On differentiability of functions in $L^p$, $0<p<1$

In this paper the author studies the connection between smoothness, expressed in terms of the integral modulus of continuity, and the existence of a derivative, understood in some sense, for functions in $L^p$, $0<p<1$; an analogous question is considered for boundary values of analytic functions in the Hardy classes $H^p$, $0<p<1$. A connection is established between the derivatives of an analytic function in $H^p$ and the derivatives of its boundary value; both global and pointwise derivatives are considered.
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