Maximal Properties of the Normalized Cauchy Transform
نویسنده
چکیده
where m is the normalized Lebesgue measure on T. By the Fatou Theorem, Kf can also be defined on T by its non-tangential boundary values. After such an extension one can view the Cauchy integral as a “transform”, i.e., an operator in L(m) which sends f into the boundary values of Kf . It is well known, and not difficult to see, that the Cauchy transform is unbounded for p = 1 or p = ∞. However, the classical theorem by M. Riesz says that the Cauchy transform is bounded in L when 1 < p < ∞. Further progress is due to R. Hunt, B. Muckenhoupt and R. Wheeden, who studied spaces with absolutely continuous weights L(w). It was shown that the Cauchy transform is bounded in L(w), 1 < p <∞, if and only if the weight w satisfies the celebrated Ap-condition. Another classical object of complex function theory is the non-tangential maximal function. For any function g in D we can define its non-tangential maximal function Mg as
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