  Fine regularity of stochastic processes is usually measured in a local way by
local H\"older exponents and in a global way by fractal dimensions. Following a
previous work of Adler, we connect these two concepts for multiparameter
Gaussian random fields. More precisely, we prove that almost surely the
Hausdorff dimensions of the range and the graph in any ball $B(t_0,\rho)$ are
bounded from above using the local H\"older exponent at $t_0$. We define the
deterministic local sub-exponent of Gaussian processes, which allows to obtain
an almost sure lower bound for these dimensions. Moreover, the Hausdorff
dimensions of the sample path on an open interval are controlled almost surely
by the minimum of the local exponents.
  Then, we apply these generic results to the cases of the multiparameter
fractional Brownian motion, the multifractional Brownian motion whose
regularity function $H$ is irregular and the generalized Weierstrass function,
whose Hausdorff dimensions were unknown so far.
