Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces. (31st December 2017) - Yang, Yanqi; Tao, Shuangping
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.