Critical regularity for elliptic equations from Littlewood-Paley theory

Using simple facts from harmonic analysis, namely Bernstein inequality and Plansherel isometry, we prove that the pseudodifferential equation ${\mathrm{\Delta }}^{\alpha }u+Vu=0$ improves the Sobolev regularity of solutions provided the potential $V$ is integrable with the critical power $n\mathrm{/}2\alpha >1$.