On an analogue of cauchy's formula for H^p, 1/2 ≤p<1, and the cauchy type integral of a singular measure

Abstract

We put forward a direct analog of the Cauchy formula for analytic functions f in the unit disk from Hardy classes H ^p with 1/2≤p>1. In this way, one has to consider a weakened Stieltjes type integration with respect to radial means of f which are in general not measures. A suitable concept of integration is found which also lets us obtain a generalization of the result–every Cauchy type integral is the Cauchy A-integral of its boundary values–to the case of a singular measure.

Keywords:

• Cauchy integral
• Hardy classes
• Generalized integral
• Singular measure

1991 Mathematics Subject Classifications::

• Primary: 30E20
• Primary: 30D55
• Primary: 26A39
• Secondary: 26A30
• Secondary: 26A45
• Secondary: 28A50

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Notes

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