Some Estimates for Commutators of Fractional Integrals Associated to Operators with Gaussian Kernel Bounds on Weighted Morrey Spaces

Abstract

Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) with Gaussian kernel bound, and let L −α/2 be the fractional integrals of L for 0 < α < n. In this paper, we will obtain some boundedness properties of commutators b, L −α/2 on the weighted Morrey spaces L p,κ (w) when the symbol b belongs to BM O(R n ) or homogeneous Lipschitz space. MSC(2010): 42B20; 42B35

Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces

“…Our results not only extend the results of [12] from (−△) to a general operator L, but also characterize the (weighted) Lipschitz spaces by means of the boundedness of [b, L −α/2 ] on the weighted Morrey spaces, which extend the results of [12] and [13]. The basic tool is based on a modification of sharp maximal function M ♯ L introduced by [6].…”

“…The weighted Morrey space was first introduced by Komori and Shirai [5]. For the general case, Wang [13]…”

Multilinear Singular and Fractional Integral Operators on Weighted Morrey Spaces

“…In 2009, Komori and Shirai [20] first defined the weighted Morrey spaces L p,κ (w) which could be viewed as an extension of weighted Lebesgue spaces, and studied the boundedness of the above classical operators in Harmonic Analysis on these weighted spaces. Recently, in [31][32][33][34][35][36][37][38], we have established the continuity properties of some other operators and their commutators on the weighted Morrey spaces L p,κ (w).…”

Weighted local Morrey spaces

“…Many researchers investigated the boundedness properties of the linear operators acting on weighted Morrey spaces. such as sublinear operator [4,12,15,35], singular integral operators [14,35,63], commutators [17,12,35,59,61], pseudo-differential operators [26], the square functions [11], Toeplitz operators [56], the fractional integral operators [12,28,31,32] and fractional integrals associated to operators [51,54,55] including the related commutators. Applications to partial differential equations can be found in [8,19,50].…”