ISSN 0001-4346, Mathematical Notes, 2017, Vol. 101, No. 3, pp. 429–442. © Pleiades Publishing, Ltd., 2017. Sublinear Operators with Rough Kernel Generated by Calder ´ on–Zygmund Operators and Their Commutators on Generalized Morrey Spaces * F. G ¨ urb ¨ uz ** Department of Mathematics, Ankara University, Ankara, Turkey Received June 6, 2016 Abstract—The aim of this paper is to establish the boundedness of certain sublinear operators with rough kernel generated by Calder ´ on–Zygmund operators and their commutators on generalized Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. The Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example. DOI: 10.1134/S0001434617030051 Keywords: sublinear operator, Calder ´ on–Zygmund operator, rough kernel, generalized Mor- rey space, commutator, BMO. 1. INTRODUCTION AND MAIN RESULTS The classical Morrey spaces M p,λ were introduced by Morrey in [24] to study the local behavior of solutions of second order elliptic partial differential equations (PDEs). Later, there were many applications of Morrey space to the Navier-Stokes equations (see [22]), the Schr ¨ odinger equations (see [30]) and the elliptic problems with discontinuous coefficients (see [3], [10], [27]). We recall the definition of classical Morrey spaces M p,λ as M p,λ (R n )=  f : ‖f ‖ M p,λ (R n ) = sup x∈R n ,r>0 r −λ/p ‖f ‖ Lp(B(x,r)) < ∞  , where f ∈ L loc p (R n ), 0 ≤ λ ≤ n and 1 ≤ p< ∞. Note that M p,0 = L p (R n ) and M p,n = L ∞ (R n ). If λ< 0 or λ>n, then M p,λ =Θ, where Θ is the set of all functions equivalent to 0 on R n . We also denote by WM p,λ ≡ WM p,λ (R n ) the weak Morrey space of all functions f ∈ WL loc p (R n ) for which ‖f ‖ WM p,λ ≡‖f ‖ WM p,λ (R n ) = sup x∈R n ,r>0 r −λ/p ‖f ‖ WLp(B(x,r)) < ∞, where WL p (B(x, r)) denotes the weak L p -space of measurable functions f for which ‖f ‖ WLp(B(x,r)) ≡‖fχ B(x,r) ‖ WLp(R n ) = sup t>0 t |{y ∈ B(x, r): |f (y)| >t}| 1/p = sup 0<t≤|B(x,r)| t 1/p  fχ B(x,r)  ∗ (t) < ∞, where g ∗ denotes the nonincreasing rearrangement of a function g. Throughout the paper we assume that x ∈ R n and r> 0 and also let B(x, r) denotes the open ball centered at x of radius r, B C (x, r) denotes its complement and |B(x, r)| is the Lebesgue measure of the ball B(x, r) and |B(x, r)| = v n r n , where v n = |B(0, 1)|. ∗ The article was submitted by the author for the English version of the journal. ** E-mail: feritgurbuz84@hotmail.com 429