Weak type (1,1) estimates for some integral operators related to rough maximal functions

1996 ◽  
Vol 95 (1) ◽  
pp. 211-229
Author(s):  
Peter Sjögren ◽  
Fernando Soria
Keyword(s):  
Weak Type ◽  
Integral Operators ◽  
Maximal Functions

scholarly journals On the Limiting Weak-type Behaviors for Maximal Operators Associated with Power Weighted Measure

2019 ◽  
Vol 62 (02) ◽  
pp. 313-326 ◽  
Author(s):  
Xianming Hou ◽  
Huoxiong Wu
Keyword(s):  
Fractional Integral ◽  
Weak Type ◽  
Integral Operators ◽  
Unit Vector ◽  
Maximal Functions ◽  
Maximal Operators ◽  
Fractional Integral Operators ◽  
Littlewood Maximal Function ◽  
Image Position ◽  
Weighted Measure

AbstractLet $\unicode[STIX]{x1D6FD}\geqslant 0$ , let $e_{1}=(1,0,\ldots ,0)$ be a unit vector on $\mathbb{R}^{n}$ , and let $d\unicode[STIX]{x1D707}(x)=|x|^{\unicode[STIX]{x1D6FD}}dx$ be a power weighted measure on $\mathbb{R}^{n}$ . For $0\leqslant \unicode[STIX]{x1D6FC}<n$ , let $M_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6FC}}$ be the centered Hardy-Littlewood maximal function and fractional maximal functions associated with measure $\unicode[STIX]{x1D707}$ . This paper shows that for $q=n/(n-\unicode[STIX]{x1D6FC})$ , $f\in L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})$ , $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D706}\rightarrow 0+}\unicode[STIX]{x1D706}^{q}\unicode[STIX]{x1D707}(\{x\in \mathbb{R}^{n}:M_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6FC}}f(x)>\unicode[STIX]{x1D706}\})=\frac{\unicode[STIX]{x1D714}_{n-1}}{(n+\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D707}(B(e_{1},1))}\Vert f\Vert _{L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})}^{q}, &amp; &amp; \displaystyle \nonumber\\ \displaystyle \lim _{\unicode[STIX]{x1D706}\rightarrow 0+}\unicode[STIX]{x1D706}^{q}\unicode[STIX]{x1D707}\left(\left\{x\in \mathbb{R}^{n}:\left|M_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6FC}}f(x)-\frac{\Vert f\Vert _{L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})}}{\unicode[STIX]{x1D707}(B(x,|x|))^{1-\unicode[STIX]{x1D6FC}/n}}\right|>\unicode[STIX]{x1D706}\right\}\right)=0, &amp; &amp; \displaystyle \nonumber\end{eqnarray}$$ which is new and stronger than the previous result even if $\unicode[STIX]{x1D6FD}=0$ . Meanwhile, the corresponding results for the un-centered maximal functions as well as the fractional integral operators with respect to measure $\unicode[STIX]{x1D707}$ are also obtained.

scholarly journals Weak type estimates for the Hardy-Littlewood maximal functions

1974 ◽  
Vol 49 (3) ◽  
pp. 217-223 ◽  
Author(s):  
Luis Caffarelli ◽  
Calixto Calderón
Keyword(s):  
Weak Type ◽  
Maximal Functions ◽  
Weak Type Estimates

On a Class of Singular Integral Operators With Rough Kernels

2006 ◽  
Vol 49 (1) ◽  
pp. 3-10 ◽  
Author(s):  
Ahmad Al-Salman
Keyword(s):  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Maximal Functions ◽  
Rough Kernels ◽  
Block Spaces ◽  
Size Condition

AbstractIn this paper, we study the Lp mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on Lp provided that their kernels satisfy a size condition much weaker than that for the classical Calderón–Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions.

scholarly journals Weak Type Estimates of Variable Kernel Fractional Integral and Their Commutators on Variable Exponent Morrey Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Xukui Shao ◽  
Shuangping Tao
Keyword(s):  
Lebesgue Space ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Weak Type ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Variable Kernel ◽  
Fractional Integral Operators ◽  
Weak Type Estimates ◽  
Weak Morrey Spaces

In this paper, the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent weak Morrey spaces based on the results of Lebesgue space with variable exponent as the infimum of exponent function p(·) equals 1. The corresponding commutators generated by BMO and Lipschitz functions are considered, respectively.

scholarly journals Estimates of Fractional Integral Operators on Variable Exponent Lebesgue Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Canqin Tang ◽  
Qing Wu ◽  
Jingshi Xu
Keyword(s):  
Integral Operator ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Weak Type ◽  
Type Inequality ◽  
Integral Operators ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Variable Exponent Lebesgue Spaces

By some estimates for the variable fractional maximal operator, the authors prove that the fractional integral operator is bounded and satisfies the weak-type inequality on variable exponent Lebesgue spaces.

scholarly journals The weak-type (1,1) of Fourier integral operators of order –(n–1)/2

2004 ◽  
Vol 76 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Terence Tao
Keyword(s):  
Hardy Space ◽  
Integral Operator ◽  
Weak Type ◽  
Fourier Integral Operator ◽  
Integral Operators ◽  
Fourier Integral ◽  
Fourier Integral Operators ◽  
Main Ideas

AbstractLet T be a Fourier integral operator on Rn of order–(n–1)/2. Seeger, Sogge, and Stein showed (among other things) that T maps the Hardy space H1 to L1. In this note we show that T is also of weak-type (1, 1). The main ideas are a decomposition of T into non-degenerate and degenerate components, and a factorization of the non-degenerate portion.

scholarly journals Lp estimates for maximal functions along surfaces of revolution on product spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 1361-1373 ◽  
Author(s):  
Mohammed Ali ◽  
Musa Reyyashi
Keyword(s):  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Maximal Functions ◽  
Maximal Operators ◽  
Product Spaces ◽  
Surfaces Of Revolution ◽  
Lp Estimates ◽  
Block Space

Abstract This paper is concerned with establishing Lp estimates for a class of maximal operators associated to surfaces of revolution with kernels in Lq(Sn−1 × Sm−1), q > 1. These estimates are used in extrapolation to obtain the Lp boundedness of the maximal operators and the related singular integral operators when their kernels are in the L(logL)κ(Sn−1 × Sm−1) or in the block space $\begin{array}{} B^{0,\kappa-1}_ q \end{array}$(Sn−1 × Sm−1). Our results substantially improve and extend some known results.

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