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On harmonic conjugates with exponential mean growth. (English) Zbl 1009.30031
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 \(\alpha >0\). A. L. Shields and 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 \(\alpha _1<0\) and \(\alpha _2>0\). 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 \(f'\in h_p(\varphi ')\).

MSC:
30H05 Spaces of bounded analytic functions of one complex variable
31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
46E15 Banach spaces of continuous, differentiable or analytic functions
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References:
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