  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 $1<p<\infty$, 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.
