In this paper we study maximal L p -regularity for evolution equations with timedependent 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 Hörmander 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 ∈ (1, ∞). 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.

Abstract. In this paper we prove an ℓ s -boundedness result for integral operators with operator-valued kernels. The proofs are based on extrapolation techniques with weights due to Rubio de Francia. The results will be applied by the first and third author in a subsequent paper where a new approach to maximal L p -regularity for parabolic problems with time-dependent generator is developed. IntroductionIn the influential work [34,35], Weis has found a characterization of maximal L p -regularity in terms of R-sectoriality, which stands for R-boundedness of a family of resolvents on a sector. The definition of R-boundedness is given in Definition 3.15. It is a random boundedness condition on a family of operators which is a strengthening of uniform boundedness. Maximal regularity of solution to PDEs is important to know as it provides a tool to solve nonlinear PDEs using linearization techniques (see [4,23,25]). An overview on recent developments on maximal L pregularity can be found in [7,21]. Maximal L p -regularity means that for all f ∈ L p (0, T ; X), where X is a Banach space, the solution u of the evolution problemhas the "maximal" regularity in the sense that u ′ , Au are both in L p (0, T ; X). Using a mild formulation one sees that to prove maximal L p -regularity one needs to bound a singular integral with operator-valued kernel Ae (t−s)A . In [11] the first and third author have developed a new approach to maximal L pregularity for the case that the operator A in (1.1) depends on time in a measurable way. In this new approach R-boundedness plays a central rôle again. Namely, the R-boundedness of the family of integral operators

We prove mixed Lp(Lq)-estimates, with p, q ∈ (1, ∞), for higherorder elliptic and parabolic equations on the half space R d+1 + with general boundary conditions which satisfy the Lopatinskii-Shapiro condition. We assume that the elliptic operators A have leading coefficients which are in the class of vanishing mean oscillations both in the time variable and the space variable. In the proof, we apply and extend the techniques developed by Krylov [24] as well as Dong and Kim in [13] to produce mean oscillation estimates for equations on the half space with general boundary conditions.

We prove weighted mixed Lp(Lq)-estimates, with p, q ∈ (1, ∞), for higher-order elliptic and parabolic equations on the half space R d+1 + and on domains with general boundary conditions which satisfy the Lopatinskii-Shapiro condition. We assume that the elliptic operators A have leading coefficients which are in the class of vanishing mean oscillations both in the time and the space variables, and that the boundary conditions have variable leading coefficients. The proofs are based on and generalize the estimates recently obtained by the authors in [5].

3003 Background: MAPK and PI3K/AKT signaling pathways regulate proliferation, differentiation and cell death in human cancers. Known interaction between the 2 pathways provides the rationale for combining both inhibitors in a phase I study. Methods: The objective is to determine the maximum tolerated dose (MTD) and/or recommended phase II dose (RP2D) for oral, daily administered, BKM120 + GSK1120212, mainly in pts with tumors with RAS/RAF mutations (mt). A Bayesian logistic regression model with overdose control guides the dose escalation of the treatment. Secondary objectives include safety, tolerability, PK and efficacy. Results: As of 22.09.11, 49 pts were treated with BKM120 + GSK1120212 as follows: 30mg + 0.5mg, 60mg + 0.5mg, 60mg + 1.0mg, 60mg + 1.5mg, 60mg + 2.0mg, 70mg + 1.5mg, 80mg + 1.0mg, 80mg + 1.5mg. 6 pts had dose-limiting toxicities (DLTs); all were reversible. Grade 3 DLTs were: 3 x stomatitis, 1 x dysphagia, 1 x LVEF decrease, 1 x CK increase, 1 x nausea, 1 x anorexia, 1 x decreased oral intake. MTD and/or RP2D for the combination have not been reached. Most common adverse events (AEs) (>25%), all grades and causality, were dermatitis acneiform, diarrhea (51% each); nausea (41%); vomiting (37%); rash (33%); asthenia (31%); CK increase, decreased appetite, pyrexia or stomatitis (29% each) and hyperglycemia (27%). There were 4 on-treatment deaths unrelated to treatment. AEs led to treatment discontinuation, 17 pts (35%) and interruptions/dose reductions, 25 pts (51%). Apparent steady-states of BKM120 and GSK1120212 were reached by day 28. Plasma concentrations of BKM120 in combination with GSK1120212 were lower than for monotherapy. Exposure to GSK1120212 with BKM120 was similar to that observed in monotherapy studies. 3 confirmed partial responses have been observed in pts with KRAS mt ovarian cancer; 2 lasting >9 months. 2 patients with BRAF mt melanoma, who had previously progressed on BRAF inhibitors, had stable disease, for 1 of whom treatment is still ongoing in cycle 6. Conclusions: BKM120 and GSK1120212 can be safely combined. Signs of clinical activity have been seen in pts with RAS/RAF mt tumors.