Sharp mutliplicative inequalities with $\mathrm{BMO}$ $\mathrm{I}$

Weighted little bmo and two-weight inequalities for Journé commutators

We characterize the boundedness of the commutators $[b, T]$ with biparameter Journé operators $T$ in the two-weight, Bloom-type setting, and express the norms of these commutators in terms of a weighted little $bmo$ norm of the symbol $b$. Specifically, if $μ$ and $λ$ are biparameter $A_p$ weights, $ν:= μ^{1/p}λ^{-1/p}$ is the Bloom weight, and $b$ is in $bmo(ν)$, then we prove a lower bound and testing condition $\|b\|_{bmo(ν)} \lesssim \sup \| [b, R_k^1 R_l^2]: L^p(μ) \rightarrow L^p(λ)\|$, where $R_k^1$ and $R_l^2$ are Riesz transforms acting in each variable. Further, we prove that for suc…

John-Nirenberg inequalities for parabolic BMO

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 c…

Two Weight Inequalities for Iterated Commutators with Calderón-Zygmund Operators

Given a Calderón-Zygmund operator $T$, a classic result of Coifman-Rochberg-Weiss relates the norm of the commutator $[b, T]$ with the BMO norm of $b$. We focus on a weighted version of this result, obtained by Bloom and later generalized by Lacey and the authors, which relates $\| [b, T] : L^p(\mathbb{R}^n; μ) \to L^p(\mathbb{R}^n; λ) \|$ to the norm of $b$ in a certain weighted BMO space determined by $A_p$ weights $μ$ and $λ$. We extend this result to higher iterates of the commutator and recover a one-weight result of Chung-Pereyra-Perez in the process.

Two-Weight Inequalities for Commutators with Fractional Integral Operators

In this paper we investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $μ,λ\in A_{p,q}$ and $α/n+1/q=1/p$, the norm $\| [b,I_α]:L^p(μ^p)\to L^q(λ^q) \|$ is equivalent to the norm of $b$ in the weighted BMO space $BMO(ν)$, where $ν=μλ^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.

Quantitative John-Nirenberg inequalities at different scales

We provide an abstract estimate of the form \[ \|f-f_{Q,μ}\|_{X \left(Q,\frac{\mathrm{d} μ}{Y(Q)}\right)}\leq c(μ,Y)ψ(X)\|f\|_{\mathrm{BMO}(\mathrm{d}μ)} \] for all cubes $Q$ in $\mathbb{R}^n$ and every function $f\in \mathrm{BMO}(\mathrm{d}μ)$, where $μ$ is a doubling measure in $\mathbb{R}^n$, $Y$ is some positive functional defined on cubes, $\|\cdot \|_{X \left(Q,\frac{\mathrm{d} w}{w(Q)}\right)}$ is a sufficiently good quasi-norm and $c(μ,Y)$ and $ψ(X)$ are positive constants depending on $μ$ and $Y$, and $X$, respectively. That abstract scheme allows us to recover the sharp estimate \[ \…

Several inequalities concerning interpolation in classical Fourier analysis

In this note, we establish several interpolation inequalities in $\mathbb R^n$ in the Lebesgue spaces and Morrey spaces. By using the classical Calderon--Zygmund decomposition, we will reprove that $L^{p}(\mathbb R^n)\cap\mathrm{BMO}(\mathbb R^n)\subset L^{q}(\mathbb R^n)$ for all $q$ with $p<q<\infty$, where $1\leq p<\infty$. We also reprove that there exists a constant $C(p,q,n)$ depending on $p,q,n$ such that the following inequality \begin{equation*} \|f\|_{L^q}\leq C(p,q,n)\cdot\big(\|f\|_{L^p}\big)^{p/q}\cdot\big(\|f\|_{\mathrm{BMO}}\big)^{1-p/q} \end{equation*} holds for all $f\in L^{p}…

Lorentz、Morrey 与 BMO 空间中的若干插值不等式

Some interpolation inequalities in Lorentz, Morrey and BMO spaces

本文研究 Lorentz、Morrey 与 BMO 空间之间的插值不等式，引入 Lorentz–Morrey 空间并建立其与 BMO 交空间的连续嵌入及最优增长阶，进而得到可用于二阶椭圆和抛物方程弱解研究的双线性估计。

Parabolic weighted norm inequalities and partial differential equations

We investigate parabolic Muckenhoupt weights and functions of bounded mean oscillation (BMO) related to nonlinear parabolic partial differential equations. The main result gives a full characterization of weak and strong type weighted norm inequalities for parabolic forward in time maximal operators. In addition, we give a Jones type factorization result for the parabolic Muckenhoupt weights and a Coifman-Rochberg type characterization of the parabolic BMO from Moser's seminal paper through parabolic Muckenhoupt weights and maximal functions.

