Prove or disprove the following claim.
"For all continuous $f : S(0, 1) \to R$, there is a holomorphic $g : B(0, 1) \to C$ which extends to a continuous ${h : \overline {B(0, 1)}} \to C$ such that $\operatorname{Re}(h(z)) = f (z)$ for all $z \in S(0, 1)$?"
$S(0,1)$ is sphere of radius $1$ about the origin, and $B(0,1)$ is open ball of radius $1$ about the origin.$ \overline {B(0, 1)}$ is the closed ball of radius one about the origin.
Based on my attempts, this assertion appears to be true, but I am uncertain. Any assistance will be greatly appreciated. I initially tried to use contradiction to show that it was false, but all of my counterexamples had shortcomings.
Yes, this claim indeed is true. This is due to the fact that the Dirichlet Problem on the disk is solvable. See here for all the necessary details - http://people.reed.edu/~jerry/311/dirichlet.pdf