scholarly journals New Herz Type Besov and Triebel-Lizorkin Spaces with Variable Exponents

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Baohua Dong ◽  
Jingshi Xu
Keyword(s):  
Maximal Operator ◽  
Maximal Functions ◽  
Variable Exponents ◽  
Herz Spaces ◽  
Vector Valued ◽  
Littlewood Maximal Operator

The authors establish the boundedness of vector-valued Hardy-Littlewood maximal operator in Herz spaces with variable exponents. Then new Herz type Besov and Triebel-Lizorkin spaces with variable exponents are introduced. Finally, characterizations of these new spaces by maximal functions are given.

Variable exponent Herz type Besov and Triebel–Lizorkin spaces

2018 ◽  
Vol 25 (1) ◽  
pp. 135-148 ◽  
Author(s):  
Jingshi Xu ◽  
Xiaodi Yang
Keyword(s):  
Maximal Operator ◽  
Variable Exponent ◽  
Maximal Functions ◽  
Herz Spaces ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractWe establish the boundedness of the vector-valued Hardy–Littlewood maximal operator in variable exponent Herz spaces, which were introduced by Samko in [33]. We also introduce variable exponent Herz type Besov and Triebel–Lizorkin spaces and give characterizations of these new spaces by maximal functions.

Endpoint Regularity of Multisublinear Fractional Maximal Functions

2017 ◽  
Vol 60 (3) ◽  
pp. 586-603 ◽  
Author(s):  
Feng Liu ◽  
Huoxiong Wu
Keyword(s):  
Maximal Operator ◽  
Maximal Functions ◽  
Maximal Operators ◽  
One Dimensional ◽  
Regularity Properties ◽  
The One ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractIn this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy–Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on the vector-valued function with all ƒ j being BV-functions.

scholarly journals Local Muckenhoupt class for variable exponents

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Toru Nogayama ◽  
Yoshihiro Sawano
Keyword(s):  
Maximal Operator ◽  
New Method ◽  
Variable Exponents ◽  
Weighted Inequality ◽  
Lebesgue Spaces ◽  
Extension Method ◽  
Extrapolation Technique ◽  
Muckenhoupt Class ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractThis work extends the theory of Rychkov, who developed the theory of $A_{p}^{\mathrm{loc}}$ A p loc weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class $A_{p(\cdot )}^{\mathrm{loc}}$ A p ( ⋅ ) loc is defined. The weighted inequality for the local Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponents is proven. Cruz-Uribe SFO, Fiorenza, and Neugebauer considered the Muckenhoupt class for Lebesgue spaces with variable exponents. However, due to the setting of variable exponents, a new method for extending weights is needed. The proposed extension method differs from that by Rychkov. A passage to the vector-valued inequality is realized by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. Additionally, a theory of extrapolation adapted to our class of weights is also obtained.

scholarly journals A Weak Type Vector-Valued Inequality for the Modified Hardy–Littlewood Maximal Operator for General Radon Measure on ℝn

2020 ◽  
Vol 8 (1) ◽  
pp. 261-267
Author(s):  
Yoshihiro Sawano
Keyword(s):  
Radon Measure ◽  
Maximal Operator ◽  
Weak Type ◽  
Strong Type ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractThe aim of this paper is to prove the weak type vector-valued inequality for the modified Hardy– Littlewood maximal operator for general Radon measure on ℝn. Earlier, the strong type vector-valued inequality for the same operator and the weak type vector-valued inequality for the dyadic maximal operator were obtained. This paper will supplement these existing results by proving a weak type counterpart.

A note on the behavior of the Dunkl maximal operator

2018 ◽  
Vol 9 (4) ◽  
pp. 237-246 ◽  
Author(s):  
Luc Deleaval
Keyword(s):  
Maximal Operator ◽  
Mathematical Society ◽  
Vector Valued ◽  
Littlewood Maximal Operator

Abstract This note is a contribution to the Proceedings of the Conference of the Tunisian Mathematical Society CSMT 2017. After briefly revisiting the case of the standard Hardy–Littlewood maximal operator, we will discuss the behavior of the Dunkl maximal operator in both the scalar and vector-valued cases.

scholarly journals Sparse domination implies vector-valued sparse domination

2022 ◽  
Author(s):  
Emiel Lorist ◽  
Zoe Nieraeth
Keyword(s):  
Function Space ◽  
Hilbert Transform ◽  
Maximal Operator ◽  
Banach Function Space ◽  
Orlicz Spaces ◽  
Function Spaces ◽  
Banach Function Spaces ◽  
Bilinear Hilbert Transform ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractWe prove that scalar-valued sparse domination of a multilinear operator implies vector-valued sparse domination for tuples of quasi-Banach function spaces, for which we introduce a multilinear analogue of the $${{\,\mathrm{UMD}\,}}$$ UMD condition. This condition is characterized by the boundedness of the multisublinear Hardy-Littlewood maximal operator and goes beyond examples in which a $${{\,\mathrm{UMD}\,}}$$ UMD condition is assumed on each individual space and includes e.g. iterated Lebesgue, Lorentz, and Orlicz spaces. Our method allows us to obtain sharp vector-valued weighted bounds directly from scalar-valued sparse domination, without the use of a Rubio de Francia type extrapolation result. We apply our result to obtain new vector-valued bounds for multilinear Calderón-Zygmund operators as well as recover the old ones with a new sharp weighted bound. Moreover, in the Banach function space setting we improve upon recent vector-valued bounds for the bilinear Hilbert transform.

scholarly journals Uniform boundedness of Szász-Mirakjan-Kantorovich operators in Morrey spaces with variable exponents

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2109-2121
Author(s):  
Yoshihiro Sawano ◽  
Xinxin Tian ◽  
Jingshi Xu
Keyword(s):  
Maximal Operator ◽  
Uniform Boundedness ◽  
Morrey Spaces ◽  
Variable Exponents ◽  
Lebesgue Spaces ◽  
Littlewood Maximal Operator ◽  
Kantorovich Operators ◽  
Uniformly Bounded

The Sz?sz-Mirakjan-Kantorovich operators and the Baskakov-Kantorovich operators are shown to be controlled by the Hardy-Littlewood maximal operator. The Sz?sz-Mirakjan-Kantorovich operators and the Baskakov-Kantorovich operators turn out to be uniformly bounded in Lebesgue spaces and Morrey spaces with variable exponents when the integral exponent is global log-H?lder continuous.

On the regularity of the Hardy-Littlewood maximal operator on subdomains of ℝn

2010 ◽  
Vol 53 (1) ◽  
pp. 211-237 ◽  
Author(s):  
Hannes Luiro
Keyword(s):  
Maximal Operator ◽  
Littlewood Maximal Operator

AbstractWe establish the continuity of the Hardy-Littlewood maximal operator on W1,p(Ω), where Ω ⊂ ℝn is an arbitrary subdomain and 1 < p < ∞. Moreover, boundedness and continuity of the same operator is proved on the Triebel-Lizorkin spaces Fps,q (Ω) for 1 < p,q < ∞ and 0 < s < 1.

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