Multilinear fractional integral operators on non-homogeneous metric measure spaces

Huajun Gong,Chen Xu,Rulong Xie

doi:10.1186/s13660-016-1218-6

Abstract

In this paper, the boundedness in Lebesgue spaces for multilinear fractional integral operators and commutators generated by multilinear fractional integrals with an $\operatorname{RBMO}(\mu)$ function on non-homogeneous metric measure spaces is obtained.

Highlights

Introduction and preliminariesA measure μ is called a doubling measure, if there exists a positive constant C such that μ(B(x, l)) ≤ Cμ(B(x, l)), for all x ∈ supp μ and all l >, which is the main condition in homogeneous spaces
Many researchers obtained the boundedness of operators on the non-homogeneous metric measure spaces; see, e.g., [ – ]
The bilinear theory for Calderón-Zygmund operators was studied by Coifman-Meyers [ ], the boundedness on Lebesgue spaces or Hardy spaces for multilinear singular integrals was proved by Gorafakos-Torres [, ]

Summary

It follows from

≤ C b ∗ b ∗Mr,( ) Iα, (g , g ) (x). ≤ C b ∗Mr,( ) [b , Iα, ](g , g ) (x). For F , it follows from (i) of Definition . The condition of λ, and Hölder’s inequality that. Similar to the estimate of F , G ≤ C b ∗ b ∗Mp(α ,/( ))g (x)Mp(α /,( ))g (x). Iα, b – b (z)g , g (z) dμ(z) ≤ CMr,( ) [b , Iα, ]g , g (x) It follows from Hölder’s inequality that b (z) – mR(b ) Iα, (g , g )(z) dμ(z) ≤ C b ∗Mr,( ) Iα, (g , g ) (x). For E , as z ∈ R, noting that R is a doubling ball, it follows from (i) of Definition . Similar to the estimate of mR(E ), mR(E ) + mR(E ) ≤ CKB(α,R/ ) b ∗Mp(α /,( ))g (x)Mp(α /,( ))g (x).

Proof of

Conclusion