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|)Γ(y1,y2)=Γ(|y1|,|y2|), we show that the maximal function along surfaceMΓf(x1,x2,x3)=supr1,r2>0{1r1nr2m∫|y2|⩽r2∫|y1|⩽r1|f(x1−y1,x2−y2,x3−Γ(|y1|,|y2|))|dy1dy2} is bounded in Lp(Rn×Rm×R)Lp(Rn×Rm×R) for all p>1p>1 and n,m⩾1n,m⩾1.