# bounded and uniformly continous harmonic function on the upper half plane

If $$f(x)$$ is bounded and uniformly continous on $$\mathbb{R}$$, and $$u(x+iy)$$ is harmonic function on the upper half plane $${\mathbb{H}}$$ defined by the convolution of the possion kernel and f(x), then $$u(x+iy)$$ is uniformly convergent to $$f(x)$$ when $$y\to 0$$.

In Axler, Bourdon and Ramey's book “Harmonic Function Theory”, they said that the converse is also true. The accurate statement is exericise 7.4.

Suppose that $$u$$ is a harmonic function on the upper half plane $${\mathbb{H}}$$, and in the harmoninc Hardy space $$h^p(\mathbb{H})$$ for $$p\in [1,\infty]$$. If $$u_y(x)=u(x+iy)$$ converge uniformly on $$\mathbb{R}$$ when $$y \to 0$$, then $$u$$ can be extended to a bounded and uniformly continous harmonic function on the closed upper half plane $$\overline{\mathbb{H}}$$.

In the case $$p=\infty$$, clearly $$u$$ is extended to a bounded continous harmonic function on $$\overline{\mathbb{H}}$$.

But I can't prove that $$u$$ is uniformly continous.

Could you show me some hints or references? Thanks a lot.