Weighted Hardy-Littlewood inequality

for -harmonic tensors

Author:
Shusen Ding

Journal:
Proc. Amer. Math. Soc. **125** (1997), 1727-1735

MSC (1991):
Primary 30C65; Secondary 31B05, 58A10

MathSciNet review:
1372027

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove a local weighted integral inequality for conjugate -harmonic tensors similar to the Hardy and Littlewood integral inequality for conjugate harmonic functions. Then by using the local weighted integral inequality, we prove a global weighted integral inequality for conjugate -harmonic tensors in John domains.

**[B]**John M. Ball,*Convexity conditions and existence theorems in nonlinear elasticity*, Arch. Rational Mech. Anal.**63**(1976/77), no. 4, 337–403. MR**0475169****[BM]**J. M. Ball and F. Murat,*𝑊^{1,𝑝}-quasiconvexity and variational problems for multiple integrals*, J. Funct. Anal.**58**(1984), no. 3, 225–253. MR**759098**, 10.1016/0022-1236(84)90041-7**[G]**J. B. Garnett,*Bounded Analytic Functions*, New York, Academic Press, 1970.**[HL]**G. H. Hardy and J. E. Littlewood,*Some properties of conjugate functions*, J. Reine Angew. Math.**167**(1932), 405-423.**[I]**Tadeusz Iwaniec,*𝑝-harmonic tensors and quasiregular mappings*, Ann. of Math. (2)**136**(1992), no. 3, 589–624. MR**1189867**, 10.2307/2946602**[IL]**Tadeusz Iwaniec and Adam Lutoborski,*Integral estimates for null Lagrangians*, Arch. Rational Mech. Anal.**125**(1993), no. 1, 25–79. MR**1241286**, 10.1007/BF00411477**[IM]**Tadeusz Iwaniec and Gaven Martin,*Quasiregular mappings in even dimensions*, Acta Math.**170**(1993), no. 1, 29–81. MR**1208562**, 10.1007/BF02392454**[IN]**T. Iwaniec and C. A. Nolder,*Hardy-Littlewood inequality for quasiregular mappings in certain domains in 𝑅ⁿ*, Ann. Acad. Sci. Fenn. Ser. A I Math.**10**(1985), 267–282. MR**802488**, 10.5186/aasfm.1985.1030**[N1]**Craig A. Nolder,*A characterization of certain measures using quasiconformal mappings*, Proc. Amer. Math. Soc.**109**(1990), no. 2, 349–356. MR**1013976**, 10.1090/S0002-9939-1990-1013976-2**[N2]**Craig A. Nolder,*A quasiregular analogue of a theorem of Hardy and Littlewood*, Trans. Amer. Math. Soc.**331**(1992), no. 1, 215–226. MR**1036007**, 10.1090/S0002-9947-1992-1036007-3**[N3]**C. A. Nolder,*Hardy-Littlewood theorems for -harmonic tensors*, Illinois J. Math., to appear.**[S]**B. Stroffolini,*On weakly -harmonic tensors*, Studia Math., 3**114**(1995), 289-301. CMP**95:14**

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Additional Information

**Shusen Ding**

Email:
sding@d.umn.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-97-03762-3

Keywords:
Conjugate harmonic tensors,
differential forms and the $A$-harmonic equation

Received by editor(s):
May 15, 1995

Received by editor(s) in revised form:
December 8, 1995

Communicated by:
Theodore W. Gamelin

Article copyright:
© Copyright 1997
American Mathematical Society