Parabolic BMO and global integrability of supersolutions to doubly nonlinear parabolic equations

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.Comment: 19 page

On regularity of weak solutions to linear parabolic systems with measurable coefficients

We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H{\"o}lder continuous Lp valued functions for some p > 2.Comment: 23 pages. Proof of Lemma 3.3 corrected. Final version to appear in J. Math. Pures App

Lp−LqL^p-L^q local smoothing estimates for the wave equation via kk-broad Fourier restriction

We explore the connection between $k$-broad Fourier restriction estimates and sharp regularity $L^p-L^q$ local smoothing estimates for the solutions of the wave equation in $\mathbb{R}^{n}\times \mathbb{R}$ for all $n \geq 3$ via a Bourgain--Guth broad-narrow analysis. An interesting feature is that local smoothing estimates for $e^{i t \sqrt{-\Delta}}$ are not invariant under Lorentz rescaling.Comment: Final version with referee's comments incorporated; to appear in J. Fourier Anal. App