Characterization of fractional maximal operator and its commutators on Orlicz spaces in the Dunkl setting
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Characterization of fractional maximal operator and its commutators on Orlicz spaces in the Dunkl setting Vagif Guliyev1,2,3
· Yagub Mammadov4,5 · Fatma Muslumova5
Received: 26 February 2020 / Revised: 26 February 2020 / Accepted: 1 September 2020 © Springer Nature Switzerland AG 2020
Abstract d On the Rd the Dunkl operators Dk, j j=1 are the differential-difference operators associated with the reflection group Zd2 on Rd . In this paper, in the setting Rd we find necessary and sufficient conditions for the boundedness of the fractional maximal operator Mα,k on Orlicz spaces L Φ,k (Rd ). As an application of this result we show that b ∈ BMOk (Rd ) if and only if the maximal commutator Mb,k is bounded on Orlicz spaces L Φ,k (Rd ). Keywords Fractional maximal operator · Orlicz space · Dunkl operator · Commutator · BMO Mathematics Subject Classification 42B20 · 42B25 · 42B35
1 Introduction It is well known that maximal operators play an important role in harmonic analysis (see [1]). Harmonic analysis associated to the Dunkl transform and the Dunkl differential-difference operator gives rise to convolutions with a relevant generalized
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Vagif Guliyev [email protected] Yagub Mammadov [email protected] Fatma Muslumova [email protected]
1
Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan
2
Department of Mathematics, Dumlupinar University, Kutahya, Turkey
3
S.M. Nikolskii Institute of Mathematics at RUDN University, Moscow, Russia
4
Department of Informatics, Nakhchivan State University, Nakhchivan, Azerbaijan
5
Nakhchivan Teacher-Training Institute, Nakhchivan, Azerbaijan
V. Guliyev et al.
d translation. In the setting Rd the Dunkl operators Dk, j j=1 , which are the differentialdifference operators introduced by Dunkl in [2]. These operators are very important in pure mathematics and in physics. They provide useful tools in the study of special functions with root systems. Dunkl operators are differential reflection operators associated with finite reflection groups which generalize the usual partial derivatives as well as the invariant differential operators of Riemannian symmetric spaces. They play an important role in harmonic analysis and the study of special functions of several variables. Among other applications, Dunkl operators are employed in the description of quantum integrable models of Calogero-Moser type. Also, there are stochastic processes associated with Dunkl Laplacians which generalize Dyson’s Brownian motion model. The Dunkl fractional maximal operator is of particular interest for harmonic analysis associated with root systems. However, the structure of the Dunkl translation makes the study difficult to which the heavy machinery of real analysis cannot be applied, such as covering methods, weighted inequalities, etc. The harmonic analysis of the Dunkl operator and Dunkl transform was developed in [3–8]. The fractional maximal function, the fractional integral and related topics associated with the Dunkl differential-difference operator have be
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