Maximal operators and singular integral operators along submanifolds

In this paper we prove, for certain values of p, the Lp boundedness of the maximal operator Γ f (¯x) = sup h p.v. Rm h(|y|)Ω(y ) |y|m f ¯x − Γ (y) dy ( ¯x ∈ Rn; n>m 2), where the supremum is taken over all measurable radial functions h with h Ls (R+, dr r ) 1 and 1 s 2. Here Ω ∈ H1(Sm−1), Γ (y) = (φ(|y|)y ,Ψ (|y|)). We also obtain the range of p for which the maximal operator above is unbounded. Moreover, we show that the singular integral TΓ f (¯x) = p.v. Rm h(|y|)Ω(y ) |y|m f ¯x − Γ (y) dy and its associated maximal function T ∗ Γ f (x) are bounded in Lp(Rn) for 1