Let n > m ≥ 1 be two integers. At first we obtain general results for the multifractal analysis of the orthogonal projections on m-dimensional linear subspaces of singular measures µ on R n satisfying the multifractal formalism. The results hold for γ n,m -almost every such subspace, where γ n,m is the uniform measure on the Grassmannian manifold G n,m . Let µ be such a measure and suppose that its upper Hausdorff dimension is less than or equal to m. Let I stand for the interval over which the singularity spectrum of µ is increasing. We prove that there exists a non-trivial subinterval I of I such that for every α ∈ I , for γ n,m -almost every m-dimensional subspace V , the multifractal formalism holds at α for µ V , the orthogonal projection of µ on V . Moreover, in some cases the result is optimal in the sense that the interval I is maximal in I . Also, we determine the L qspectrum τ µ V (q) on the minimal interval J necessary to recover the singularity spectrum of µ V over I as the Legendre transform of τ µ V . The interval J and the function τ µ V (q) do not depend on V , and τ µ V (q) can differ from τ µ on a non-trivial interval. For Gibbs measures and some of their discrete counterparts, we show the stronger uniform result: for γ n,m -almost every m-dimensional subspace V , the multifractal formalism holds for µ V over the whole interval I . As an application, we obtain a part of the singularity spectrum of some self-similar measures on attractors of iterated function systems which do not satisfy the weak separation condition.
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