Abstract
In this paper, we study the properties of a bilinear multiplier space. We give a necessary condition for a continuous bounded function to be a bilinear multiplier on variable exponent Lebesgue spaces, and we prove the localization theorem of multipliers on variable exponent Lebesgue spaces. Moreover, we present a Mihlin-Hormander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces and give some applications.
Highlights
Given a non-empty open set p(x) such that⊂ Rn, we denote by P( ) the set of exponent functions≤ p– ≤ p+ < ∞, where p–( ) := essinf{p(x) : x ∈ } and p+( ) := esssup{p(x) : x ∈ }
In the case that p(·) ∈ P ( ), it is defined to be the set of all functions f satisfying |f (x)|p ∈ Lq(·)( ), q(x) = p(x)/p ∈ P( ) for some < p < p
We study some properties of the space of bilinear Fourier multipliers and the Mihlin-Hörmander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces
Summary
≤ p– ≤ p+ < ∞, where p–( ) := essinf{p(x) : x ∈ } and p+( ) := esssup{p(x) : x ∈ }. In [ ], Chen and Lu proved a Hörmander type multilinear theorem on weighted Lebesgue spaces when the Fourier multipliers were only assumed with limited smoothness. Huang and Xu [ ] obtained the boundedness of multilinear Calderón-Zygmund operators on variable exponent Lebesgue spaces. We study the weighted estimates of Tm with nearly the same conditions as in [ ], but on variable exponent Lebesgue spaces. Let m (p (·),p (·),p (·)) = Bm. A similar function space is defined in the following. ([ ]) Given a function K ∈ L loc(Rn \ { }), it is called a standard singular kernel if there exists a constant C > such that:. Let BM(Rn)(p(·), p(·)) denote the space of multipliers which correspond to bounded operators from Lp(·)(Rn) to Lp(·)(Rn). M–uf (ξ )g(η)M(ξ – η)e πi ξ+η,x dξ dηe πi u,x φ(u) du
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