Making the case for Cauchy transforms
Making the case for Cauchy transforms
Making the case for Cauchy transforms
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Theorem (Smirnov - 1929)<br />
For almost every (Lebesgue measure) t ∈ [0, 2π],<br />
Kµ(e it ) = lim<br />
r→1 −(Kµ)(reit )<br />
exists and is finite. Moreover, <strong>for</strong> each 0 < p < 1,<br />
and in fact 2π<br />
0<br />
2π<br />
|Kµ(e<br />
0<br />
it )| p dt < ∞<br />
|Kµ(e it )| p dt = O( 1<br />
1 − p ).<br />
William T. Ross (University of Richmond) <strong>Making</strong> <strong>the</strong> <strong>case</strong> <strong>for</strong> <strong>Cauchy</strong> trans<strong>for</strong>ms 16 / 42