Noncoercive quasilinear elliptic operators with singular lower order terms

We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular.

On boundedness of the Hilbert transform on Marcinkiewicz spaces

We study boundedness properties of the classical (singular) Hilbert transform (Hf)(t) = p.v.1/π \int_R f(s)/(t − s)ds acting on Marcinkiewicz spaces. The Hilbert transform is a linear operator which arises from the study of boundary values of the real and imaginary parts of analytic functions. Questions involving the H arise therefore from the utilization of complex methods in Fourier analysis, for example. In particular, the H plays the crucial role in questions of norm-convergence of Fourier series and Fourier integrals. We consider the problem of what is the least rearrangement-invariant Banach function space F(R) such that H : Mφ(R) → F(R) is bounded for a fixed Marcinkiewicz space Mφ(R). We also show the existence of optimal rearrangement-invariant Banach function range on Marcinkiewicz spaces. We shall be referring to the space F(R) as the optimal range space for the operator H restricted to the domain Mφ(R) ⊆ Λϕ0(R). Similar constructions have been studied by J.Soria and P.Tradacete for the Hardy and Hardy type operators [1]. We use their ideas to obtain analogues of their some results for the H on Marcinkiewicz spaces.

Noncommutative weighted individual ergodic theorems with continuous time

We show that ergodic flows in the noncommutative [Formula: see text]-space (associated with a semifinite von Neumann algebra) generated by continuous semigroups of positive Dunford–Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly. The corresponding local ergodic theorem is also proved. We then extend these results to arbitrary noncommutative fully symmetric spaces and present applications to noncommutative Orlicz (in particular, noncommutative [Formula: see text]-spaces), Lorentz, and Marcinkiewicz spaces. The commutative counterparts of the results are derived.

Continuity properties of solutions to the p-Laplace system

A sharp integrability condition on the right-hand side of the p-Laplace system for all its solutions to be continuous is exhibited. Their uniform continuity is also analyzed and estimates for their modulus of continuity are provided. The relevant estimates are shown to be optimal as the right-hand side ranges in classes of rearrangement-invariant spaces, such as Lebesgue, Lorentz, Lorentz–Zygmund, and Marcinkiewicz spaces, as well as some customary Orlicz spaces.