Treffer: On harmonic conjugates with exponential mean growth

Let \(\varphi \) be a positive continuous function defined on some interval \([r_0,1)\). For a function \(u\) harmonic in the unit disc \(D\) let \(M_p(u,r)= ((1/2\pi)\int _{0}^{2\pi } |u(re^{i\theta })|^p d\theta )^{1/p}\). The space \(h_p(\varphi)\) consists of all functions \(u\) harmonic in the unit disc, which satisfy \(M_p(u,r)=O(\varphi (r))\) as \(r\to 1_{-}\), and \(h_p(\varphi)\) is called self-conjugate if the Riesz projection maps \(h_p(\varphi)\) into itself. The well-known theorem due to Hardy and Littlewood states that \(h_p((1-r)^{-\alpha })\) is self-conjugate whenever \(p>0\) and \(\alpha >0\). \textit{A. L. Shields} and \textit{D. L. Williams} [Mich. Math. J. 29, 3-25 (1982; Zbl 0508.31001)] extended this to a finer scale of functions \(\varphi \) such that \((1-r)^{\alpha _1}\varphi (r)\to 0\) as \(r\to 1_{-}\) and \((1-r)^{\alpha _2}\varphi (r)\to \infty \) as \(r\to 1_{-}\) for some \(\alpha _10\). This means e.g. a fine logarithmic tuning of functions \((1-r)^{-\alpha }\), \(\alpha >0\). The authors now consider functions of type \((1-r)^{-\alpha } (\log (1/(1-r)))^{\beta }\exp (c/(1-r))\) and prove that \(h_{p}(\varphi)\) is self-conjugate if \(\varphi ^{-m}\) is almost convex on some interval \([r_0,1)\). Furthermore, they show that under this assumption on \(\varphi \) it is \(f\in h_p(\varphi)\) if and only if \(f'\in h_p(\varphi ')\).