We introduce a class of random …eld models with variable regularity/singularity order on multifractal\ud domains. The derivation of results on atomic decomposition of functions in fractional\ud Sobolev spaces of variable order, and traces of these spaces on multifractal domains plays a key\ud role. The non-triviality of the multifractal spectrum of the measure de…ning the multifractal domain\ud ensures the non-triviality of the multifractal spectrum of the functions in the trace Sobolev\ud spaces of variable order de…ned. The generalized reproducing kernel Hilbert space theory is considered\ud to introduce a class of random …elds with non-trivial mean-square multifractal spectrum.\ud In the Gaussian case, the random …eld models studied have multifractal sample paths
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