s
< u y u B < s w r p dx
HB
F C < B < Ž1r qy1r p.r n
ž
/ 1 < B<
1rp t
H B
< ¨ y c < t w t r q s dx
/
.
Ž 4.2. ⬘
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INTEGRAL INEQUALITIES
Proof. Let k s psrŽ p y 1.. Since p ) 1, then k ) 0 and k ) s. Applying Holder’s inequality, we have ¨ 1rs
žH
< u y u B < s w r p dx
B
s
/
žH
B
1rs
s
Ž < u y u B < w r p s . dx
F 5 u y uB 5 k , B s 5 u y uB 5 k , B
žH žH
/
w k r pŽ kys. dx
B
Ž kys .rk s
/
1rp s
w dx
B
/
.
Ž 4.3.
Choose m s qstrŽ qs q t Ž r y 1.., then m - t. By Theorem D we have qr p 5 u y u B 5 k , B F C1 < B <  5 ¨ y c 5 m ,B,
Ž 4.4.
where  s 1rk q 1rn y Ž1rm q 1rn . qrp. Since 1rm s 1rt q Ž t y m.rmt, by Holder’s inequality again, we obtain ¨ 5¨ y c5 m, B s F s
1rm
m
žH žH žH
B
Ž < ¨ y c < w 1r q s wy1r q s . dx
/ Ž tym .rm t
1rt
B
< ¨ y c < t w t r q s dx
/ ž / žH Ž
H B Ž 1rw .
m trq s Ž tym .
dx
Ž ry1 .rq s
1rt
B
< ¨ y c < t w t r q s dx
B
/
1rw .
1r Ž ry1 .
dx
/
.
Ž 4.5. Hence 5 ¨ y c 5 qmr, p B F
Ž ry1 .rp s
ž
H B Ž 1rw .
1r Ž ry1 .
dx
/
qrp t
ž
H B< ¨ y c < w t
tr qs
dx
/
.
Ž 4.6. Combining Ž4.3., Ž4.4., and Ž4.6. yields 1rs
ž
HB< u y u
B
< s w r p dx
F C1 < B <  =
ž
/ Ž ry1 .rp s
1rp s
žH
w dx
B
/ žH /
H B< ¨ y c < w t
Ž
B
tr qs
dx
Ž 1rw . 1r ry1 dx .
/
qrp t
.
Ž 4.7.
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GEJUN BAO
Using the condition that w g A r Ž ., we obtain Ž ry1 .rp s
1rp s
ž
HBw
dx
/ ž
H B Ž 1rw .
F < B < rr ps
žž
1 < B<
1r Ž ry1 .
HBw
dx
dx
/ž
/ 1rs p
1 < B<
H B Ž 1rw .
1r Ž ry1 .
dx
//
F C2 < B < r r p s s C3 < B < r r p s .
Ž 4.8.
Putting Ž4.8. into Ž4.7. and noting that  q rrps s 1rk q 1rn y Ž1rm q 1rn. qrp q rrps s 1rs q 1rn y Ž1rt q 1rn. qrp, we have 1rs
ž
HB< u y u
< s r p dx B w
/
F C < B<␥
qrp t
ž
H B< ¨ y c < w t
tr qs
dx
/
,
where ␥ s 1rs q 1rn y Ž1rt q 1rn. qrp. We have completed the proof of Theorem 4.1. As an application of Theorem 4.1, we have the following example. EXAMPLE. Let f Ž x . s Ž f 1 , f 2 , . . . , f n . be K-quasiregular in R n. Then u s f l df 1 n df 2 n ⭈⭈⭈ n df ly1 and ¨ s 夹 f lq1 df lq2 n ⭈⭈⭈ n df n ,
l s 1, 2, . . . , n y 1, are conjugate A-harmonic tensors with p s nrl and q s nrŽ n y l ., where A is some operator satisfying Ž1.2.. Then by Theorem 4.1, we obtain
žH
B
f l df 1 n df 2 n ⭈⭈⭈ n df ly1 s
y Ž f l df 1 n df 2 n ⭈⭈⭈ n df ly1 . B w r p dx F C < B<␥
1rs
/ qrp t
žH
B
<夹 f lq1 df lq2 n ⭈⭈⭈ n df n y c < t w t r q s dx
/
,
where C is independent of f, ␥ s 1rs q 1rn y Ž1rt q 1rn. qrp and d 夹 c s 0.
