Related papers: Mean oscillation bounds on rearrangements
We derive bounds on the mean oscillation of the decreasing rearrangement $f^*$ on $\mathbb{R}_+$ in terms of the mean oscillation of $f$ on a suitable measure space $X$. In the special case of a doubling metric measure space, the bound…
We study the decreasing rearrangement of functions in VMO, and show that for rearrangeable functions, the mapping f -> f* preserves vanishing mean oscillation. Moreover, as a map on BMO, while bounded, it is not continuous, but continuity…
We consider a very general definition of BMO on a domain in $\mathbb{R}^n$, where the mean oscillation is taken with respect to a basis of shapes, i.e. a collection of open sets covering the domain. We examine the basic properties and…
In this paper we explore several applications of the recently introduced spaces of functions of bounded $\beta$-dimensional mean oscillation for $\beta \in (0,n]$ to regularity theory of critical exponent elliptic equations. We first show…
In the two-parameter setting, we say a function belongs to the mean little $BMO$, if its mean over any interval and with respect to any of the two variables has uniformly bounded mean oscillation. This space has been recently introduced by…
This announcement considers the following problem. We produce a bounded mean oscillation theorem for small distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$. A revision of this announcement is in the memoir preprint:…
We introduce the space of dyadic bounded mean oscillation functions $f$ defined on $[0,1]^n$ and study the behavior of the nonincreasing rearrangement of $f$, as an element of the space $BMO((0,1])$. We also study the analogous class of…
In this note, we study a quantitative extension of the John-Nirenberg inequality for the Hardy-Littlewood maximal function of a $\operatorname{BMO}$ function. More precisely, for every nonconstant locally integrable function $f$ such that…
The traditional approach to analyzing mean motion resonances is through canonical perturbation theory. While this is a powerful method, its generality leads to complicated combinations of variables that are challenging to interpret and…
This paper investigates the concept of harmonic functions of bounded mean oscillation, starting from John-Nirenberg's pioneering studies, under a renewed formalism, suitable for bringing out some fundamental properties inherent in it. In…
We consider various notions of vanishing mean oscillation on a (possibly unbounded) domain $\Omega \subset \mathbb{R}^n$, and prove an analogue of Sarason's theorem, giving sufficient conditions for the density of bounded Lipschitz…
In this paper we study sharp pointwise inequalities for maximal operators. In particular, we strengthen DeVore's inequality for the moduli of smoothness and a logarithmic variant of Bennett--DeVore--Sharpley's inequality for rearrangements.…
This paper studies functions of bounded mean oscillation (BMO) on metric spaces equipped with a doubling measure. The main result gives characterizations for mappings that preserve BMO. This extends the corresponding Euclidean results by…
We obtain sharp bounds for the monotonic rearrangement operator from "dyadic-type" classes to "continuous". In particular, for the $\mathrm{BMO}$ space and Muckenhoupt classes. The idea is to connect the problem with a simple geometric…
We prove the rigidity of rectifiable boundaries with constant distributional mean curvature in the Brendle class of warped product manifolds (which includes important models in General Relativity, like the deSitter--Schwarzschild and…
This paper starts by introducing results from geometric measure theory to prove symmetric decreasing rearrangement inequalities on $\mathbb{R}^n$, which give multiple proofs of the isoperimetric and P\'{o}lya-Szeg\H{o} inequalities. Then we…
We shall prove a rearrangement inequality in probability measure spaces in order to obtain sharp Leibniz-type rules of mean oscillations in Lp-spaces and rearrangement invariant Banach function spaces.
We prove a scalar-mean rigidity theorem for compact Riemannian manifolds with boundary in dimension less than five by developing a dimension reduction argument for mean curvature, which extends Schoen-Yau's dimension reduction argument for…
We introduce another notion of bounded logarithmic mean oscillation in the N-torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little BMO of Cotlar-Sadosky to the product BMO of…
In this paper, we define a notion of $\beta$-dimensional mean oscillation of functions $u: Q_0 \subset \mathbb{R}^d \to \mathbb{R}$ which are integrable on $\beta$-dimensional subsets of the cube $Q_0$: \begin{align*}…