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

• Published in: Journal of Differential Equations Vol. 394; pp. 98 - 119
• Main Authors: Carbonaro, Andrea, Dragičević, Oliver
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.06.2024
• Subjects: 47D03; 42B25; 35R15; 47A60
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2024.02.032

Let LA=−div(A∇) be an elliptic divergence form operator with bounded complex coefficients subject to mixed boundary conditions on an arbitrary open set Ω⊆Rd. We prove that the maximal operator MAf=supt>0⁡|exp⁡(−tLA)f| is bounded in Lp(Ω) whenever A is p-elliptic in the sense of [11]. The relevance of this result is that, in general, the semigroup generated by −LA is neither contractive in L∞ nor positive, therefore neither the Hopf–Dunford–Schwartz maximal ergodic theorem [16, Chap. VIII] nor Akcoglu's maximal ergodic theorem [2] can be used. We also show that if d⩾3 and the domain of the sesquilinear form associated with LA embeds into L2⁎(Ω) with 2⁎=2d/(d−2), then the range of Lp-boundedness of MA improves to (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 sups,t>0⁡|TsA1TtA2f|.