Carlos Perez, Sheldy Ombrosi
We study mixed weighted weak-type inequalities for families of functions, which can be applied to study classical operators in harmonic analysis. Our main theorem extends the key result from D. Cruz-Uribe, J.M. Martell and C. Perez, Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math. Res. Not., V. 30, 2005, 1849-1871.
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Sawyer estimates of mixed type for operators associated to a critical radius function
本文研究临界半径函数框架下极大函数与 Schrödinger–Calderón–Zygmund 算子的混合加权不等式，并推广 Cruz-Uribe–Martell–Pérez 外推定理。
We study mixed weak estimates of Sawyer type for maximal operators associated to the family of Young functions $Φ(t)=t^r(1+\log^+t)^δ$, where $r\geq 1$ and $δ\geq 0$. More precisely, if $u$ and $v^r$ are $A_1$ weights, and $w$ is defined as $w=1/Φ(v^{-1})$ then the following estimate \[uw\left(\left\{x\in \mathbb{R}^n: \frac{M_Φ(fv)(x)}{v(x)} > t\right\}\right) \leq C\int_{\mathbb{R}^n} Φ\left(\frac{|f(x)|v(x)}{t}\right)u(x) \,dx\] holds for every positive $t$. This extends mixed estimates to a wider class of maximal operators, since when we put $r=1$ and $δ=0$ we recover a previous result for…
We study mixed weak type inequalities for the commutator $[b,T]$, where $b$ is a BMO function and $T$ is a Calderón-Zygmund operator. More precisely, we prove that for every $t>0$ \begin{equation*}%\label{tesis_teo2.2} uv(\{x\in\R^n: |\frac{[b,T](fv)(x)}{v(x)}|>t\})\leq C\int_{\R^n}ϕ(\frac{|f(x)|}{t})u(x)v(x)\,dx, \end{equation*} where $ϕ(t)=t(1+\log^{+}{t})$, $u\in A_1$ and $v\in A_{\infty}(u)$. Our technique involves the classical Calderón-Zygmund decomposition, which allow us to give a direct proof. We use this result to prove an analogous inequality for higher order commutators. We also ob…
We show that if $v\in A_\infty$ and $u\in A_1$, then there is a constant $c$ depending on the $A_1$ constant of $u$ and the $A_{\infty}$ constant of $v$ such that $$\Big\|\frac{ T(fv)} {v}\Big\|_{L^{1,\infty}(uv)}\le c\, \|f\|_{L^1(uv)},$$ where $T$ can be the Hardy-Littlewood maximal function or any Calderón-Zygmund operator. This result was conjectured in [IMRN, (30)2005, 1849--1871] and constitutes the most singular case of some extensions of several problems proposed by E. Sawyer and Muckenhoupt and Wheeden. We also improve and extends several quantitative estimates.
In this paper we prove mixed inequalities for the maximal operator $M_Φ$, for general Young functions $Φ$ with certain additional properties, improving and generalizing some previous estimates for the Hardy-Littlewood maximal operator proved by E. Sawyer. We show that given $r\geq 1$, if $u,v^r$ are weights belonging to the $A_1$-Muckenhoupt class and $Φ$ is a Young function as above, then the inequality \[uv^r\left(\left\{x\in \mathbb{R}^n: \frac{M_Φ(fv)(x)}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\] holds for every positive $t$. A motivation…
We study the boundedness problem for maximal operators in 3-dimensional Euclidean space associated to hypersurfaces given as the graph of $c+f$, where $f$ is a mixed homogeneous function which is smooth away from the origin and $c$ is a constant. Our result generalizes a corresponding theorem on mixed homogeneous polynomial functions by A. Iosevich and E. Sawyer.
We improve on several mixed weak type inequalities both for the Hardy-Littlewood maximal function and for Calderón-Zygmund operators. These type of inequalities were considered by Muckenhoupt and Wheeden and later on by Sawyer estimating the $L^{1, \infty}(uv)$ norm of $v^{-1}T(fv)$ for special cases. The emphasis is made in proving new and more precise quantitative estimates involving the $A_p$ or $A_\infty$ constants of the weights involved.
