Abstract
This paper is concerned with the norm estimates for the multilinear singular integral operators and their commutators formed by BMO functions on the weighted amalgam spacesLvw→q,Lpαℝn. Some criterions of boundedness for such operators inLvw→q,Lpαℝnare given. As applications, the norm inequalities for the multilinear Calderón-Zygmund operators and multilinear singular integrals with nonsmooth kernels as well as the corresponding commutators onLvw→q,Lpαℝnare obtained.
Highlights
Let Rn (n ≥ 2) be the n-dimensional Euclidean space equipped with the Euclidean norm | ⋅ | and the Lebesgue measure dx
We remark that the amalgam spaces (Lq, Lp)(Rn) were introduced by Fofana in [1] in connection with the study of the continuity of the fractional maximal operator of Hardy-Littlewood and of the Fourier transformation in Rn
In [1], Fofana considered the subspace (Lq, Lp)α(Rn) of (Lq, Lp)(Rn), which consists of measurable functions f such that for 1 ≤ α ≤ ∞, f(Lq,Lp)α(Rn) sup r>0
Summary
Let Rn (n ≥ 2) be the n-dimensional Euclidean space equipped with the Euclidean norm | ⋅ | and the Lebesgue measure dx. For 1 ≤ p, q ≤ ∞; the amalgam spaces (Lq, Lp)(Rn) of Lp(Rn) and Lq(Rn) are denoted by the set of all measurable functions f : Rn → C, which are locally in Lq(Rn) and satisfy. A suitable modification version for p = ∞ or q = ∞ By the definitions, it is clear ( see [1]) that (Lq, Lq)(Rn) = Lq(Rn), (Lq, L∞)α(Rn) = Lq, (nq/α)(Rn), where Lq,λ(Rn), with 1 ≤ q < ∞ and 0 < λ < n, is the classical Morrey space that consists of measurable functions f : Rn → C such that. The authors in [11] showed that the AP⃗ conditions are the largest classes of weights because all mlinear Calderon-Zygmund operators are bounded on the weighted Lebesgue spaces.
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