Characterizations for the fractional maximal operator and its commutators in generalized weighted Morrey spaces on Carno
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Characterizations for the fractional maximal operator and its commutators in generalized weighted Morrey spaces on Carnot groups V. S. Guliyev1,2,3 Received: 29 January 2020 / Revised: 20 February 2020 / Accepted: 28 February 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we shall give a characterization for the strong and weak type Spanne type boundedness of the fractional maximal operator Mα , 0 ≤ α < Q on Carnot group G on generalized weighted Morrey spaces M p,ϕ (G, w), where Q is the homogeneous dimension of G. Also we give a characterization for the Spanne type boundedness of the fractional maximal commutator operator Mb,α on generalized weighted Morrey spaces. Keywords Carnot group · Fractional maximal operator · Generalized weighted Morrey space · Commutator · BM O · Homogeneous dimension Mathematics Subject Classification Primary 42B25 · 42B35 · 43A15 · 43A80
1 Introduction The classical Morrey spaces were introduced by Morrey [29] to study the local behavior of solutions to second-order elliptic partial differential equations. Moreover, various Morrey spaces are defined in the process of study. The author, Mizuhara and Nakai [9,28,32] introduced generalized Morrey spaces M p,ϕ (Rn ) (see, also [11,12,20,37]). Komori and Shirai [27] defined weighted Morrey spaces L p,κ (w). The author in [14] gave a concept of the generalized weighted Morrey spaces M p,ϕ (Rn , w) which could be viewed as extension of both M p,ϕ (Rn ) and L p,κ (w). In [14], the boundedness of
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V. S. Guliyev [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, RUDN University, Moscow, Russia 0123456789().: V,-vol
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V. S. Guliyev
the classical operators and their commutators in spaces M p,ϕ (Rn , w) was also studied, see also [4,15–18,22,23,26,33,35,36]. The spaces M p,ϕ (Rn , w) defined by the norm f M p,ϕ (Rn ,w) ≡
sup
x∈Rn ,r >0
ϕ(x, r )−1 w(B(x, r ))−1/ p f L p (B(x,r ),w) ,
where the function ϕ is a positive measurable function on Rn × (0, ∞) and w is a non-negative measurable function on Rn . Carnot groups appear in quantum physics and many parts of mathematics, including Fourier analysis, several complex variables, geometry and topology. Analysis on the groups is also motivated by their role as the simplest and the most important model in the general theory of vector fields satisfying Hormander’s condition. The simplest examples of the Carnot groups are Euclidean space Rn , Heisenberg group Hn and (Heisenberg)-type groups introduced by Kaplan [25]. For x ∈ G and r > 0, let D(x, r ) denote the G- ball centered at x of radius r and D(x, r ) denote its complement. Let f ∈ L loc 1 (G). The fractional maximal operator Mα is defined by α
Mα f (x) = sup |D(x, r )|−1+ Q r >0
D(x,r )
| f (y)|dy,
where |D(x, t)| is the Haar measure of the G- ball D(x, t). The fractional maximal commutator generated by b ∈ L loc 1 (G) and the o
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