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Paraexponentials, Muckenhoupt weights,
and resolvents of paraproducts

Authors: María C. Pereyra and Lesley A. Ward
Journal: Proc. Amer. Math. Soc. 126 (1998), 135-144
MSC (1991): Primary 42B99; Secondary 47A10
MathSciNet review: 1452819
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Abstract: We analyze the stability of Muckenhoupt's $\bold{ RH} ^d_p$ and $\bold{ A} _p^d$ classes of weights under a nonlinear operation, the $\lambda $-operation. We prove that the dyadic doubling reverse Hölder classes $\bold{ RH} ^d_p$ are not preserved under the $\lambda $-operation, but the dyadic doubling $A_p$ classes $\bold{ A} _p^d$ are preserved for $0\leq \lambda \leq 1$. We give an application to the structure of resolvent sets of dyadic paraproduct operators.

References [Enhancements On Off] (What's this?)

  • [B] S. Buckley, Summation conditions on weights. Michigan Math. J. 40 #1, 153-170 (1993). MR 94d:42021
  • [Ch] M. Christ, Lectures on singular integral operators. Regional Conference Series in Math, AMS # 77 (1990). MR 92f:42021
  • [D] G. David, Wavelets and singular integrals on curves and surfaces. Springer Verlag Lecture Notes in Math. # 1465 (1991). MR 92k:42021
  • [FKP] R. Fefferman, C. Kenig, J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations. Annals of Math. # 134, 65-124 (1991). MR 93h:31010
  • [GC-RF] J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted norm inequalities and related topics. North Holland (1985). MR 87d:42023
  • [H] A. Haar, Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. # 69, 331-371 (1910).
  • [JN] R. Johnson, C. J. Neugebauer, Homeomorphisms preserving $\bold{ A} _p^d$. Revista Matemática Iberoamericana Vol. 3, No. 2, 249-273 (1987). MR 90d:42013
  • [M] Y. Meyer, Ondelettes et Opérateurs II. Herman (1990). MR 93i:42003
  • [P1] M. C. Pereyra, On the resolvents of dyadic paraproducts. Revista Matemática Iberoamericana Vol. 10 No. 3, 627-664 (1994). MR 96e:42021
  • [P2] M. C. Pereyra, On the resolvent of the dyadic paraproduct, and a nonlinear operation on $RH_p$ weights. Cont. Math. of AMS. Vol 189, 461-471 (1995). MR 96h:42016

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Additional Information

María C. Pereyra
Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131-1141

Lesley A. Ward
Affiliation: Department of Mathematics, Harvey Mudd College, Claremont, California 91711

Keywords: Muckenhoupt weights, reverse H\"older $RH_p$, $A_p$, doubling weights, dyadic paraproducts, paraexponentials
Received by editor(s): May 9, 1996
Additional Notes: This research was supported in part by (CP) NSF grant #DMS-93-04580 and (LW) at MSRI by NSF grant #DMS-90-22140.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society

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