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 $\Delta^\alpha u+Vu=0$ improves the Sobolev regularity of solutions provided the potential $V$ is integrable with the critical power $n/2\alpha>1$. 
Denis A. Labutin