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In this paper, we are interested in providing lower estimations for themaximum number of limit cycles $H(n)$ that planar piecewise linear differentialsystems with two zones separated by the curve $y=x^n$ can have, where $n$ is apositive integer. For this, we perform a higher order Melnikov analysis forpiecewise linear perturbations of the linear center. In particular, we obtainthat $H(2)\geq 4,$ $H(3)\geq 8,$ $H(n)\geq7,$ for $n\geq 4$ even, and $H(n)\geq9,$ for $n\geq 5$ odd. This improves all the previous results for $n\geq2.$ Ouranalysis is mainly based on some recent results about Chebyshev systems withpositive accuracy and Melnikov theory, which will be developed at any order fora class of nonsmooth differential systems with nonlinear switching manifold.

Higher order Melnikov analysis for planar piecewise linear vector fields with nonlinear switching curve