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Suppose u is a harmonic function in disc $|z|<1$, and u can be extended continuously to boundary, what about its harmonic conjugate v? Can it also be extended continuous to boundary?

I know v can be represented by integral use the boundary value of u, but I don't think this can help.

Can anyone prove it or give a counterexample? Thank all of you.

Kira Yamato
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1 Answers1

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Try the real and imaginary parts of a conformal map from the disc to a region such as $-1/(1+y^2) < x < 1/(1+y^2)$.

Robert Israel
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