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INTEGRAL INEQUALITIES
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REFERENCES 1. J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 Ž1977., 337᎐403. 2. S. Ding, Some examples of conjugate p-harmonic differential forms, J. Math. Anal. Appl. 227 Ž1998., 251᎐270. 3. S. Ding, Weighted Hardy᎐Littlewood inequality for A-harmonic tensors, Proc. Amer. Math. Soc. 125 Ž1997., 1727᎐1735. 4. S. Ding and P. Shi, Weighted Poincare-type inequalities for differential forms in Ls Ž .´ averaging domains, J. Math. Anal. Appl. 227 Ž1998., 200᎐215. 5. G. H. Hardy and J. E. Littlewood, Some properties of conjugate functions, J Reine Angew. Math. 167 Ž1932., 405᎐423. 6. T. Iwaniec and A. Lutoborski, Integral estimates for null Lagrangians, Arch. Rational Mech. Anal. 125 Ž1993., 25᎐79. 7. C. A. Nolder, Hardy᎐Littlewood theorems for A-harmonic tensors, Illinois J. Math., in press. 8. G. De Rham, ‘‘Differential Manifolds,’’ Springer-Verlag, New YorkrBerlin, 1980. 9. B. Stroffolini, On weakly A-harmonic tensors, Studia Math. 3, No. 114 Ž1995., 289᎐301.
A r Ž . -Weighted Integral Inequalities for A-Harmonic Tensors Gejun Bao Department of Mathematics, Harbin Institute of Technology, Harbin, People’s Republic of China Submitted by William F. Ames Received August 17, 1999 In this paper we prove the A r Ž .-weighted Caccioppoli-type inequality and weak reverse Holder inequality for A-harmonic tensors. We also obtain the A r Ž .¨ weighted Hardy᎐Littlewood inequality for conjugate A-harmonic tensors. These inequalities can be considered as extensions of the classical results. 䊚 2000 Academic Press
1. INTRODUCTION A-harmonic tensors are important extensions of p-harmonic tensors, p-harmonic functions, and harmonic functions, which have various applications in many fields, such as potential theory, quasiregular mappings, and the theory of elasticity. Many interesting results about A-harmonic tensors have been established recently; see w1᎐4, 6, 7, 9x. S. Ding and P. Shi introduce A r Ž ., a new class of weighted functions and obtain some basic properties of this class in w4x. In this paper we first obtain the A r Ž .weighted Caccioppoli-type estimate and the A r Ž .-weighted weak reverse Holder inequality for A-harmonic tensors. Then, we prove the A r Ž .¨ weighted Hardy᎐Littlewood inequality for conjugate A-harmonic tensors. These results can be used to study the integrability of A-harmonic tensors and estimate the integrals for A-harmonic tensors. Let ⍀ be a connected open subset of R n throughout this paper. Let e1 , e2 , . . . , e n be the standard unit basis of R n. For l s 0, 1, . . . , n, the linear space of l-vectors, spanned by the exterior products e I s e i1 n e i 2 n ⭈⭈⭈ e i l , corresponding to all ordered l-tuples I s Ž i1 , i 2 , . . . , i l ., 1 F i1 i 2 - ⭈⭈⭈ i l F n, is denoted by nl s nl ŽR n .. The Grassman algebra ns [n l is a graded algebra with respect to the exterior products. For ␣ s Ý ␣ I e I g n and  s Ý I e I g n, the inner product in n is given by 466 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
Ar Ž .-WEIGHTED
467
INTEGRAL INEQUALITIES
² ␣ ,  : s Ý ␣ I I with summation over all l-tuples I s Ž i1 , i 2 , . . . , i l . and all integers l s 0, 1, . . . , n. We define the Hodge star operator 夹: nª n by the rule 夹1 s e1 n e2 n ⭈⭈⭈ n e n and ␣ n 夹 s  n 夹␣ s ² ␣ ,  :Ž夹1. for all ␣ ,  g n. The norm of ␣ g n is given by the formula < ␣ < 2 s ² ␣ , ␣ : s 夹Ž ␣ n 夹␣ . g n 0 s R. The Hodge star is an isometric isomorphism on n with 夹: nl ª n ny l and 夹夹Žy1. lŽ nyl . : n l ª n l. Let 0 - p - ⬁; we denote the weighted L p-norm of a measurable function f over E by 5 f 5 p, E, s
1rp
p
žH
f Ž x . w Ž x . dx
E
/
.
