On semigroup maximal operators associated with divergence-form operators with complex coefficients

Let L-A = -div(A del) be an elliptic divergence form operator with bounded complex coefficients subject to mixed boundary conditions on an arbitrary open set Omega subset of R-d. We prove that the maximal operator M(A )f = sup(t>0)|exp(-tL(A))f| is bounded in L-p(Omega), whenever A is p-elliptic in the sense of [10]. The relevance of this result is that, in general, the semigroup generated by -L-A is neither contractive in L-infinity nor positive, therefore neither the Hopf--Dunford--Schwartz maximal ergodic theorem [15, Chap. VIII] nor Akcoglu's maximal ergodic theorem [1] can be used. We also show that if d >= 3 and the domain of the sesquilinear form associated with LA embeds into L-2 & lowast;(Omega) with 2(& lowast;)=2d/(d-2), then the range of L-p-boundedness of M-A improves into the interval (rd/((r-1)d+2),rd/(d-2)), where r >= 2 is such that A is r-elliptic. With our method we are also able to study the boundedness of the two-parameter maximal operator sup(s,t>0)|T(s)(A1)T(t)(A2)f|.