Error bound for a generalized M. A. Lavrentiev's formula via the norm in a fractional Sobolev space

We generalize M. A. Lavrentiev's approximate formula for the conformal mapping of the perturbed half-plane onto the half-plane. The generalization concerns harmonic functions and their derivatives in locally perturbed half-spaces (Lipschitz epigraphs). For both formulas, we obtain remainder estimates involving the square of the norm of the perturbing function in the fractional homogeneous Sobolev space $\dot{H}^{1/2}$. By the Kashin–Besov–Kolyada inequality, these estimates imply pointwise stability bounds in terms of the Lebesgue measure. Moreover, we prove the joint analyticity of the above-named harmonic functions with respect to the perturbing parameter and the space variables and justify a result on the interpolation between $L^1$ and homogeneous Slobodetskii spaces which is essentially due to A. Cohen.