A differential l-form on ⍀ is a de Rham current Žsee w7, Chap. IIIx. on ⍀ with values in n l ŽR n .. We denote the space of differential l-forms by D⬘Ž ⍀, n l .. We write L p Ž ⍀, n l . for the l-forms Ž x . s Ý I I Ž x . dx I s Ý i1 i 2 ⭈⭈⭈ i lŽ x . dx i1 n dx i 2 n ⭈⭈⭈ n dx i l with I g L p Ž ⍀, R. for all ordered l-tuples I. Thus L p Ž ⍀, n l . is a Banach space with norm 5 5 p, ⍀ s
žH
⍀
Ž x.
p
1rp
dx
/
s
žH ž Ý ⍀
I Ž x .
I
2
1rp
pr2
/
dx
/
.
Similarly, Wp1 Ž ⍀, n l . are those differential l-forms on ⍀ whose coefficients are in Wp1 Ž ⍀, R.. The notations Wp,1 loc Ž ⍀, R. and Wp,1 loc Ž ⍀, n l . are self-explanatory. We denote the exterior derivative by d : D⬘Ž ⍀, n l . ª D⬘Ž ⍀ , n lq 1 . for l s 0, 1, . . . , n. Its formal adjoint operator d 夹 : DŽ ⍀, n lq1 . ª D⬘Ž ⍀, n l . is given by d夹 s Žy1. nlq1夹 d夹 on D⬘Ž ⍀, n lq1 ., l s 0, 1, . . . , n. In recent years there have been remarkable advances made in the study of the A-harmonic equation d 夹A Ž x, d . s 0
Ž 1.1.
for differential forms, where A : ⍀ = n l ŽR n . ª n l ŽR n . satisfies the conditions A Ž x, . F a < < py 1
and
² A Ž x, . , : G < < p
Ž 1.2.
for almost every x g ⍀ and all g n l ŽR n .. Here a ) 0 is a constant and 1 - p - ⬁ is a fixed exponent associated with Ž1.1.. A solution to Ž1.1. is an element of the Sobolev space Wp,1 loc Ž ⍀, n ly1 . such that
H⍀² AŽ x, d . , d : s 0 for all g Wp1 Ž ⍀, n ly1 . with compact support.
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GEJUN BAO
DEFINITION 1.3. We call u an A-harmonic tensor in ⍀ if u satisfies the A-harmonic equation Ž1.1. in ⍀. A differential l-form u g D⬘Ž ⍀, n l . is called a closed form if du s 0 in ⍀. A differential form u is called a p-harmonic tensor if d 夹 Ž < du < py 2 du . s 0
d 夹 u s 0,
and
where 1 - p - ⬁. The equation A Ž x, du . s d 夹 ¨
Ž 1.4.
is called the conjugate A-harmonic equation. For example, du s d 夹 ¨ is an analogue of a Cauchy᎐Riemann system in R n. Clearly, the A-harmonic equation is not affected by adding a closed form to u and coclosed form to ¨ . Therefore, any type of estimates between u and ¨ must be modulo such forms. Suppose that u is a solution to Ž1.1. in ⍀. Then, at least locally in a ball B, there exists a form ¨ g Wq1 Ž B, n lq1 ., 1rp q 1rq s 1, such that Ž1.4. holds. DEFINITION 1.5. When u and ¨ satisfy Ž1.4. in ⍀, and Ay1 exists in ⍀, we call u and ¨ conjugate A-harmonic tensors in ⍀. DEFINITION 1.6. We call u a p-harmonic function if u satisfies the p-harmonic equation div Ž ⵜu < ⵜu < py 2 . s 0 with p ) 1. Its conjugate in the plane is a q-harmonic function ¨ , py1 q qy1 s 1, which satisfies ⵜu < ⵜu < py 2 s
ž
⭸¨ ⭸y
,y
⭸¨ ⭸x
/
.
