Báo cáo toán học: A Characterization of Morrey Type Besov and Triebel-Lizorkin Spaces

Trong bài báo này tác giả đưa ra một đặc tính chức năng tối đa của s, s β, β Besov Morrey loại và Triebel Lizorkin không gian, M Bp, q (Rn) và M fp, q (Rn),là khái quát của Morrey loại nổi tiếng...
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Báo cáo toán học: "A Characterization of Morrey Type Besov and Triebel-Lizorkin Spaces" 9LHWQDP -RXUQDOVietnam Journal of Mathematics 33:4 (2005) 369–379 RI 0$7+(0$7,&6 ‹ 9$67 A Characterization of Morrey Type Besov and Triebel-Lizorkin Spaces* Jingshi Xu Department of Mathematics, Hunan Normal University, Changsha, 410081, China Received September 25, 2003 Revised June 1, 2005Abstract. In this paper the author gives a maximal function characterization of theMorrey-type Besov and Triebel-Lizorkin spaces, M Bp,q (Rn ) and M Fp,q (Rn ), which s,β s,βare the generalizations of the well-known Morrey-type spaces and the inhomogeneousBesov and Triebel-Lizorkin spaces.1. IntroductionIn recent years, the Morrey-type space continues to attract the attention ofmany authors. Many problems of partial differential equation based on Morreyspace and Morrey type Besov space have been considered in [1 - 6, 11, 16].Many results obtained parallel with the theory of standard Besov and Triebel-Lizorkin spaces and new applications have also been given. Actually, in [7]Mazzuato established some decompositions of Morrey type Besov spaces (in [7],they were called Besov-Morrey spaces) in terms of smooth wavelets, moleculesconcentrated on dyadic cubes, and atoms supported on dyadic cubes. In [10],Tang Lin and the author obtained some properties including lift properties anda Fourier multiplier theorem on Morrey type Besov and Triebel-Lizorkin spaces,and a discrete characterization of these spaces. Moreover, in [10] the authorsalso considered the boundedness of a class pseudo-differential operators on thesespaces.∗ The project was supported by the NNSF(60474070) of China.370 Jingshi Xu For readers interesting in standard Besov and Triebel-Lizorkin spaces andtheir applications, we recommend them Triebel’s books [12 - 15]. Motivated by [8], our purpose is to give a maximal function inequality onMorrey-type Besov and Triebel-Lizorkin spaces, which is a characterization ofMorrey-type Besov and Triebel-Lizorkin spaces. Before stating it, we recall somenotations and the definition of Morrey-type Besov and Triebel-Lizorkin spaces(see, e.g., [10]). Let Rn be the n-dimensional real Euclidean space. Let S (Rn ) be the Schwartzspace of all complex-valued rapidly decreasing infinitely differentiable functionson Rn . Let S (Rn ) be the set of all the tempered distribution on Rn . If ϕ ∈ S (Rn ),then ϕ denotes the Fourier transform of ϕ, and ϕ∨ denotes the inverse Fouriertransform of ϕ. p < ∞ and f ∈ Lq (Rn ), we say f ∈ Mq (Rn ) pDefinition 1. If 0 < q Locprovided that, for any ball BR,x centered at x with radius R, 1/q Rn(1/p−1/q) |f (y )|q dy < ∞. f =: sup p Mq x∈Rn ,R>0 BR,x Morrey spaces can be seen as a complement to Lp spaces. In fact, Mq ≡ Lp p p pand L ⊂ Mq . For j ∈ N we put ϕj (x) = 2nj ϕ(2j x), x ∈ Rn . Let functions A, θ ∈ S (Rn )satisfy the following conditions: |A(ξ )| > 0 on {|ξ | < 2}, supp A ⊂ {|ξ | < 4}, |θ(ξ )| > 0 on {1/2 < |ξ | < 2}, supp θ ⊂ {1/4 < |ξ | < 4}. Now the Morrey type Besov and Triebel-Lizorkin spaces can be defined asfollows.Definition 2. Let −∞ < s < ∞, 0 < q p < ∞, 0 < β ∞, and A, θ be asabove, then we define (i) The Morrey type Besov spaces asM Bp,q (Rn ) = f ∈ S (Rn ) : s,β {2sj θj ∗f }∞ = A∗f A Characterization of Morrey Type Besov and Triebel-Lizorkin Spaces 371 To make these space meaningful, the key point is to show that Definition 2is independent of the choice of functions A and θ. Actually, by the method ofTriebel’s book [12] we ...