New Weighted Norm Inequalities for Pseudodifferential Operators and Their Commutators

This paper is dedicated to study weighted inequalities for pseudodifferential operators with amplitudes and their commutators by using the new class of weights and the new BMO function space BMO∞ which are larger than the Muckenhoupt class of weights and classical BMO space BMO, respectively. The obtained results therefore improve substantially some well-known results. 1. Introduction and the Main Results For a pseudodifferential operator given formally by where the amplitude satisfies certain growth conditions. The boundedness of pseudodifferential operators has been studied extensively by many mathematicians; see, for example, [1–7] and the references therein. One of the most interesting problems is studying the weighted norm inequalities for pseudodifferential operators and their commutators with BMO function; see, for example, [5–9]. In this paper we consider the following classes of symbols and amplitudes (in what follows we set ). Definition 1. Let and , and .(a)We say when for each triple of multi-indices , , and there exists a constant such that (b)We say when for each triple of multi-indices , , and there exists a constant such that Definition 2. Let and , and .(a)We say when for each pair of multi-indices and there exists a constant such that (b)We say when for each multi-indices there exists a constant such that It is easy to see that , , , and . The classes and were studied in [3, 8]. For further information about these two classes, we refer the reader to, for example, [3, 10]. The class was introduced by [11], and it is the natural generalization of the class . This class is much rougher than that considered in [6, 7]. The amplitude class in Definition 1 is rough in the variable, but smooth in the variable. This is smaller than the class introduced in [5] but includes the class . The aim of this paper is to study the weighted norm inequalities for pseudodifferential operators and their commutators by using the new BMO functions and the new class of weights. Firstly, we would like to give brief definitions on the new class of weights and the new BMO function space (we refer to Section 2 for details). The new classes of weights for , where , , is the set of those weights satisfying for all ball . We denote that . It is easy to see that the new class is strictly larger than the Muckenhoupt class . Indeed, for example, the weight with belongs to the class , but it is not in , for , see, for example, [12]. The new BMO space with is defined as a set of all locally integrable functions satisfying where and . A norm for , denoted by , is given by