Note that if p s q s 2, we get the usual conjugate harmonic functions. We write R s R1. Balls are denoted by B and B is the ball with the same center as B and with diamŽ B . s diamŽ B .. The n-dimensional Lebesgue measure of a set E : R n is denoted by < E <. We call w a weight if w g L1loc ŽR n . and w ) 0 a.e. Also in general d s w dx where w is a weight. The following result appears in w6x. Let Q ; R n be a cube or a ball. To each y g Q there corresponds a linear operator K y : C⬁Ž Q, n l . ª C⬁Ž Q, n ly1 . defined by 1
Ž K y . Ž x ; 1 , . . . , l . s H t ly1 Ž tx q y y ty ; x y y, 1 , . . . , ly1 . dt 0
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469
INTEGRAL INEQUALITIES
and the decomposition
s dŽ K y . q K y Ž d . . We define another linear operator TQ C⬁Ž Q, n l . ª C⬁Ž Q, n ly1 . by averaging K y over all points y in Q TQ s
HQ Ž y . K
y
dy,
where g C0⬁Ž Q . is normalized by HQ Ž y . dy s 1. We define the l-form Q g D⬘Ž Q, n l . by
Q s < Q
HQ Ž y . dy, l s 0,
Q s d Ž TQ . , l s 1, 2, . . . , n,
and
for all g L p Ž Q, n l .,k 1 F p - ⬁. 2. THE A r Ž .-WEIGHTED CACCIOPPOLI-TYPE ESTIMATE DEFINITION 2.1. We say the weight w Ž x . satisfies the A r Ž . condition for r ) 1 and 1 - - ⬁, write w g A r Ž ., if w Ž x . ) 0 a.e., and sup B
ž
1 < B<
HBw
dx
1
1
/ž Hž / < B<
B
ry1
1r Ž ry1 .
w
dx
/
-⬁
Ž 2.2.
for any ball B ; R n. The above A r Ž .-weights are introduced in w4x. See w4x for the properties of A r Ž .-weights. We will also need the following generalized Holder’s inequality. ¨ LEMMA 2.3. Let 0 - ␣ - ⬁, 0 -  - ⬁, and sy1 s ␣y1 q y1 . If f and g are measurable functions on R n, then 5 fg 5 s, ⍀ F 5 f 5 ␣ , ⍀ ⭈ 5 g 5  , ⍀
Ž 2.4.
for any ⍀ g R . n
In w7x, C. A. Nolder obtains the following Caccioppoli-type estimate. THEOREM A. Let u be an A-harmonic tensor in ⍀ and let ) 1. Then there exists a constant C, independent of u, such that 5 du 5 s, B F C diam Ž B . y1 5 u y c 5 s, B for all balls or cubes B with B ; ⍀ and all closed forms c. Here 1 - s - ⬁.
470
GEJUN BAO
The following weak reverse Holder inequality appears in w7x. ¨ THEOREM B. Let u be an A-harmonic tensor in ⍀, ) 1, and 0 - s, t - ⬁. Then there exists a constant C, independent of u, such that 5 u 5 s, B F C < B < Ž tys.r st 5 u 5 t , B for all balls or cubes B with B ; ⍀. We prove the following A r Ž .-weighted Caccioppoli-type estimate for A-harmonic tensors. THEOREM 2.5. Let u g D⬘Ž ⍀, n l ., l s 0, 1, . . . , n, be an A-harmonic tensor in a domain ⍀ ; R n and ) 1. Assume that 1 - s - ⬁ is a fixed exponent associated with the A-harmonic equation and w g A r Ž . for some r ) 1 and ) 1. Then there exists a constant C, independent of u, such that 1rs
žH
< du < s w 1r dx
B
/
F
1rs
C
2
žH
diam Ž B .
< u y c < s w 1r dx
B
/
Ž 2.6.
for all balls B with B ; ⍀ and all closed forms c. Proof. Choose t s s 2rŽ 2 y 1.; then 1 - s - t. Since 1rs s 1rt qŽ t y s .rst, by Holder’s inequality and Theorem A, we have ¨ 1rs
ž
HB< du< w s
1r
dx
/
s F
HB Ž < du< w
ž žH
1r s
1rs
s
. dx
/ Ž tys .rst
1rt
< du < t dx
B
F 5 du 5 t , B ⭈
/ ž
žH
y1
1r s str Ž tys .
