Maximal functions along surfaces in product spaces

Under certain natural conditions of a measurable radial function Γ :Rn × Rm→R, Γ (y1, y2) = Γ (|y1|, |y2|), we show that the maximal function along surface MΓ f (x1, x2, x3) = sup r1,r2>0 1 rn 1 rm 2 |y2| r2 |y1| r1 f x1 −y1, x2 − y2, x3 −Γ |y1|, |y2| dy1 dy2 is bounded in Lp(Rn ×Rm ×R) for all p >1 and n,m 1.  2005 Elsevier Inc. All rights reserved