Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces. (31st December 2017)

• Record Type: Journal Article
• Authors: Yang, Yanqi; Tao, Shuangping
• Other Names: Hudzik Henryk Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2017(2017)
• Publication Date: 2017-12-31
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2017/4340805
• Languages: English
• ISSNs: 2314-8896

Abstract

Abstract : Let T be the singular integral operator with variable kernel defined by T f ( x ) = p.v. ∫ R n ( Ω ( x, x - y ) / x - y n ) f ( y ) d y and let D γ ( 0 ≤ γ ≤ 1 ) be the fractional differentiation operator. Let T ⁎ and T ♯ be the adjoint of T and the pseudoadjoint of T, respectively. In this paper, the authors prove that T D γ - D γ T and ( T ⁎ - T ♯ ) D γ are bounded, respectively, from Morrey-Herz spaces M K ˙ p, 1 α, λ ( R n ) to the weak Morrey-Herz spaces W M K ˙ p, 1 α, λ ( R n ) by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product T 1 T 2 and the pseudoproduct T 1 ∘ T 2 are also given.