.
dx
5 u y c5 t, B
w dx
w dx
B
s C1 diam Ž B .
HB Ž w
/
/ žH
B
1r 2 s
/
Ž 2.7.
for all balls B with B ; ⍀ and all closed forms c. Since c is a closed form and u is an A-harmonic tensor, then u y c is still an A-harmonic tensor. Taking m s 2 srŽ 2 q r y 1., we find that m - s - t. Applying Theorem B yields 5 u y c 5 t , B F C2 < B < Ž myt .r m t 5 u y c 5 m , 2 B s C2 < B < Ž myt .r m t 5 u y c 5 m , B ,
Ž 2.8.
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471
INTEGRAL INEQUALITIES
where s 2 . Substituting Ž2.8. in Ž2.7., we have 1rs
ž
HB< du< w s
1r
dx
F C3 diam Ž B .
/
y1
< B < Ž myt .r m t
= 5 u y c5 m, B
žH
w dx
B
1r 2 s
/
.
Ž 2.9.
Now 1rm s 1rs q Ž s y m.rsm, by Holder’s inequality again, we obtain ¨ 1rm
5 u y c5 m, B s s
F
< u y c < m dx
žH žH Ž žH B
B
B
/
< u y c < w 1r s wy1r 2
2
s
m
1rm
/ / žH ž / .
dx
1rs
< u y c < w 1r dx 2
s
1
dx
w
B
Ž ry1 .r 2 s
1r Ž ry1 .
/ Ž 2.10.
for all balls B with B ; ⍀ and all closed forms c. Combining Ž2.9. and Ž2.10., we obtain 1rs
žH
< du < s w 1r dx
B
/
F C3 diam Ž B .
y1
< B < Ž myt .r m t 1rs
s 1r s = 5 w 5 1r , B 51rw 5 1rŽ ry1., B 2
žH
B
< u y c < s w 1r dx 2
/
. Ž 2.11.
Since w g A r Ž ., then we have s 1r s 5 w 5 1r , B ⭈ 51rw 5 1rŽ ry1., B 2
2 ry1 1r s
F
ž žH / žH B
ž
w dx
s < B< r
ž
1 < B<
F C4 < B < r r s . 2
B
/ / /ž H ž /
Ž 1rw .
H Bw
dx
1r Ž ry1 .
dx
1
< B<
1
B
w
2 ry1 1r s
1r Ž ry1 .
dx
/ / Ž 2.12.
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GEJUN BAO
Substituting Ž2.12. in Ž2.11., we find that 1rs
žH
< du < s w 1r dx
B
F
/
1rs
C diam Ž B .
< u y c < s w 1r dx 2
žH
B
/
for all balls B with B ; ⍀ and all closed forms c. This ends the proof of Theorem 2.5. 3. THE A r Ž .-WEIGHTED WEAK REVERSE ¨ HOLDER INEQUALITY We prove the following A r Ž .-weighted weak reverse Holder inequality. ¨ THEOREM 3.1. Let u g D⬘Ž ⍀, n l ., l s 0, 1, . . . , n, be an A-harmonic tensor in a domain ⍀ ; R n, ) 1. Assume that 0 - s, t - ⬁ and w g A r Ž . for some r ) 1 and ) 1. Then there exists a constant C, independent of u, such that 1rs
žH
< u < s w r2 dx
B
1rt
F C < B < Ž tys.r st
/
žH
B
< u < t w t r2 s dx
/
,
Ž 3.2.
for all balls B with B ; ⍀. Note that Ž3.2. can be written as
ž
1rs
1
< u< w < B < HB s
r2
dx
FC
/
1rt
1
< u< w < B < H B
ž
t
t r2 s
dx
/
.
Ž 3.2. ⬘
The proof of Theorem 3.1 is similar to that of Theorem 2.5. For completion of the paper, we prove Theorem 3.1 as follows. Proof. Applying Holder’s inequality, we have ¨ 1rs
ž
HB< u < w s
r2
dx
/
s F
HB Ž < u < w
ž žH
r2 s
s
1rs
. dx
/
1r2 s
< u < 2 s dx
B
/ žH žH /
Ž w r2 s .
