We discuss a parabolic version of the space of functions of bounded mean oscillation related to a doubly nonlinear parabolic partial differential equation. Parabolic John–Nirenberg inequalities, which give exponential decay estimates for the oscillation of a function, are shown in the natural geometry of the partial differential equation. Chaining arguments are applied to change the time lag in the parabolic John–Nirenberg inequality. We also show that the quasihyperbolic boundary condition is a necessary and sufficient condition for a global parabolic John–Nirenberg inequality. Moreover, we consider John–Nirenberg inequalities with medians instead of integral averages and show that this approach gives the same class of functions as the original definition.

We study the so-called John-Nirenberg space that is a generalization of functions of bounded mean oscillation in the setting of metric measure spaces with a doubling measure. Our main results are local and global John-Nirenberg inequalities, which give weak type estimates for the oscillation of a function. We consider medians instead of integral averages throughout, and thus functions are not a priori assumed to be locally integrable. Our arguments are based on a Calderón-Zygmund decomposition and a good-λ inequality for medians. A John-Nirenberg inequality up to the boundary is proven by using chaining arguments. As a consequence, the integral-type and the median-type John-Nirenberg spaces coincide under a Boman-type chaining assumption.

We study the so-called John–Nirenberg space that is a generalization of functions of bounded mean oscillation in the setting of metric measure spaces with a doubling measure. Our main results are local and global John–Nirenberg inequalities, which give weak-type estimates for the oscillation of a function. We consider medians instead of integral averages throughout, and thus functions are not a priori assumed to be locally integrable. Our arguments are based on a Calderón–Zygmund decomposition and a good-$$\lambda $$ λ inequality for medians. A John–Nirenberg inequality up to the boundary is proven by using chaining arguments. As a consequence, the integral-type and the median-type John–Nirenberg spaces coincide under a Boman-type chaining assumption.

We discuss the dyadic John-Nirenberg space that is a generalization of functions of bounded mean oscillation. A John-Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John-Nirenberg space and provide a method to construct nontrivial functions in the dyadic John-Nirenberg space. Moreover, we prove that the John-Nirenberg space is complete. Several open problems are also discussed.2020 Mathematics Subject Classification. 42B25, 42B35.

We discuss the dyadic John–Nirenberg space that is a generalization of functions of bounded mean oscillation. A John–Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John–Nirenberg space and provide a method to construct nontrivial functions in the dyadic John–Nirenberg space. Moreover, we prove that the John–Nirenberg space is complete. Several open problems are also discussed.