Bilinear Multiplier Research Articles

Sobolev smoothing estimates for bilinear maximal operators with fractal dilation sets

Given a hypersurface S⊂R2d, we study the bilinear averaging operator that averages a pair of functions over S, as well as more general bilinear multipliers of limited decay and various maximal analogs. Of particular interest are bilinear maximal operators associated to a fractal dilation set E⊂[1,2]; in this case, the boundedness region of the maximal operator is associated to the geometry of the hypersurface and various notions of the dimension of the dilation set. In particular, we determine Sobolev smoothing estimates at the exponent L2×L2→L2 using Fourier-analytic methods, which allow us to deduce additional Lp improving bounds for the operators and sparse bounds and their weighted corollaries for the associated multi-scale maximal functions. We also extend the method to study analogues of these questions for the triangle averaging operator and biparameter averaging operators. In addition, some necessary conditions for boundedness of these operators are obtained.

Weighted bilinear multiplier theorems in Dunkl setting via singular integrals

Abstract The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood–Paley type theorems and weighted inequalities for multilinear Calderón–Zygmund operators in Dunkl setting are also proved.

Linear and bilinear Fourier multipliers on Orlicz modulation spaces

Let Φi,Ψi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi _i, \\Psi _i$$\\end{document} be Young functions, ωi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _i$$\\end{document} be weights and MωiΦi,Ψi(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{\\Phi _i,\\Psi _i}_{\\omega _i}(\\mathbb {R} ^{d})$$\\end{document} be the corresponding Orlicz modulation spaces for i=1,2,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1,2,3$$\\end{document}. We consider linear (respect. bilinear) multipliers on Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R} ^{d}$$\\end{document}, that is bounded measurable functions m(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m(\\xi )$$\\end{document} (respect. m(ξ,η)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m(\\xi ,\\eta )$$\\end{document}) on Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R} ^{d}$$\\end{document} (respect. R2d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R} ^{2d}$$\\end{document}) such that Tm(f)(x)=∫Rdf^(ξ)m(ξ)e2πi⟨ξ,x⟩dξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} T_m(f)(x)=\\int _{\\mathbb {R} ^{d}}{\\hat{f}}(\\xi ) m(\\xi )e^{2\\pi i \\langle \\xi , x\\rangle }d\\xi \\end{aligned}$$\\end{document}(respect. Bm(f1,f2)(x)=∫Rd∫Rdf1^(ξ)f2^(η)m(ξ,η)e2πi⟨ξ+η,x⟩dξdη\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} B_m(f_1,f_2)(x)=\\int _{\\mathbb {R} ^{d}}\\int _{\\mathbb {R} ^{d}} \\hat{f_1}(\\xi ) \\hat{f_2}(\\eta )m(\\xi ,\\eta )e^{2\\pi i \\langle \\xi +\\eta , x\\rangle }d\\xi d\\eta \\end{aligned}$$\\end{document}define a bounded linear (respect. bilinear) operator from Mω1Φ1,Ψ1(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{\\Phi _1,\\Psi _1}_{\\omega _1}(\\mathbb {R} ^{d})$$\\end{document} to Mω2Φ2,Ψ2(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{\\Phi _2,\\Psi _2}_{\\omega _2}(\\mathbb {R} ^{d})$$\\end{document} (respect. Mω1Φ1,Ψ1(Rd)×Mω2Φ2,Ψ2(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{\\Phi _1,\\Psi _1}_{\\omega _1}(\\mathbb {R} ^{d})\ imes M^{\\Phi _2,\\Psi _2}_{\\omega _2}(\\mathbb {R} ^{d})$$\\end{document} to Mω3Φ3,Ψ3(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{\\Phi _3,\\Psi _3}_{\\omega _3}(\\mathbb {R} ^{d})$$\\end{document}). In this paper we study some properties of these spaces and give methods to generate linear and bilinear multipliers between Orlicz modulation spaces.

