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Journal of the London Mathematical Society 1996 54(2):297-310; doi:10.1112/jlms/54.2.297
© 1996 by London Mathematical Society
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© The London Mathematical Society

Weighted Weak Type (1,1) Bounds for Rough Operators

Ana M. Vargas

Departamento de Matemáticas, Facultad de Ciencias Universidad Autónoma de Madrid, 28049 Madrid, Spain E-mail: avargas{at}ccuam3.sdi.uam.es

Received 20 July 1993. Revision received 24 November 1994.

In this work, we study three, by now classical, operators from the point of view of their weak type (1,1) behaviour with respect to weights. The first one is the Bochner-Riesz multiplier operator of index Formula(n – 1). We show that it is of weak type (1,1) with respect to any weight in the A1-Muckenhoupt class.

The other two are the maximal operator and the homogeneous singular integral (the former in any dimension and the latter in dimension 2) with rough kernel. We prove that they are of weak-type (1,1) with respect to weights ({omega} [Formula({omega}ß)]1/ß) for ß > 1, where Formula is a composition of the Hardy-Littlewood maximal operator and the maximal operator with rough kernel.

Research partially supported by Spanish DGICYT grant no. PB90-0187.


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