The distribution of spatially aggregated data from a stochastic process may exhibit tail behaviour different from that of its marginal distributions. For a large class of aggregating functionals we introduce the -extremal coefficient, which quantifies this difference as a function of the extremal spatial dependence in . We also obtain the joint extremal dependence for multiple aggregation functionals applied to the same process. Formulae for the -extremal coefficients and multivariate dependence structures are derived in important special cases. The results provide a theoretical link between the extremal distribution of the aggregated data and the corresponding underlying process, which we exploit to develop a method for statistical downscaling. We apply our framework to downscale daily temperature maxima in the south of France from a gridded dataset and use our model to generate high-resolution maps of the warmest day during the heatwave.