Bilinear multipliers on Orlicz spaces on locally compact groups

Bilinear multipliers on Orlicz spaces on locally compact groups

Local and multilinear noncommutative de Leeuw theorems

Let Γ<G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma < G$$\\end{document} be a discrete subgroup of a locally compact unimodular group G. Let m∈Cb(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m\\in C_b(G)$$\\end{document} be a p-multiplier on G with 1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le p < \\infty $$\\end{document} and let Tm:Lp(G^)→Lp(G^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{m}: L_p({\\widehat{G}}) \\rightarrow L_p({\\widehat{G}})$$\\end{document} be the corresponding Fourier multiplier. Similarly, let Tm|Γ:Lp(Γ^)→Lp(Γ^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{m \\vert _\\Gamma }: L_p({\\widehat{\\Gamma }}) \\rightarrow L_p({\\widehat{\\Gamma }})$$\\end{document} be the Fourier multiplier associated to the restriction m|Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m|_{\\Gamma }$$\\end{document} of m to Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}. We show that c(supp(m|Γ))‖Tm|Γ:Lp(Γ^)→Lp(Γ^)‖≤‖Tm:Lp(G^)→Lp(G^)‖,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} c( {{\\,\ extrm{supp}\\,}}( m|_{\\Gamma } ) ) \\Vert T_{m \\vert _\\Gamma }: L_p({\\widehat{\\Gamma }}) \\rightarrow L_p({\\widehat{\\Gamma }}) \\Vert \\le \\Vert T_{m }: L_p({\\widehat{G}}) \\rightarrow L_p({\\widehat{G}}) \\Vert , \\end{aligned}$$\\end{document}for a specific constant 0≤c(U)≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le c(U) \\le 1$$\\end{document} that is defined for every U⊆Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U \\subseteq \\Gamma $$\\end{document}. The function c quantifies the failure of G to admit small almost Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, c(Γ)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(\\Gamma ) =1$$\\end{document} when G has small almost Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-invariant neighbourhoods. Our result thus extends the de Leeuw restriction theorem from Caspers et al. (Forum Math Sigma 3(e21):59, 2015) as well as de Leeuw’s classical theorem (Ann Math 81(2):364–379, 1965). For real reductive Lie groups G we provide an explicit lower bound for c in terms of the maximal dimension d of a nilpotent orbit in the adjoint representation. We show that c(BρG)≥ρ-d/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(B_\\rho ^G) \\ge \\rho ^{-d/4}$$\\end{document} where BρG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_\\rho ^G$$\\end{document} is the ball of g∈G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\in G$$\\end{document} with ‖Adg‖<ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert {{\\,\ extrm{Ad}\\,}}_g \\Vert < \\rho $$\\end{document}. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear de Leeuw restriction theorem for pairs Γ<G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma <G$$\\end{document} with c(Γ)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(\\Gamma ) = 1$$\\end{document}. We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.

Paraproducts for bilinear multipliers associated with convex sets

We prove bounds in the local L^2 range for exotic paraproducts motivated by bilinear multipliers associated with convex sets. One result assumes an exponential boundary curve. Another one assumes a higher order lacunarity condition.

Bilinear multipliers of small Lebesgue spaces

Let $G$ be a locally compact abelian metric group with Haar measure $\lambda $ and $\hat{G}$ its dual with Haar measure $\mu ,$ and $\lambda ( G) $ is finite. Assume that$~1<p_{i}<\infty $, $p_{i}^{\prime }=\frac{ p_{i}}{p_{i}-1}$, $( i=1,2,3) $ and $\theta \geq 0$. Let $ L^{(p_{i}^{\prime },\theta }( G) ,$ $( i=1,2,3) $ be small Lebesgue spaces. A bounded measurable function $m( \xi ,\eta ) $ defined on $\hat{G}\times \hat{G}$ is said to be a bilinear multiplier on $G$ of type $[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] _{\theta }$ if the bilinear operator $B_{m}$ associated with the symbol $m$, \begin{equation} B_{m}(f,g) ( x) =\sum_{s\in \hat{G} }\sum_{t\in \hat{G}}\hat{f}(s) \hat{g}(t) m(s,t) \langle s+t,x\rangle \end{equation} defines a bounded bilinear operator from $L^{(p_{1}^{\prime },\theta }( G) \times L^{(p_{2}^{\prime },\theta }( G) $ into $ L^{(p_{3}^{\prime },\theta }(G) $. We denote by $BM_{\theta } [ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] $ the space of all bilinear multipliers of type $[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] _{\theta }$. In this paper, we discuss some basic properties of the space $BM_{\theta }[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] $ and give examples of bilinear multipliers.

