Free interpolation of bounded harmonic functions by analytic ones

Let $L^p$ (resp. $H^p$) denote the space of harmonic (analytic) functions in the unit disk $\mathbb D$ with the norm $\|f\|_p=\lim_{r\to1-2}(\int_{\mathbb T}|f(re^{it}|^p\,dt)^{1/p}$, $1\le p\le\infty$. A complete characterization of subsets $E$, $E\subset\mathbb D$, satisfying $L^\infty|_E=H^\infty|_E$ is given. There are some results about sets $E$, $E\subset\mathbb D$ with $L^p|_E=H^p|_E$, $1\le p<\infty$.