Log-integrability of the boundary values of a Hardy function

If $p<\mathrm{\infty }$, [F1] gives ${\sup }_r\int {\log }^{+}\mathrm{\mid }{f}_r\mathrm{\mid }dm\le \mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}^p\mathrm{/}p<\mathrm{\infty }$. If $p=\mathrm{\infty }$, ${\log }^{+}\mathrm{\mid }f\mathrm{\mid }\le {\log }^{+}\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$, a constant harmonic majorant. In both cases $f\in N(\mathbb{D})$ by [F1], including functions with origin zeros or no zeros; no zero enumeration or general Hardy boundary theorem is used.

Apply [F2] to the nonzero f. It gives the finite nontangential boundary function $f^{\ast }$ and $\log \mathrm{\mid }{f}^{\ast }\mathrm{\mid }\in L^1$. For finite p, [F3] applied along radii to $\mathrm{\mid }{f}_r{\mathrm{\mid }}^p$ also gives $\int \mathrm{\mid }{f}^{\ast }{\mathrm{\mid }}^{p}dm\le \mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}^p$; for p infinity, limits preserve the bound $\mathrm{\mid }{f}^{\ast }\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$. Thus the same boundary function is in the asserted Lp class under CC alone.

The positive boundary logarithmic part is integrable by the Lp bound and ${\log }^{+}t\le t^p\mathrm{/}p$ (or the uniform bound at infinity). Hence finiteness of the lower logarithmic integral is equivalent to integrability of its negative part, and so to absolute logarithmic integrability. Both are given by step 2.1.

An integrable real logarithm is finite almost everywhere, so $f^{\ast }\mathrm{\ne }0$ almost everywhere. If the boundary function of a Hardy function vanishes on a positive-measure set, it cannot be nonzero by step 2.1; it must therefore be the zero function. This proves all original logarithmic and uniqueness claims under the existing CC class conventions.