Maximal functions along twisted surfaces on product domains

In this paper, we introduce a class of maximal functions along twisted surfaces in Rⁿ ×R^m of the form {(φ(|v|)u, ϕ(|u|)v): (u, v) ∈ Rⁿ ×R^m }. We prove L^p bounds when the kernels lie in the space L^q (Sⁿ⁻¹ ×S^m−1). As a consequence, we establish the L^p boundedness for such class of operators provided that the kernels are in L log L(Sⁿ⁻¹ ×S^m−1) or in the Block spaces Bq^0,0 (Sⁿ⁻¹ ×S^m−1) (q > 1).