Vol. 9, No. 7, 2016

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Parabolic weighted norm inequalities and partial differential equations

Juha Kinnunen and Olli Saari

Vol. 9 (2016), No. 7, 1711–1736
DOI: 10.2140/apde.2016.9.1711

We introduce a class of weights related to the regularity theory of nonlinear parabolic partial differential equations. In particular, we investigate connections of the parabolic Muckenhoupt weights to the parabolic BMO. The parabolic Muckenhoupt weights need not be doubling and they may grow arbitrarily fast in the time variable. Our main result characterizes them through weak- and strong-type weighted norm inequalities for forward-in-time maximal operators. In addition, we prove a Jones-type factorization result for the parabolic Muckenhoupt weights and a Coifman–Rochberg-type characterization of the parabolic BMO through maximal functions. Connections and applications to the doubly nonlinear parabolic PDE are also discussed.

parabolic BMO, weighted norm inequalities, parabolic PDE, doubly nonlinear equations, one-sided weight
Mathematical Subject Classification 2010
Primary: 42B25, 42B37, 35K55
Received: 15 February 2016
Revised: 20 June 2016
Accepted: 28 August 2016
Published: 7 November 2016
Juha Kinnunen
Department of Mathematics
%, School of Science
Aalto University
P.O. Box 11100
FI-00076 Aalto
Olli Saari
Department of Mathematics
Aalto University
P.O. Box 11100
FI-00076 Aalto