Parabolic BMO and the forward-in-time maximal operator

Abstract

We study if the parabolic forward-in-time maximal operator is bounded on parabolic BMO. It turns out that for nonnegative functions the answer is positive, but the behaviour of sign changing functions is more delicate. The class parabolic BMO and the forward-in-time maximal operator originate from the regularity theory of nonlinear parabolic partial differential equations. In addition to that context, we also study the question in dimension one.2010 Mathematics Subject Classification. Primary: 42B37, 42B25, 42B35. Secondary: 35K92.

Poincaré inequalities for the maximal function

“…Similarly, the role of the non-local part is almost negligible in the case of BMO. Compare to [1] and the slightly more complicated setting of [36]. The following theorem finds a connection between these extreme cases at the level of generalized Poincaré inequalities.…”

Parabolic John–Nirenberg spaces with time lag

“…This allows us to conclude that the parabolic John-Nirenberg space is independent of the size of the time lag, see Corollary 4.2. For more about chaining techniques in the parabolic geometry, see [7,19,20].…”

John–Nirenberg inequalities for parabolic BMO

“…A chaining argument in the proof of Theorem 4.1 shows that the size of the time lag can be changed in parabolic BMO. Parabolic chaining arguments have been previously studied by Saari [22,23] and our approach complements these techniques. We also discuss the John-Nirenberg inequality up to the spatial boundary of a space-time cylinder by applying results of Saari [22] and Smith and Stegenga [24].…”