scholarly journals On logarithmic bounds of maximal sparse operators

2019 ◽  
Vol 294 (3-4) ◽  
pp. 1271-1281
Author(s):  
Grigori A. Karagulyan ◽  
Michael T. Lacey
Keyword(s):  
Sparse Operators

WEIGHTED WEAK TYPE ENDPOINT ESTIMATES FOR THE COMPOSITIONS OF CALDERÓN–ZYGMUND OPERATORS

2019 ◽  
Vol 109 (3) ◽  
pp. 320-339 ◽  
Author(s):  
GUOEN HU
Keyword(s):  
Weak Type ◽  
Composite Operator ◽  
Weighted Estimate ◽  
Sparse Operators ◽  
Image Position

AbstractLet $T_{1}$, $T_{2}$ be two Calderón–Zygmund operators and $T_{1,b}$ be the commutator of $T_{1}$ with symbol $b\in \text{BMO}(\mathbb{R}^{n})$. In this paper, by establishing new bilinear sparse dominations and a new weighted estimate for bilinear sparse operators, we prove that the composite operator $T_{1}T_{2}$ satisfies the following estimate: for $\unicode[STIX]{x1D706}>0$ and weight $w\in A_{1}(\mathbb{R}^{n})$, $$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log \bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx,\nonumber\end{eqnarray}$$ while the composite operator $T_{1,b}T_{2}$ satisfies $$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1,b}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}^{2}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log ^{2}\bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx.\nonumber\end{eqnarray}$$

scholarly journals Data-Driven Corrections of Partial Lotka–Volterra Models

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1313
Author(s):  
Rebecca E. Morrison
Keyword(s):  
Chemical Species ◽  
Active Species ◽  
Data Driven ◽  
Volterra Models ◽  
Discrepancy Model ◽  
Partial Models ◽  
Sparse Operators ◽  
Partial Interaction ◽  
Interacting Systems ◽  
Physical Information

In many applications of interacting systems, we are only interested in the dynamic behavior of a subset of all possible active species. For example, this is true in combustion models (many transient chemical species are not of interest in a given reaction) and in epidemiological models (only certain subpopulations are consequential). Thus, it is common to use greatly reduced or partial models in which only the interactions among the species of interest are known. In this work, we explore the use of an embedded, sparse, and data-driven discrepancy operator to augment these partial interaction models. Preliminary results show that the model error caused by severe reductions—e.g., elimination of hundreds of terms—can be captured with sparse operators, built with only a small fraction of that number. The operator is embedded within the differential equations of the model, which allows the action of the operator to be interpretable. Moreover, it is constrained by available physical information and calibrated over many scenarios. These qualities of the discrepancy model—interpretability, physical consistency, and robustness to different scenarios—are intended to support reliable predictions under extrapolative conditions.

Global regularity estimates for non-divergence elliptic equations on weighted variable Lebesgue spaces

2020 ◽  
pp. 2050014
Author(s):  
The Quan Bui ◽  
The Anh Bui ◽  
Xuan Thinh Duong
Keyword(s):  
Elliptic Equations ◽  
Global Regularity ◽  
Order Elliptic Equation ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Regularity Estimates ◽  
Sparse Operators ◽  
Second Order Elliptic Equation ◽  
Bmo Coefficients ◽  
Weighted Variable

This paper is to prove global regularity estimates for solutions to the second-order elliptic equation in non-divergence form with BMO coefficients in a [Formula: see text] domain on weighted variable exponent Lebesgue spaces. Our approach is based on the representations for the solutions to the non-divergence elliptic equations and the domination technique by sparse operators in harmonic analysis.

Stein’s Square Function $$G_\alpha $$ G α and Sparse Operators

2016 ◽  
Vol 27 (2) ◽  
pp. 1624-1635
Author(s):  
María J. Carro ◽  
Carlos Domingo-Salazar
Keyword(s):  
Square Function ◽  
Sparse Operators

scholarly journals Weighted estimates for the Calderón commutator

2019 ◽  
Vol 63 (1) ◽  
pp. 169-192
Author(s):  
Jiecheng Chen ◽  
Guoen Hu
Keyword(s):  
Weak Type ◽  
Weighted Estimates ◽  
Sparse Operators ◽  
Image Position

AbstractIn this paper the authors consider the weighted estimates for the Calderón commutator defined by \mathcal{C}_{m+1, A}(a_1,\ldots,a_{m};f)(x)={\rm p. v.} \displaystyle\int_{\mathbb{R}}\displaystyle\frac{P_2(A; x, y)\prod\nolimits_{j=1}^m(A_j(x)-A_j(y))}{(x-y)^{m+2}}f(y){\rm d}y,with P2(A;x, y) = A(x) − A(y) − A′(y)(x − y) and A′ ∈ BMO(ℝ). Dominating this operator by multi(sub)linear sparse operators, the authors establish the weighted bounds from $L^{p_1}(\mathbb {R},w_1) \times \cdots \times L^{p_{m+1}}(\mathbb {R},w_{m+1})$ to $L^{p}(\mathbb {R},\nu _{\vec {\kern 1pt w}})$, with p1, …, pm+1 ∈ (1, ∞), 1/p = 1/p1 + · · · + 1/pm+1, and $\vec {\kern 1pt w}=(w_1, \ldots , w_{m+1})\in A_{\vec {P}}(\mathbb {R}^{m+1})$. The authors also obtain the weighted weak type endpoint estimates for $\mathcal {C}_{m+1, A}$.

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