Regularity and continuity of the multilinear strong maximal operators

Let m≥1, in this paper, our object of investigation is the regularity and continuity properties of the following multilinear strong maximal operatorMR(f→)(x)=supR∋xR∈R⁡∏i=1m1|R|∫R|fi(y)|dy, where x∈Rd and R denotes the family of all rectangles in Rd with sides parallel to the axes. When m=1, denote MR by MR. Then, MR coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that MR is bounded and continuous from the product Sobolev spaces W1,p1(Rd)×⋯×W1,pm(Rd) to W1,p(Rd), from the product Besov spaces Bsp1,q(Rd)×⋯×Bspm,q(Rd) to Bsp,q(Rd), from the product Triebel-Lizorkin spaces Fsp1,q(Rd)×⋯×Fspm,q(Rd) to Fsp,q(Rd). As a consequence, we further showed that MR is bounded and continuous from the product fractional Sobolev spaces to fractional Sobolev space. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to prove the p-quasicontinuity of MR. In addition, we proved that MR(f→) is approximately differentiable a.e. when f→=(f1,⋯,fm) with each fj∈L1(Rd) being approximately differentiable a.e.