Extrapolation via Sawyer-type inequalities
本研究提出不依赖Rubio de Francia迭代算法的多变量受限弱型外推理论，依托加权Hardy-Littlewood极大算子的Sawyer型不等式实现，可恢复含端点结果的弱型外推格式，还拓展了特征函数弱型等价范围、填补了多变量算子权估计的文献空白。
It is known that that the endpoint mixed norm estimate $|| \, ||Tf(x,y)||_{L_{x}^{p}}||_{L_{y}^{\infty}} \lesssim || \, ||f(x,y)||_{L_{x}^{p}}||_{L_{y}^{\infty}}$ in general does not hold for Calderón-Zygmund operator $T$. In this article, we show that when $p=2$, even if we make the right hand side of the above estimate larger by replacing it with $ || \, ||e^{x^2+y^2}f(x,y)||_{L_{y}^{\infty}}||_{L_{x}^{\infty}} $, the estimate does not hold for the double Riesz transform given by the kernel $K(x,y)=\frac{xy}{2π(x^2+y^2)^{2}}$. As a consequence we will show that the mixed norm estimate $|| \,…
In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let $\vec{w}=(w_1,...,w_m)$ and $ν= w_1^\frac{1}{m}...w_m^\frac{1}{m}$, the main result of the paper sentences that under different conditions on the weights we can obtain $$\Bigg\| \frac{T(\vec f\,)(x)}{v}\Bigg\|_{L^{\frac{1}{m}, \infty}(νv^\frac{1}{m})} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(w_i)}}, $$ where $T$ is a multilinear Calderón-Zygmund operator. To obtain this result we first prove it for the $m$-fold product of the Hardy-Littlewood maximal operator…
A formulation of the Carleson embedding theorem in the multilinear setting is proved which allows to obtain a multilinear analogue of Sawyer's two weight theorem for the multisublinear maximal function \mathcal{M} introduced in Lerner et al. A multilinear version of the B_p theorem from Hytönen and Pérez is also obtained and a mixed A_{\vec{P}}-W_{\vec{P}}^{\infty} bound for \mathcal{M} is proved as well.
Mixed Weak type inequalities for pairs of weights related to the Hardy-Littlewood maximal funcion, Calderón-Zygmund operators and their commutators
该研究针对Hardy-Littlewood极大算子、Calderón-Zygmund算子及其交换子，推导了对应算子$S_v$的双权弱型估计，拓展了权不等式相关研究成果。
A Mixed-Resonance Counterexample to the Lukic Conjecture
针对Lukic猜想，通过GPT-5.6构造了重数为3的两临界点混合共振反例，该序列满足分解条件但加权熵为-∞，推翻了原猜想。
Let $L$ be a linear operator in $L^2(\mathbb{R}^n)$ which generates a semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy the Gaussian upper bound. In this paper, we investigate several kinds of weighted norm inequalities for the conical square function $S_{α,L}$ associated with an abstract operator $L$. We first establish two-weight inequalities including bump estimates, and Fefferman-Stein inequalities with arbitrary weights. We also present the local decay estimates using the extrapolation techniques, and the mixed weak type estimates corresponding Sawyer's conjecture by means of a Coifman…
We characterize strong type and weak type inequalities with two weights for positive operators on filtered measure spaces. These estimates are probabilistic analogues of two-weight inequalities for positive operators associated to the dyadic cubes in $\mathbb R^n$ due to Lacey, Sawyer and Uriarte-Tuero \cite{LaSaUr}. Several mixed bounds for the Doob maximal operator on filtered measure spaces are also obtained. In fact, Hytönen-Pérez type and Lerner-Moen type norm estimates for Doob maximal operator are established. Our approaches are mainly based on the construction of principal sets.
Sharp weighted inequalities for iterated commutators of a class of multilinear operators
本文在点态稀疏控制估计的一般假设下，系统研究了一类多线性算子迭代交换子的精确量化加权界，建立了局部衰减、弱$A_\infty$衰减及相关不等式，并应用于多种多线性算子。
Weighted variational inequalities for heat semigroups associated with Schrödinger operators related to critical radius functions
本文研究与临界半径函数相关的薛定谔算子热半群变分算子，建立其定量加权L^p界、更弱的双权bump条件、Sawyer猜想对应的混合弱型不等式以及新权重类下的限制弱型界，并给出相关权重类的特征刻画。
The initial-to-final-state inverse problem with unbounded potentials and Strichartz estimates
本文研究了具有无界势的初始到最终态逆问题，通过证明Strichartz估计和Carleman不等式，探讨了数据驱动预测在量子力学中的可行性，并指出Bourgain空间的局限性。
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