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

Aydin S Balakishiyev,Vagif S Guliyev,Ayhan Serbetci,Ferit Gurbuz

doi:10.1186/s13660-015-0582-y

Abstract

In this paper, we will study the boundedness of a large class of sublinear operators with rough kernel $T_{\Omega}$ on the generalized local Morrey spaces $LM_{p,\varphi}^{\{x_{0}\}}$ , for $s' \le p$ , $p \neq1$ or $p < s$ , where $\Omega\in L_{s}(S^{n-1})$ with $s>1$ are homogeneous of degree zero. In the case when $b \in LC_{p,\lambda}^{\{x_{0}\}}$ is a local Campanato spaces, $1< p<\infty$ , and $T_{\Omega,b}$ be is a sublinear commutator operator, we find the sufficient conditions on the pair $(\varphi_{1},\varphi_{2})$ which ensures the boundedness of the operator $T_{\Omega,b}$ from one generalized local Morrey space $LM_{p,\varphi_{1}}^{\{x_{0}\}}$ to another $LM_{p,\varphi_{2}}^{\{x_{0}\}}$ . In all cases the conditions for the boundedness of $T_{\Omega}$ are given in terms of Zygmund-type integral inequalities on $(\varphi_{1},\varphi_{2})$ , which do not make any assumptions on the monotonicity of $\varphi_{1}$ , $\varphi_{2}$ 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.

Highlights

For x ∈ Rn and r >, let B(x, r) denote the open ball centered at x of radius r, B(x, r) denote its complement and |B(x, r)| is the Lebesgue measure of the ball B(x, r)
Let b be a locally integrable function on Rn, we shall define the commutators generated by singular integral operators with rough kernels and b as follows:
We prove the boundedness of the operators T from one generalized local Morrey space LMp{x,φ } to another LMp{x,φ }, < p < ∞, and from the space LM {x,φ } to the weak space WLM {x,φ }

Summary

Introduction

For x ∈ Rn and r > , let B(x, r) denote the open ball centered at x of radius r, B(x, r) denote its complement and |B(x, r)| is the Lebesgue measure of the ball B(x, r). Let b be a locally integrable function on Rn, we shall define the commutators generated by singular integral operators with rough kernels and b as follows:. Recall that in the doctoral thesis [ ] by Guliyev (see [ – ]) introduced the local Morrey-type space LMpθ,w given by f LMpθ,w = w(r) f Lp(B( ,r)) Lθ ( ,∞) < ∞, where w is a positive measurable function defined on ( , ∞). The main purpose of [ ] ( of [ – ]) is to give some sufficient conditions for the boundedness of fractional integral operators and singular integral operators defined on homogeneous Lie groups in the local Morrey-type space LMpθ,w. (see [ – ]), some necessary and sufficient conditions for the boundedness of fractional maximal operators, fractional integral operators, and singular integral operators in local Morrey-type spaces LMpθ,w were given. We recover the local Morrey space LMp{x,λ } and weak local

Morrey space

Bq sup kn q f χk

Morrey spaces

By the Fubini theorem we have

As in the proof of