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Stimulated by Malliavin calculus, the theory of quasi sure analysis of Wiener functionals has been extensively developed (cf. [4, 6, 7, 8, 9, 12, 13, 16, 17, 18, 19], etc). Recently, J. Ren (cf. [17]) has studied quasi sure properties of quadratic variation of "smooth martingales", a no tion introduced by P . M alliavin and D. N ualart [8], and his results are concentrated on studying one parameter smooth martingales and tw o param eter sm ooth strong m artinga les. In this paper we generalize his results in m ore general setting and study the quasi sure properties of two parameter smooth martingales which are not necessary strong martingale in genera l. This situation is m uch m ore difficulty to handle, when Malliavin calculus is involved, because th e two parameter stochastic differentiation rules (cf. [1, 21, 24]) and the representation of two parameter square integrable martingales involve "stochastic integral of the second type", i.e., f(c,n)dw4ciw„ (cf. [1]). N ow le t u s state our results in m ore details. L e t N b e a two param eter sm ooth martingale, th e n b y [1 ] and [21], for each z E H = [0, 1] 2 , N can be represented a s a sum o f stochastic integral of the first type and stochastic integral of the second type, = 0( 11)dW , + if ti)dfrylW n JR. Rzx.tt

Quasi Sure quadratic variations of two parameter smooth martingales on the Wiener space