The boundedness of bilinear Fourier multiplier operators in Besov spaces with variable exponents

The boundedness of bilinear Fourier multiplier operators in Besov spaces with variable exponents

Maximal operators associated with bilinear multipliers of limited decay

Results analogous to those proved by Rubio de Francia [28] are obtained for a class of maximal functions formed by dilations of bilinear multiplier operators of limited decay. We focus our attention on L2 × L2 → L1 estimates. We discuss two applications: the boundedness of the bilinear maximal Bochner-Riesz operator and of the bilinear spherical maximal operator. For the latter we improve the known results in [1] by reducing the dimension restriction from n ≥ 8 to n ≥ 4.

Bilinear Fourier multipliers and the rate of decay of their derivatives

We investigate two types of boundedness criteria for bilinear Fourier multiplier operators with symbols with bounded partial derivatives of all (or sufficiently many) orders. Theorems of the first type explicitly prescribe only a certain rate of decay of the symbol itself while theorems of the second type require, in addition, the same rate of decay of all derivatives of the symbol. We show that even though these two types of bilinear multiplier theorems are closely related, there are some fundamental differences between them which arise in limiting cases. Also, since theorems of the latter type have so far been studied mainly in connection with the more general class of bilinear pseudodifferential operators, we revisit them in the special case of bilinear Fourier multipliers, providing also some improvements of the existing results in this setting.

New Young Inequalities and Applications

We establish upper bounds for the convolution operator acting between interpolation spaces. This gives new Young inequalities in the context of Lorentz–Karamata spaces, grand Lebesgue spaces and small Lebesgue spaces besides many other known results. Furthermore, we use this abstract Young inequality to prove a bilinear interpolation theorem for limit interpolation methods. Finally, we show applications to the study of bilinear multipliers.

Notes on bilinear multipliers on Orlicz spaces

AbstractLet and Φ3 be Young functions and let , and be the corresponding Orlicz spaces. We say that a function defined on is a bilinear multiplier of type if defines a bounded bilinear operator from to . We denote by the space of all bilinear multipliers of type and investigate some properties of such a class. Under some conditions on the triple we give some examples of bilinear multipliers of type . We will focus on the case and get necessary conditions on to get non‐trivial multipliers in this class. In particular we recover some of the known results for Lebesgue spaces.

Bilinear Multipliers on Banach Function Spaces

Let X1,X2,X3 be Banach spaces of measurable functions in L0(R) and let m(ξ,η) be a locally integrable function in R2. We say that m∈BM(X1,X2,X3)(R) if Bm(f,g)(x)=∫R∫Rf^(ξ)g^(η)m(ξ,η)e2πi&lt;ξ+η,x&gt;dξdη, defined for f and g with compactly supported Fourier transform, extends to a bounded bilinear operator from X1×X2 to X3. In this paper we investigate some properties of the class BM(X1,X2,X3)(R) for general spaces which are invariant under translation, modulation, and dilation, analyzing also the particular case of r.i. Banach function spaces. We shall give some examples in this class and some procedures to generate new bilinear multipliers. We shall focus on the case m(ξ,η)=M(ξ-η) and find conditions for these classes to contain nonzero multipliers in terms of the Boyd indices for the spaces.

Approaching Bilinear Multipliers via a Functional Calculus

We propose a framework for bilinear multiplier operators defined via the (bivariate) spectral theorem. Under this framework, we prove Coifman–Meyer type multiplier theorems and fractional Leibniz rules. Our theory applies to bilinear multipliers associated with the discrete Laplacian on mathbb {Z}^d, general bi-radial bilinear Dunkl multipliers, and to bilinear multipliers associated with the Jacobi expansions.

The Hörmander multiplier theorem, II: The bilinear local $$L^2$$ L 2 case

We use wavelets of tensor product type to obtain the boundedness of bilinear multiplier operators on $$\mathbb R^n\times \mathbb R^n$$ associated with Hormander multipliers on $$\mathbb R^{2n}$$ with minimal smoothness. We focus on the local $$L^2$$ case and we obtain boundedness under the minimal smoothness assumption of n / 2 derivatives. We also provide counterexamples to obtain necessary conditions for all sets of indices.