B
2s
1r2 s
dx
/
1r2 s
s 5 u 5 2 s, B
w dx
Ž 3.3.
B
for all balls B with B ; ⍀. Choosing m s 2 strŽ2 s q t Ž r y 1.., by Theorem B we obtain 5 u 5 2 s, B F C3 < B < Ž my2 s.r2 m s 5 u 5 m , B .
Ž 3.4.
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473
INTEGRAL INEQUALITIES
Since 1rm s 1rt q Ž t y m.rmt, by the Holder’s inequality again, we have ¨ 5 u5 m , B s F s
H B Ž < u < w
ž žH žH
1r2 s
wy1r2 s .
1rm
m
dx
/ Ž tym .rm t
1rt
B
< u < t w t r2 s dx
/ žH Ž / žH Ž B
1rw .
m tr2 s Ž tym .
1rw .
1r Ž ry1 .
B
B
/
Ž ry1 .r2 s
1rt
< u < t w t r2 s dx
dx
dx
/
.
Ž 3.5.
Combining Ž3.3., Ž3.4., and Ž3.5. yields 1rs
ž
HB< u < w s
r2
dx
/ 1r2 s
F C1 < B < Ž my2 s.r2 m s
ž
HBw
dx
/
Ž ry1 .r2 s
=
žH
B
Ž 1rw . 1r ry1 dx Ž
.
/
1r6
žH
B
< u < t w t r2 s dx
/
. Ž 3.6.
Since w g A r Ž ., then we have Ž ry1 .r2 s
1r2 s
žH
w dx
B
/ žH
B
F < B < r r2 s
žž
Ž 1rw . 1r ry1 dx Ž
1 < B<
.
w dx
HB
/ž
/ 1r2 s
1 < B<
H B
Ž 1rw . 1r ry1 dx Ž
.
//
F C2 < B < r r2 s s C3 < B < r r2 s .
Ž 3.7.
Substituting Ž3.7. into Ž3.6. yields 1rs
žH
< u < s w r2 dx
B
/
1rt
F C < B < Ž tys.r st
žH
B
We have completed the proof of Theorem 3.1.
< u < t w t r2 s dx
/
.
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GEJUN BAO
4. THE A r Ž .-WEIGHTED HARDY᎐LITTLEWOOD INEQUALITY Hardy and Littlewood in w5x proved the following result. THEOREM C.
For each p ) 0, there is a constant C such that
HD u y u Ž 0.
p
dx dy F C
HD ¨ y ¨ Ž 0.
p
dx dy
for all analytic functions f s u q i¨ in the unit disk D. C. A. Nolder proves the following Theorem in w7x. THEOREM D. Let u and ¨ be conjugate A-harmonic tensors in ⍀ ; R n, ) 1, and 0 - s, t - ⬁. Then there exists a constant C, independent of u and ¨ , such that 5 u y u B 5 s, B F C < B <  5 ¨ y c 5 tq,rpB for all balls B and B ; ⍀. Here c is any form in Wp,1 loc Ž ⍀, ⌳ . with d 夹 c s 0 and  s 1rs q 1rn y Ž1rt q 1rn. qrp. Now we prove the following A r Ž .-weighted Hardy᎐Littlewood inequality. THEOREM 4.1. Let u and ¨ be conjugate A-harmonic tensors in a domain ⍀ ; R n and w g A r Ž . for some r ) 1. Let 0 - s, t - ⬁. Then there exists a constant C, independent of u and ¨ , such that 1rs
žH
< u y u B < s w r p dx
B
/
F C < B<␥
qrp t
žH
B
< ¨ y c < t w t r q s dx
/
Ž 4.2.
for all balls B with B ; ⍀ ; R n and ) 1. Here c is any form in Wq,1 loc Ž ⍀, ⌳ . with d 夹 c s 0 and ␥ s 1rs q 1rn y Ž1rt q 1rn. qrp. Note that Ž4.2. can be written as the symmetric form
ž
1 < B<
1rq s
< u y u B < s w r p dx
HB
F C < B < Ž1r qy1r p.r n
ž
/ 1 < B<
1rp t
H B
< ¨ y c < t w t r q s dx
/
.
