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