Bilinear Multipliers of Weighted Lorentz Spaces and Variable Exponent Lorentz Spaces

Let ω1, ω2 be slowly increasing functions and let ω3 be weight function on ℝn. In section 2 we define a bilinear multiplier from L(p1, q1, ω1dμ) (ℝn) × L(p2, q2, ω2dμ) (ℝn) to L(p3, q3, ω3dμ) (ℝn) by a bounded operator Bm, where 1≤ p1, p2, p3, q1, q2, q3 < ∞ and m (ξ,η) is a bounded, measurable function on ℝn × ℝn. We denote the space of bilinear multipliers of this type by BM (L(p1, q1, ω1dμ) × L(p2, q2, ω2dμ), L(p3, q3, ω3dμ)), and study of the basic properties of this space. We give methods of construction examples of bilinear multipliers. Similarly in section 3, by using variable exponent Lorentz space, we define the bilinear multipliers from L( p1 (x), q1 (x)) × L( p2 (x), q2 (x)) to L( p3 (x), q3 (x)) and discuss basic properties of the space of bilinear multipliers BM (L( p1 (x), q1 (x)) × L( p2 (x), q2 (x)), L( p3 (x), q3 (x))).

Kato-Ponce inequalities on weighted and variable Lebesgue spaces

We prove fractional Leibniz rules and related commutator estimates in the settings of weighted and variable Lebesgue spaces. Our main tools are uniform weighted estimates for sequences of square-function-type operators and a bilinear extrapolation theorem. We also give applications of the extrapolation theorem to the boundedness on variable Lebesgue spaces of certain bilinear multiplier operators and singular integrals.

Resolution of Peller's problem concerning Koplienko–Neidhardt trace formulae

A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product S 2 ×S 2 of two copies of the Hilbert-Schmidt classes into the trace class S 1 is established in terms of linear Schur multipliers acting on the space S 1 of all compact operators. Using this formula, we resolve Peller's problem on Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function f with a bounded second derivative, a self-adjoint (unbounded) operator A and a self-adjoint operator B 2 S 2 such that f(A + B) f(A) d dt (f(A + tB)) � t=0 / 2 S 1 .

A class of bilinear multipliers given by Littlewood–Paley decomposition

In this paper we study a particular class of bilinear multipliers which are given by Littlewood–Paley decompositions. In the first part of the paper, we show that if ϕ(ξ−η) is a bilinear multiplier for (p,q,r), 1≤p,q≤∞ satisfying the Hölder's condition 1p+1q=1r and have support inside [0,1), then its periodization ϕ♯(ξ)=∑j∈Zϕ(ξ−j) is also a bilinear multiplier for the same triplet (p,q,r). Further, we show that for a given triplet (p,q,r) of exponents outside the local L2-range, there exists sequence {ϕj}j∈Z of uniformly bounded bilinear multipliers so that the function σ(ξ)=∑j∈Zϕj(ξ) is not a bilinear multiplier for the triplet (p,q,r). In the second part, we describe several results for bilinear multipliers of the type m(ξ,η) which are similar to the first part in nature. In particular, we point out that the results described by P. Honzik (2014) [13] can be generalized to a more general setting.

