Extension of Zygmund's classical conjugate-function theorem from analytic to harmonic K-quasiregular mappings on the unit disk: if f = u + iv is harmonic K-quasiregular with u bounded below and u in h log+ h, then f (hence v) lies in the harmonic Hardy space h1, with an explicit K-squared bound proved via a Laplacian comparison and Green's theorem. A partial converse -- f in h1 with Re f bounded below and Im h non-vanishing forces u into h log+ h -- shows the growth condition is best possible, and the classical Riesz and Kolmogorov theorems yield a Hardy-Littlewood-type coefficient theorem for planar harmonic functions. Continues the Riesz-type theorem of Liu-Zhu and the Kolmogorov-type theorem of Kalaj.
Given an analytic function $f=u+iv$ in the unit disk $\mathbb{D}$, Zygmund's theorem gives the minimal growth restriction on $u$ which ensures that $v$ is in the Hardy space $h^1$. This need not be t…