The Carleman–Goluzin–Krylov formula and analytic functions smooth up to the boundary

The classical Carleman–Goluzin–Krylov formula recovers an $H^1$-function from its boundary values on an arc. We study this formula when it is applied to Lipschitz spaces of order $\alpha\le1$ and to higher order smoothness spaces. The rate of convergence is estimated and some (counter-) examples are given.