Abstract
LetTbe a multilinear square function with a kernel satisfying Dini(1) condition and letT⁎be the corresponding multilinear maximal square function. In this paper, first, we showed thatTis bounded fromL1×⋯×L1toL1/m,∞.Secondly, we obtained that if eachpi>1, thenTandT⁎are bounded fromLp1(ω1)×⋯×Lpm(ωm)toLp(νω→)and if there ispi=1, thenTandT⁎are bounded fromLp1(ω1)×⋯×Lpm(ωm)toLp,∞(νω→), whereνω→=∏i=1mωip/pi.Furthermore, we established the weighted strong and weak type boundedness forTandT⁎on weighted Morrey type spaces, respectively.
Highlights
Introduction and Main ResultsLet m(t) ∈ L∞ and a ∈ BMO
First, we showed that T is bounded from L1 × ⋅ ⋅ ⋅ × L1 to L1/m,∞
In 1978, Coifman and Meyer [1] introduced a class of multilinear operators as a multilinearization of Littlewood-Paley g-function as follows: B
Summary
In 2015, Xue et al [4] introduced and studied the weighted estimates for the following multilinear Littlewood-Paley g-function with convolution type kernel:. It should be pointed out that the methods used in [4, 11, 12] do not work for Littlewood-Paley operators with more general nonconvolution type kernels, for the reason that the estimates there rely heavily on the convolution type kernels and the well-known Marcinkiewicz function studied in [14]. Let T be a multilinear square function of type ω(t) and ω ∈ Dini(1). Let T∗ be a multilinear maximal square function of type ω(t) and ω ∈ Dini(1).
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