Maximal operators associated with some singular submanifolds

Let U be a bounded open subset of ℝ^d and let Ω be a Lebesgue measurable subset of U. Let γ = (γ₁, · · ·, γ_n): U \ Ω → ℝⁿ be a Lebesgue measurable function, and let µ be a Borel measure on ℝ^d+n defined by where ψ is a smooth function supported in U. In this paper we give some conditions under which the Fourier decay estimates |µ(ξ)| ≤ C(1+|ξ|)^-ε hold for some ε > 0. As a corollary we obtain the L^p-boundedness properties of the maximal operators MS associated with a certain class of possibly non-smooth n-dimensional submanifolds of ℝ^d+n, i.e., where Ω_sym is a radially symmetric Lebesgue measurable subset of ℝ^d, γ(y) = (γ₁(y), · · ·, γ_n(y)), γ_i(ty) = tai^γi(y) for each t > 0 where ai ∈ ℝ, and the function γ_i: ℝ^d \Ω_sym → ℝ satisfies some singularity conditions over a certain subset of ℝ^d. Also we investigate the endpoint (parabolic H¹, L^1,∞) mapping properties of the maximal operators M_H associated with a certain class of possibly non-smooth hypersurfaces, i.e., where the function γ: ℝ^d → ℝ satisfies some singularity conditions over a certain subset of ℝ^d and γ(ty) = t^mγ(y) for each t > 0 where m > 0.