Abstract. It is shown that the realizations of the gaussian field $\{X(a), a\in{\msbm R}^d\} $ with $E(X(a)-X(b))^2=|a-b|^\beta $, $0<\beta < 2$, and mean zero, are locally in the Hölder class of exponent $\beta /2$ in the Orlicz norm corresponding to the Young function with tail $\exp(u^2)$. Other support function spaces are treated as well. In the proofs a crucial role is played by the theorem on the equivalence of modified Franklin and diamond bases in some generalized Hölder classes on a cube. Moreover, it is shown that the box dimension of the graph of $X(\cdot )$ over a cube is almost surely equal to $d+1-\beta /2$.

AMS Subject Classification (1991): 28C20, 41A15, 60B01, 60G15, 60G17

Received November 23, 1994; in revised form March 30, 1995. (Registered under 5621/2009.)