Sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized local Morrey spaces (original) (raw)

2015, Journal of Inequalities and Applications

In this paper, we will study the boundedness of a large class of sublinear operators with rough kernel T on the generalized local Morrey spaces LM {x 0 } p,ϕ , for s ≤ p, p = 1 or p < s, where ∈ L s (S n-1) with s > 1 are homogeneous of degree zero. In the case when b ∈ LC {x 0 } p,λ is a local Campanato spaces, 1 < p < ∞, and T ,b be is a sublinear commutator operator, we find the sufficient conditions on the pair (ϕ 1 , ϕ 2) which ensures the boundedness of the operator T ,b from one generalized local Morrey space LM {x 0 } p,ϕ 1 to another LM {x 0 } p,ϕ 2. In all cases the conditions for the boundedness of T are given in terms of Zygmund-type integral inequalities on (ϕ 1 , ϕ 2), which do not make any assumptions on the monotonicity of ϕ 1 , ϕ 2 in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudo-differential operators, Littlewood-Paley operators, Marcinkiewicz operators, and Bochner-Riesz operators. MSC: 42B20; 42B25; 42B35