Consider a bounded strictly pseudoconvex domain \(\Omega \) with smooth boundary \(\partial \Omega \) . In this paper, we establish the higher-order Taylor expansion in terms of the horizontal vector fields via the Folland–Stein approximation of \(\partial \Omega \) by the Heisenberg group in terms of the vector fields, and then provide an explicit construction of the Alpert wavelet with general higher-order cancellation. These results are of independent interest. As a direct application of this Alpert wavelet, we obtain the characterization of the endpoint Schatten estimate for the commutator of the Cauchy–Szegö projection.
