scholarly journals Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function

2021 ◽  
Vol 47 (1) ◽  
pp. 203-235
Author(s):  
Feng Liu ◽  
Qingying Xue ◽  
Kôzô Yabuta
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Maximal Function ◽  
Pointwise Estimates ◽  
Boundary Values ◽  
First Order ◽  
Littlewood Maximal Function ◽  
Weak Derivatives ◽  
Derivatives Of

Let \(\Omega\) be a subdomain in \(\mathbb{R}^n\) and \(M_\Omega\) be the local Hardy-Littlewood maximal function. In this paper, we show that both the commutator and the maximal commutator of \(M_\Omega\) are bounded and continuous from the first order Sobolev spaces \(W^{1,p_1}(\Omega)\) to \(W^{1,p}(\Omega)\) provided that \(b\in W^{1,p_2}(\Omega)\), \(1<p_1,p_2,p<\infty\) and \(1/p=1/p_1+1/p_2\). These are done by establishing several new pointwise estimates for the weak derivatives of the above commutators. As applications, the bounds of these operators on the Sobolev space with zero boundary values are obtained.

scholarly journals A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function

2002 ◽  
Vol 65 (2) ◽  
pp. 253-258 ◽  
Author(s):  
Hitoshi Tanaka
Keyword(s):  
Sobolev Space ◽  
Maximal Function ◽  
60Th Birthday ◽  
One Dimensional ◽  
Partial Derivatives ◽  
First Order ◽  
Littlewood Maximal Function ◽  
The One ◽  
Derivatives Of

Dedicated to Professor Kôzô Yabuta on the occasion of his 60th birthdayJ. Kinnunen proved that of P > 1, d ≤ 1 and f is a function in the Sobolev space W1,P(Rd), then the first order weak partial derivatives of the Hardy-Littlewood maximal function ℳf belong to LP(Rd). We shall show that, when d = 1, Kinnunen's result can be extended to the case where P = 1.

On the Regularity of the Multisublinear Maximal Functions

2015 ◽  
Vol 58 (4) ◽  
pp. 808-817 ◽  
Author(s):  
Feng Liu ◽  
Huoxiong Wu
Keyword(s):  
Sobolev Spaces ◽  
Maximal Function ◽  
Maximal Operator ◽  
Maximal Functions ◽  
Partial Derivatives ◽  
First Order ◽  
Derivatives Of

AbstractThis paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.

scholarly journals Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

2017 ◽  
Vol 5 (1) ◽  
pp. 98-115 ◽  
Author(s):  
Eero Saksman ◽  
Tomás Soto
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Metric Space ◽  
Besov Spaces ◽  
Metric Spaces ◽  
Function Spaces ◽  
First Order ◽  
Trace Theorems ◽  
Euclidean Setting ◽  
Regular Subset

Abstract We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the first order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces defined intrinsically on F. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.

The Hilbert-Schmidt Property for Embedding Maps between Sobolev Spaces

1966 ◽  
Vol 18 ◽  
pp. 1079-1084 ◽  
Author(s):  
Colin Clark
Keyword(s):  
Euclidean Space ◽  
Sobolev Space ◽  
Sobolev Spaces ◽  
Dimensional Euclidean Space ◽  
Generalized Derivatives ◽  
Bounded Region ◽  
Complete Continuity ◽  
Completely Continuous ◽  
Natural Embedding ◽  
Derivatives Of

Let H0m(Ω) denote the so-called Sobolev space consisting of functions denned on a region Ω in n-dimensional Euclidean space, which together with their generalized derivatives of all orders ⩽m belong to , and which vanish in a certain sense on the boundary ∂Ω. (Precise definitions are given in the next section.) For each pair m, k of non-negative integers the inclusion H0m+k(Ω) ⊂ H0m(Ω) defines a natural “embedding” map. For the case of a bounded region Ω it is well known that these maps are completely continuous, and even, for sufficiently large k, of Hilbert-Schmidt type. We have discussed complete continuity in the case of unbounded regions in an earlier paper; here we consider conditions on Ω which imply the Hilbert-Schmidt property for embeddings.

scholarly journals Weighted composition operators on Orlicz-Sobolev spaces

2007 ◽  
Vol 83 (3) ◽  
pp. 327-334
Author(s):  
Subhash C. Arora ◽  
Gopal Datt ◽  
Satish Verma
Keyword(s):  
Euclidean Space ◽  
Sobolev Space ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Weighted Composition Operators ◽  
Sufficient Condition ◽  
First Order ◽  
Singular Transformation ◽  
Weighted Composition ◽  
Derivatives Of

AbstractFor an open subset Ω of the Euclidean space Rn, a measurable non-singular transformation T: Ω → Ω and a real-valued measurable function u on Rn, we study the weighted composition operator uCτ: f ↦ u · (f º T) on the Orlicz-Sobolev space W1·Ψ (Ω) consxsisting of those functions of the Orlicz space LΨ (Ω) whose distributional derivatives of the first order belong to LΨ (Ω). We also discuss a sufficient condition under which uCτ is compact.

