Hardy–Stein identity for non-symmetric Lévy processes and Fourier multipliers

Using Itô's formula for processes with jumps, we extend the Hardy–Stein identity proved in [7] to non-symmetric pure jump Lévy processes and derive its martingale version. By a symmetrization argument for Littlewood–Paley functions, the Lp boundedness of Fourier multipliers arising from non-symmetric pure jump Lévy processes is proved.