  We establish optimal Lebesgue estimates for a class of generalized Radon
transforms defined by averaging functions along polynomial-like curves. The
presence of an essentially optimal weight allows us to prove uniform estimates,
wherein the Lebesgue exponents are completely independent of the curves and the
operator norms depend only on the polynomial degree. Moreover, our weighted
estimates possess rather strong diffeomorphism invariance properties, allowing
us to obtain uniform bounds for averages on curves satisfying a natural
nilpotency hypothesis.