scholarly journals The Poisson equation in homogeneous Sobolev spaces

2004 ◽  
Vol 2004 (36) ◽  
pp. 1909-1921
Author(s):  
Tatiana Samrowski ◽  
Werner Varnhorn
Keyword(s):  
Sobolev Space ◽  
Poisson Equation ◽  
Sobolev Spaces ◽  
Exterior Domain ◽  
Smooth Boundary ◽  
Boundary Values ◽  
Homogeneous Sobolev Space ◽  
The Poisson Equation ◽  
Homogeneous Sobolev Spaces ◽  
External Forces

We consider Poisson's equation in ann-dimensional exterior domainG(n≥2)with a sufficiently smooth boundary. We prove that for external forces and boundary values given in certainLq(G)-spaces there exists a solution in the homogeneous Sobolev spaceS2,q(G), containing functions being local inLq(G)and having second-order derivatives inLq(G)Concerning the uniqueness of this solution we prove that the corresponding nullspace has the dimensionn+1, independent ofq.

Existence of Extremal Functions for Higher-Order Caffarelli–Kohn–Nirenberg Inequalities

2018 ◽  
Vol 18 (3) ◽  
pp. 543-553 ◽  
Author(s):  
Mengxia Dong
Keyword(s):  
Sobolev Space ◽  
Weighted Sobolev Space ◽  
Higher Order ◽  
Extensive Study ◽  
Extremal Functions ◽  
Compact Embedding ◽  
First Order ◽  
Nirenberg Inequality ◽  
Higher Order Derivative ◽  
Derivatives Of

Abstract Though there has been an extensive study on first-order Caffarelli–Kohn–Nirenberg inequalities, not much is known for the existence of extremal functions for higher-order ones. The higher-order derivative of the Caffarelli–Kohn–Nirenberg inequality established by Lin [14] states \bigg{(}\int_{\mathbb{R}^{N}}\lvert D^{j}u|^{r}\frac{dx}{|x|^{s}}\bigg{)}^{% \frac{1}{r}}\leq C\bigg{(}\int_{\mathbb{R}^{N}}\lvert D^{m}u|^{p}\frac{dx}{|x|% ^{\mu}}\bigg{)}^{\frac{a}{p}}\bigg{(}\int_{\mathbb{R}^{N}}\lvert u|^{q}\frac{% dx}{|x|^{\sigma}}\bigg{)}^{\frac{1-a}{q}}, where {C=C(p,q,r,\mu,\sigma,s,m,j)} and {p,q,r,\mu,\sigma,s,m,j} are parameters satisfying some balanced conditions. The main purpose of this paper is to establish the existence of extremal functions for a family of this higher-order derivatives of Caffarelli–Kohn–Nirenberg inequalities under numerous circumstances of parameters. Moreover, we study the compactness of the weighted Sobolev space for higher-order derivatives and prove that {\dot{H}^{m,p}_{\mu}(\Omega)\cap L^{q}_{\sigma}(\Omega)\hookrightarrow\dot{H}^% {j,r}_{s}(\Omega)} is a compact embedding within some range of the parameters.

scholarly journals A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Feng Liu ◽  
Lei Xu
Keyword(s):  
Bounded Variation ◽  
Sobolev Spaces ◽  
Maximal Function ◽  
Maximal Operator ◽  
Functions Of Bounded Variation ◽  
Two Dimensional ◽  
Littlewood Maximal Function ◽  
Regularity Properties ◽  
Littlewood Maximal Operator

We investigate the regularity properties of the two-dimensional one-sided Hardy-Littlewood maximal operator. We point out that the above operator is bounded and continuous on the Sobolev spaces Ws,p(R2) for 0≤s≤1 and 1<p<∞. More importantly, we establish the sharp boundedness and continuity for the discrete two-dimensional one-sided Hardy-Littlewood maximal operator from l1(Z2) to BV(Z2). Here BV(Z2) denotes the set of all functions of bounded variation on Z2.

Complete monotonicity of entropy production in chemical reaction

1981 ◽  
Vol 46 (2) ◽  
pp. 452-456
Author(s):  
Milan Šolc
Keyword(s):  
Entropy Production ◽  
Equilibrium Constant ◽  
Chemical Reaction ◽  
Relative Entropy ◽  
Order Reaction ◽  
Complete Monotonicity ◽  
First Order Reaction ◽  
First Order ◽  
Completely Monotonic ◽  
Derivatives Of

The successive time derivatives of relative entropy and entropy production for a system with a reversible first-order reaction alternate in sign. It is proved that the relative entropy for reactions with an equilibrium constant smaller than or equal to one is completely monotonic in the whole definition interval, and for reactions with an equilibrium constant larger than one this function is completely monotonic at the beginning of the reaction and near to equilibrium.

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