Maximal functions with polynomial densities in lacunary directions

Given a real polynomial p(t) in one variable such that p(0) = 0, we consider the maximal operator in R2, Mpf(x1, x2)= sup h>0, i, jℤZ 1/h ∫ h0 |f(x1-2ip(t), x2-2jp(t))| dt. We prove that Mp is bounded on Lq(ℝ2) for q > 1 with bounds that only depend on the degree of p.