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We compute the Hausdorff dimension of the image X(E) of a non random Borelset E $\subset$ [0, 1], where X is a L\'evy multistable process in R. Thisextends the case where X is a classical stable L\'evy process by letting thestability exponent $\alpha$ be a smooth function, which leads tonon-homogeneous processes because their increments are not stationary and notnecessarily independent. Contrary to the situation where the stabilityparameter is a constant, the dimension depends on the version of themultistable L\'evy motion when the process has an infinite first moment.

The Hausdorff dimension of the range of the L\'evy multistable processes