Hardy-Stein identities and square functions for semigroups

We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump L'evy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1<p<\mathrm{\infty }$, of the corresponding Littlewood-Paley square function. The square function yields a direct proof of the $L^p$ boundedness of Fourier multipliers obtained by transforms of martingales of L'evy processes.