On the quantitative Fatou property (Q2773352)

Let the function \(F\) belong to the Hardy class \(H^p\), \(0<p<\infty,\) in the upper half plane. \textit{A. A. Solyanik} [``Approximation of functions from Hardy classes'', Ph. D. Thesis, Odessa (1986), per bibl.] obtained an estimate on the rate of convergence of \(F(x+iy)\) towards \(F(x), y\to 0^+\), in terms of certain characteristics of the \(L^p-\)modulus of continuity of \(F(x)\). The sharpness of this estimate was verified in [\textit{A. Kamaly, A. M. Stokolos} and \textit{W. Trebels}, J. Approx. Theory 10, No.~2, 240--264 (1999; Zbl 0941.42002), Theorem 2] for \(p\geq 1\). The present paper extends the result of Kamaly et al. to the remaining case \(0<p<1\).