Sharp transference principle for $\mathrm{BMO}$ and $A_p$

We provide a version of the transference principle. It says that certain optimization problems for functions on the circle, the interval, and the line have the same answers. In particular, we show that the sharp constants in the John--Nirenberg inequalities for naturally defined $\mathrm{BMO}$-spaces on the circle, the interval, and the line coincide. The same principle holds true for the Reverse Hölder inequality for Muckenhoupt weights.

Sharp norm inequalities for commutators of classical operators

We prove several sharp weighted norm inequalities for commutators of classical operators in harmonic analysis. We find sufficient $A_p$-bump conditions on pairs of weights $(u,v)$ such that $[b,T]$, $b\in BMO$ and $T$ a singular integral operator (such as the Hilbert or Riesz transforms), maps $L^p(v)$ into $L^p(u)$. Because of the added degree of singularity, the commutators require a "double log bump" as opposed to that of singular integrals, which only require single log bumps. For the fractional integral operator $I_\al$ we find the sharp one-weight bound on $[b,I_\al]$, $b\in BMO$, in ter…

A new look at the John-Nirenberg and John-Stromberg theorems for BMO. Lecture Notes

We develop some techniques for studying various versions of the function space BMO. Special cases of one of our results give alternative proofs of the celebrated John- Nirenberg inequality and of related inequalities due to John and to Wik. Our approach enables us to pose a simply formulated "geometric" question, for which an affirmative answer would lead to a version of the John-Nirenberg inequality with dimension free constants. A more detailed summary of the main ideas and results of this paper can be found at http://www.math.technion.ac.il/~mcwikel/bmo/CwikSaghShvaSummary.pdf

分数阶Gagliardo–Nirenberg不等式：逐点估计、尖锐渐近行为及最优目标空间

Fractional Gagliardo--Nirenberg Inequalities: Pointwise Estimates,Sharp Asymptotics, and Optimal Target Spaces

该研究针对分数阶差分算子建立逐点估计，推导含BMO端点的分数阶Gagliardo–Nirenberg不等式，确定其渐近因子最优阶，完全刻画重排不变目标空间并解答了开放问题，结果在非对角和BMO情形下为全新成果。

薛定谔型算子交换子的加权混合不等式

Weighted mixed inequalities for commutators of Schrödinger type operators

本文研究了薛定谔背景下奇异积分算子交换子的加权混合不等式，针对 $(s,δ)$ 型算子和 $\text{BMO}(ρ)$ 符号给出估计，并推广了先前结果，同时引入 $A_p^ρ$ 权作为 Muckenhoupt 类的扰动。

齐次树BMO中的尖锐常数与极值函数

Sharp Constants and Extremizers in Homogeneous Tree BMO

该研究针对齐次q元滤子上的实值均值中心化L¹-BMO建立有限树形式的尖锐不等式，通过归约定理等方法，在绝对值映射等应用中得到严格小于2的BMO半范数界，阐明了相关机制与差异。

矩阵加权极大算子与乘积BMO的向量值估计

Vector valued estimates for matrix weighted maximal operators and product $\mathrm{BMO}$

本文研究了矩阵加权极大算子在复数空间上的向量值估计，利用Bownik和Cruz-Uribe的凸体值函数外推结果，并应用到双参数paraproducts的上界估计中。

平坦 Muckenhoupt 权的渐近锐反向 Hölder 不等式

Asymptotically sharp reverse Hölder inequalities for flat Muckenhoupt weights

本文给出平坦 Muckenhoupt 权的渐近锐反向 Hölder 不等式，通过精确覆盖论证避免 BMO 方法，并在一维情形证明单侧与双侧权的最优性，同时推广至弱型和多参数情形。

$A_\infty$到$A_p$的渐近精确嵌入及其在皮卡-索博列夫不等式中的应用

Asymptotically sharp embedding of $A_\infty$ into $A_p$ for flat weights and applications to Poincaré-Sobolev inequalities

研究$A_\infty$到$A_p$的渐近精确嵌入，通过Fujii-Wilson常数接近1时的渐进行为，得到参数p趋近于1的结论，并应用于皮卡-索博列夫不等式。