Abstract
Let T be a Calderón-Zygmund operator in a “non-homogeneous” space (\(\mathbb{X}\), d, μ), where, in particular, the measure μ may be non-doubling. Much of the classical theory of singular integrals has been recently extended to this context by F. Nazarov, S. Treil, and A. Volberg and, independently by X. Tolsa. In the present work we study some weighted inequalities for T*, which is the supremum of the truncated operators associated with T. Specifically, for1<p<∞, we obtain sufficient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresponding Lp weighted inequality holds for T*. The main tool to deal with this problem is the theory of vector-valued inequalities for T* and some related operators. We discuss it first by showing how these operators are connected to the general theory of vector-valued Calderón-Zygmund operators in non-homogeneous spaces, developed in our previous paper [6]. For the Cauchy integral operator C, which is the main example, we apply the two-weight inequalities for C* to characterize the existence of principal values for functions in weighted Lp.
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Communicated by Carlos Kenig
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García-Cuerva, J., Martell, J.M. On the existence of principal values for the cauchy integral on weighted lebesgue spaces for non-doubling measures. The Journal of Fourier Analysis and Applications 7, 469–487 (2001). https://doi.org/10.1007/BF02511221
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DOI: https://doi.org/10.1007/BF02511221