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

Weighted Norm Inequalities for Rough Singular Integral Operators

In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals TΩ with Ω ∈ L∞(Sn-1) and the Bochner–Riesz multiplier at the critical index B(n-1)/2. More precisely, we prove qualitative and quantitative versions of Coifman–Fefferman type inequalities and their vector-valued extensions, weighted Ap- A∞ strong and weak type inequalities for 1 < p< ∞, and A1- A∞ type weak (1, 1) estimates. Moreover, Fefferman–Stein type inequalities are obtained, proving in this way a conjecture raised by the second-named author in the 1990s. As a corollary, we obtain the weighted A1- A∞ type estimates. Finally, we study rough homogenous singular integrals with a kernel involving a function Ω ∈ Lq(Sn-1) , 1 < q< ∞, and provide Fefferman–Stein inequalities too. The arguments used for our proofs combine several tools: a recent sparse domination result by Conde–Alonso et al. (Anal PDE 10(5):1255–1284, 2017), results by the first author (Collect Math 68:129–144, 2017), suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for A∞ weights (Cruz-Uribe et al. in J Funct Anal 213:412–439, 2004, Curbera et al. in Adv Math 203:256–318, 2006), and ideas contained in previous works by Seeger (J Am Math Soc 9:95–105 1996) and Fan and Sato (Tohoku Math J 53:265–284, 2001).Fil: Kangwei, Li. Basque Center for Applied Mathematics; EspañaFil: Pérez, Carlos. Universidad del País Vasco; EspañaFil: Rivera Ríos, Israel Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina. Universidad del País Vasco; EspañaFil: Roncal, Luz. Basque Center for Applied Mathematics; Españ