Local Characterizations of Besov and Triebel-Lizorkin Spaces with Variable Exponent

We introduce new Besov and Triebel-Lizorkin spaces with variable integrable exponent, which are different from those introduced by the second author early. Then we characterize these spaces by the boundedness of the local Hardy-Littlewood maximal operator on variable exponent Lebesgue space. Finally the completeness and the lifting property of these spaces are also given. 1. Introduction Variable exponent function spaces have attracted many attentions because of their applications in some aspects, such as partial differential equations with nonstandard growth [1], electrorheological fluids [2], and image restoration [3–5]. In fact, since the variable Lebesgue and Sobolev spaces were systemically studied by Ková？ik and Rákosník in [6], there are many spaces introduced, such as, Bessel potential spaces with variable exponent, Besov and Triebel-Lizorkin spaces with variable exponents, Morrey spaces with variable exponents, and Hardy spaces with variable exponent; see [7–20] and references therein. When the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue spaces, many results in classical harmonic analysis and function theory hold for the variable exponent case; see [21–23]. Let be a measurable function. Denote by the space of all measurable functions on such that for some with the norm Then is a Banach space with the norm . We will use the following notations: and . The set consists of all satisfying and . Moreover, we define to be the set of measurable functions on with the range in such that . Given , one can define the space as above. This is equivalent to defining it to be the set of all functions such that , where and . We also define a quasinorm on this space by . Let be a locally integrable function on ; the local variant of the Hardy-Littlewood maximal operator is given by for some constant . We denote the set of such that is bounded on . In 2013 Danelia et al. gave characterizations of , a vector-estimate for the local Hardy-Littlewood maximal operator if , and a Littlewood-Paley square-function characterization of the variable exponent Lebesgue spaces when belongs to in [24]. In 2001 Rychkov used the boundedness of the local Hardy-Littlewood maximal operator to prove a stronger result of the Peetre type for spaces and and gave the lifting property for these spaces in [25]. Motivated by the previous papers, the goal of this paper is to introduce new Besov and Triebel-Lizorkin spaces with variable exponent. To state our result, we need some notations. Throughout this paper denotes the Lebesgue measure for a measurable set