Ž 4.2. ⬘
Ar Ž .-WEIGHTED
475
INTEGRAL INEQUALITIES
Proof. Let k s psrŽ p y 1.. Since p ) 1, then k ) 0 and k ) s. Applying Holder’s inequality, we have ¨ 1rs
žH
< u y u B < s w r p dx
B
s
/
žH
B
1rs
s
Ž < u y u B < w r p s . dx
F 5 u y uB 5 k , B s 5 u y uB 5 k , B
žH žH
/
w k r pŽ kys. dx
B
Ž kys .rk s
/
1rp s
w dx
B
/
.
Ž 4.3.
Choose m s qstrŽ qs q t Ž r y 1.., then m - t. By Theorem D we have qr p 5 u y u B 5 k , B F C1 < B <  5 ¨ y c 5 m ,B,
Ž 4.4.
where  s 1rk q 1rn y Ž1rm q 1rn . qrp. Since 1rm s 1rt q Ž t y m.rmt, by Holder’s inequality again, we obtain ¨ 5¨ y c5 m, B s F s
1rm
m
žH žH žH
B
Ž < ¨ y c < w 1r q s wy1r q s . dx
/ Ž tym .rm t
1rt
B
< ¨ y c < t w t r q s dx
/ ž / žH Ž
H B Ž 1rw .
m trq s Ž tym .
dx
Ž ry1 .rq s
1rt
B
< ¨ y c < t w t r q s dx
B
/
1rw .
1r Ž ry1 .
dx
/
.
Ž 4.5. Hence 5 ¨ y c 5 qmr, p B F
Ž ry1 .rp s
ž
H B Ž 1rw .
1r Ž ry1 .
dx
/
qrp t
ž
H B< ¨ y c < w t
tr qs
dx
/
.
Ž 4.6. Combining Ž4.3., Ž4.4., and Ž4.6. yields 1rs
ž
HB< u y u
B
< s w r p dx
F C1 < B <  =
ž
/ Ž ry1 .rp s
1rp s
žH
w dx
B
/ žH /
H B< ¨ y c < w t
Ž
B
tr qs
dx
Ž 1rw . 1r ry1 dx .
/
qrp t
.
Ž 4.7.
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GEJUN BAO
Using the condition that w g A r Ž ., we obtain Ž ry1 .rp s
1rp s
ž
HBw
dx
/ ž
H B Ž 1rw .
F < B < rr ps
žž
1 < B<
1r Ž ry1 .
HBw
dx
dx
/ž
/ 1rs p
1 < B<
H B Ž 1rw .
1r Ž ry1 .
dx
//
F C2 < B < r r p s s C3 < B < r r p s .
Ž 4.8.
Putting Ž4.8. into Ž4.7. and noting that  q rrps s 1rk q 1rn y Ž1rm q 1rn. qrp q rrps s 1rs q 1rn y Ž1rt q 1rn. qrp, we have 1rs
ž
HB< u y u
< s r p dx B w
/
F C < B<␥
qrp t
ž
H B< ¨ y c < w t
tr qs
dx
/
,
where ␥ s 1rs q 1rn y Ž1rt q 1rn. qrp. We have completed the proof of Theorem 4.1. As an application of Theorem 4.1, we have the following example. EXAMPLE. Let f Ž x . s Ž f 1 , f 2 , . . . , f n . be K-quasiregular in R n. Then u s f l df 1 n df 2 n ⭈⭈⭈ n df ly1 and ¨ s 夹 f lq1 df lq2 n ⭈⭈⭈ n df n ,
l s 1, 2, . . . , n y 1, are conjugate A-harmonic tensors with p s nrl and q s nrŽ n y l ., where A is some operator satisfying Ž1.2.. Then by Theorem 4.1, we obtain
žH
B
f l df 1 n df 2 n ⭈⭈⭈ n df ly1 s
y Ž f l df 1 n df 2 n ⭈⭈⭈ n df ly1 . B w r p dx F C < B<␥
1rs
/ qrp t
žH
B
<夹 f lq1 df lq2 n ⭈⭈⭈ n df n y c < t w t r q s dx
/
,
where C is independent of f, ␥ s 1rs q 1rn y Ž1rt q 1rn. qrp and d 夹 c s 0.
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INTEGRAL INEQUALITIES
477
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