The spaces of bilinear multipliers of weighted Lorentz type modulation spaces

Abstract Fix a nonzero window g ∈ 𝒮 ⁢ ( ℝ n ) ${g\in\mathcal{S}(\mathbb{R}^{n})}$ , a weight function w on ℝ 2 ⁢ n ${\mathbb{R}^{2n}}$ and 1 ≤ p , q ≤ ∞ ${1\leq p,q\leq\infty}$ . The weighted Lorentz type modulation space M ⁢ ( p , q , w ) ⁢ ( ℝ n ) ${M(p,q,w)(\mathbb{R}^{n})}$ consists of all tempered distributions f ∈ 𝒮 ′ ⁢ ( ℝ n ) ${f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})}$ such that the short time Fourier transform V g ⁢ f ${V_{g}f}$ is in the weighted Lorentz space L ⁢ ( p , q , w ⁢ d ⁢ μ ) ⁢ ( ℝ 2 ⁢ n ) ${L(p,q,w\,d\mu)(\mathbb{R}^{2n})}$ . The norm on M ⁢ ( p , q , w ) ⁢ ( ℝ n ) ${M(p,q,w)(\mathbb{R}^{n})}$ is ∥ f ∥ M ⁢ ( p , q , w ) = ∥ V g ⁢ f ∥ p ⁢ q , w ${\|f\/\|_{M(p,q,w)}=\|V_{g}f\/\|_{pq,w}}$ . This space was firstly defined and some of its properties were investigated for the unweighted case by Gürkanlı in [9] and generalized to the weighted case by Sandıkçı and Gürkanlı in [16]. Let 1 &lt; p 1 , p 2 &lt; ∞ ${1&lt;p_{1},p_{2}&lt;\infty}$ , 1 ≤ q 1 , q 2 &lt; ∞ ${1\leq q_{1},q_{2}&lt;\infty}$ , 1 ≤ p 3 , q 3 ≤ ∞ ${1\leq p_{3},q_{3}\leq\infty}$ , ω 1 , ω 2 ${\omega_{1},\omega_{2}}$ be polynomial weights and ω 3 ${\omega_{3}}$ be a weight function on ℝ 2 ⁢ n ${\mathbb{R}^{2n}}$ . In the present paper, we define the bilinear multiplier operator from M ⁢ ( p 1 , q 1 , ω 1 ) ⁢ ( ℝ n ) × M ⁢ ( p 2 , q 2 , ω 2 ) ⁢ ( ℝ n ) ${M(p_{1},q_{1},\omega_{1})(\mathbb{R}^{n})\times M(p_{2},q_{2},\omega_{2})(% \mathbb{R}^{n})}$ to M ⁢ ( p 3 , q 3 , ω 3 ) ⁢ ( ℝ n ) ${M(p_{3},q_{3},\omega_{3})(\mathbb{R}^{n})}$ in the following way. Assume that m ⁢ ( ξ , η ) ${m(\xi,\eta)}$ is a bounded function on ℝ 2 ⁢ n ${\mathbb{R}^{2n}}$ , and define B m ⁢ ( f , g ) ⁢ ( x ) = ∫ ℝ n ∫ ℝ n f ^ ⁢ ( ξ ) ⁢ g ^ ⁢ ( η ) ⁢ m ⁢ ( ξ , η ) ⁢ e 2 ⁢ π ⁢ i ⁢ 〈 ξ + η , x 〉 ⁢ 𝑑 ξ ⁢ 𝑑 η for all f , g ∈ 𝒮 ⁢ ( ℝ n ) . $B_{m}(f,g)(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\hat{f}(\xi)\hat{g}(% \eta)m(\xi,\eta)e^{2\pi i\langle\xi+\eta,x\rangle}\,d\xi\,d\eta\quad\text{for % all ${f,g\in\mathcal{S}(\mathbb{R}^{n})}$. }$ The function m is said to be a bilinear multiplier on ℝ n ${\mathbb{R}^{n}}$ of type ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ${(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})}$ if B m ${B_{m}}$ is the bounded bilinear operator from M ⁢ ( p 1 , q 1 , ω 1 ) ⁢ ( ℝ n ) × M ⁢ ( p 2 , q 2 , ω 2 ) ⁢ ( ℝ n ) ${M(p_{1},q_{1},\omega_{1})(\mathbb{R}^{n})\times M(p_{2},q_{2},\omega_{2})(% \mathbb{R}^{n})}$ to M ⁢ ( p 3 , q 3 , ω 3 ) ⁢ ( ℝ n ) ${M(p_{3},q_{3},\omega_{3})(\mathbb{R}^{n})}$ . We denote by BM ⁢ ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ) ⁢ ( ℝ n ) ${\mathrm{BM}(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2})(\mathbb{R}^{n})}$ the space of all bilinear multipliers of type ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) ${(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})}$ , and define ∥ m ∥ ( p 1 , q 1 , ω 1 ; p 2 , q 2 , ω 2 ; p 3 , q 3 , ω 3 ) = ∥ B m ∥ ${\|m\|_{(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})% }=\|B_{m}\|}$ . We discuss the necessary and sufficient conditions for B m ${B_{m}}$ to be bounded. We investigate the properties of this space and we give some examples.