Maximal regularity for non-autonomous equations with measurable dependence on time

In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the $L^p$-boundedness of a class of vector-valued singular integrals which does not rely on H\"ormander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of $m$-th order elliptic operators $A$ with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an $L^p(L^q)$-theory for such equations for $p,q\in (1, \infty)$. In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.Comment: Application to a quasilinear equation added. Accepted for publication in Potential Analysi

Risk of cancer in patients on insulin glargine and other insulin analogues in comparison with those on human insulin

Aims/hypothesis Several publications suggest an association between certain types of insulin and cancer, but with conflicting results. We investigated whether insulin glargine (A21Gly,B31Arg,B32Arg human insulin) is associated with an increased risk of cancer in a large population-based cohort study. Methods Data for this study were obtained from dispensing records from community pharmacies individually linked to hospital discharge records from 2.5 million individuals in the Netherlands. In a cohort of incident users of insulin, the association between insulin glargine and other insulin analogues, respectively, and cancer was analysed in comparison with human insulin using Cox proportional hazard models with cumulative duration of drug use as a time-varying determinant. The first hospital admission with a primary diagnosis of cancer was considered as the main outcome; secondary analyses were performed with specific cancers as outcomes. Results Of the 19,337 incident insulin users enrolled, 878 developed cancer. Use of insulin glargine was associated with a lower risk of malignancies in general in comparison with human insulin (HR 0.75, 95% CI 0.71, 0.80). In contrast, an increased risk was found for breast cancer (HR 1.58, 95% CI 1.22, 2.05). Dose-response relationships could not be identified. Conclusion/interpretation Users of insulin glargine and users of other insulin analogues had a lower risk of cancer in general than those using human insulin. Both associations might be a consequence of residual confounding, lack of adherence or competing risk. However, as in previous studies, we demonstrated an increased risk of breast cancer in users of insulin glargine in comparison with users of human insulin