A functional analytic approach to obtaining self-improving properties of solutions to linear non-local elliptic equations is presented. It yields conceptually simple and very short proofs of some previous results due to Kuusi-Mingione-Sire and Bass-Ren. Its flexibility is demonstrated by new applications to non-autonomous parabolic equations with non-local elliptic part and questions related to maximal regularity.the left-hand side is associated with a sesquilinear form on the Hilbert space W α,2 (R n ) and thanks to ellipticity (1.1) the Lax-Milgram lemma applies and yields invertibility of 1 + L α,A onto the dual 1 2 PASCAL AUSCHER, SIMON BORTZ, MORITZ EGERT, AND OLLI SAARI space. Now, the main difference compared with second order elliptic equations is that we can transfer regularity requirements between u and φ without interfering with the coefficients A: Without making any further assumption we may writeThen the ubiquitous analytic perturbation lemma ofŠneȋberg [21] allows one to extrapolate invertibility to ε > 0 small enough. Compared to [3,14] we can also work in an L p -setting without hardly any additional difficulties. In this way, we shall recover some of their results on global weak solutions in Section 4 and discuss some new and sharpened local self-improvement properties in Section 5.Finally, in Section 6 we demonstrate the simplicity and flexibility of our approach by proving that for each f ∈ L 2 (0, T ; L 2 (R n )) the unique solution u ∈ H 1 (0, T ; W α,2 (R n ) * ) ∩ L 2 (0, T ; W α,2 (R n )) of the non-autonomous Cauchy problem

No abstract

Abstract. We prove that local and global parabolic BMO spaces are equal thus extending the classical result of Reimann and Rychener. Moreover, we show that functions in parabolic BMO are exponentially integrable in a general class of space-time cylinders. As a corollary, we establish global integrability for positive supersolutions to a wide class of doubly nonlinear parabolic equations.

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions n ≥ 2. We also show that the spherical fractional maximal function maps L p into a first order Sobolev space in dimensions n ≥ 5. t α |ϕ t * f (x)|.