Multilinear Fourier Multipliers Research Articles

A weighted norm inequality for multilinear Fourier multiplier operator

In this paper, a weighted norm inequality with multiple-weight is established for the multilinear Fourier multiplier operator. Mathematics subject classification (2010): 42B15, 42B20.

Compactness of the Commutator of Multilinear Fourier Multiplier Operator on Weighted Lebesgue Space

LetTσbe the multilinear Fourier multiplier operator associated with multiplierσsatisfying the Sobolev regularity thatsupl∈ZσlWs1,…,sm(Rmn)<∞for somesk∈(n/2,n] (k=1,…,m). The authors prove that ifb1,…,bm∈BMO(Rn)andw⃗∈∏k=1m‍Apk/tk(tk=n/sk), then the commutatorTσ,Σbis bounded fromLp1(Rn,w1)×⋯×Lpm(Rn,wm)toLp(Rn,νw⃗). Moreover, the authors also prove that ifb1,…,bm∈VMO(Rn)andw⃗∈∏k=1m‍Apk/tk(tk=n/sk), then the commutatorTσ,Σbis compact operator fromLp1(Rn,w1)×⋯×Lpm(Rn,wm)toLp(Rn,νw⃗).

Weighted Morrey Estimates for Multilinear Fourier Multiplier Operators

The multilinear Fourier multipliers and their commutators with Sobolev regularity are studied. The purpose of this paper is to establish that these operators are bounded on certain product Morrey spacesLp,k(ℝn). Based on the boundedness of these operators fromLp1(ω1)×⋯×Lpm(ωm)toLp(∏j=1m‍ωp/pj), we obtained that they are also bounded fromLp1,k(ω1)×⋯×Lpm,k(ωm)toLp,k(∏j=1m‍ωp/pj), with0<k<1,1<pj<∞,1/p=1/p1+⋯+1/pm, andωj∈Apj,j=1,…,m.

A counterexample to weighted estimates for multilinear Fourier multipliers with Sobolev regularity

The problem whether the weighted estimates for multilinear Fourier multipliers with Sobolev regularity hold under weak condition on weights is considered.

Weighted version of Carleson measure and multilinear Fourier multiplier

Abstract In this paper, we first establish some results about weighted Carleson measure, and then apply them to multilinear Fourier multiplier. By using a different method from the one used by Fujita and Tomita (2012), we obtain some weighted result of multilinear multiplier operators by considering the missing parts of their results in the end-point cases such as from L 2(ω) × L ∞(ω) × L ∞(ω) to L 2(ω), using weighted Carleson measure theory and employing the multilinear interpolation theory.

Multilinear Fourier multipliers related to time–frequency localization

We consider multilinear multipliers associated in a natural way with localization operators. Boundedness and compactness results are obtained. In particular, we get a geometric condition on a subset A⊂R2d which guarantees that, for a fixed synthesis window ψ∈L2(Rd), the family of localization operators Lφ,ψA obtained when the analysis window φ varies on the unit ball of L2(Rd) is a relatively compact set of compact operators.

Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers

Let T be a multilinear operator which is bounded on certain products of unweighted Lebesgue spaces of Rn. We assume that the associated kernel of T satisfies some mild regularity condition which is weaker than the usual Hölder continuity of those in the class of multilinear Calderón–Zygmund singular integral operators. We then show the boundedness for T and the boundedness of the commutator of T with BMO functions on products of weighted Lebesgue spaces of Rn. As an application, we obtain the weighted norm inequalities of multilinear Fourier multipliers and of their commutators with BMO functions on the products of weighted Lebesgue spaces when the number of derivatives of the symbols is the same as the best known result for the multilinear Fourier multipliers to be bounded on the products of unweighted Lebesgue spaces.

A Hörmander type multiplier theorem for multilinear operators

In this paper, we prove a Hörmander type multiplier theorem for multilinear operators. As a corollary, we can weaken the regularity assumption for multilinear Fourier multipliers to assure the boundedness.