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We consider a class of endomorphisms which contains a set of piecewisepartially hyperbolic skew-products with a non-uniformly expanding base map. Theaimed transformation preserves a foliation which is almost everywhere uniformlycontracted with possible discontinuity sets, which are parallel to thecontracting direction. We prove that the associated transfer operator, actingon suitable anisotropic normed spaces, has a spectral gap (on which we havequantitative estimation) and the disintegration of the unique invariantphysical measure, along the stable leaves, is $\zeta$-H\"older. We use thisfact to obtain exponential decay of correlations on the set of $\zeta$-H\"olderfunctions.

H\"older Regularity and Exponential Decay of Correlations for a class of Piecewise Partially Hyperbolic Maps