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Description:
The following two norms for holomorphic functions \(F\), defined on the right complex half-plane \(\{z \in C:\Re(z)\gt 0\}\) with values in a Banach space \(X\), are equivalent: \[\begin{eqnarray*} \lVert F \rVert _{H_p(C_+)} &=& \sup_{a\gt0}\left( \int_{-\infty}^\infty \lVert F(a+ib) \rVert ^p \ db \right)^{1/p} \mbox{, and} \\ \lVert F \rVert_{H_p(\Sigma_{\pi/2})} &=& \sup_{\lvert \theta \lvert \lt \pi/2}\left( \int_0^\infty \left \lVert F(re^{i \theta}) \right \rVert ^p\ dr \right)^{1/p}.\end{eqnarray*}\] As a consequence, we derive a description of boundary values ofsectorial holomorphic functions, and a theorem of Paley-Wiener typefor sectorial holomorphic functions.