Compact bilinear operators and commutators

Journal of Function Spaces, 2019

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<ξ+η,x>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.

Studia Mathematica, 2011

Let L = −∆ + V be a Schrödinger operator in R d and H 1 L (R d) be the Hardy type space associated to L. We investigate the bilinear operators T + and T − defined by T ± (f, g)(x) = (T1f)(x)(T2g)(x) ± (T2f)(x)(T1g)(x), where T1 and T2 are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T + or T − is bounded from L p (R d) × L q (R d) to H 1 L (R d) for 1 < p, q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.

Studia Mathematica, 2001

This article is concerned with the question of whether Marcinkiewicz multipliers on R 2n give rise to bilinear multipliers on R n ×R n. We show that this is not always the case. Moreover, we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces.

MATHEMATICA SCANDINAVICA, 2019

The behavior of bilinear operators acting on the interpolation of Banach spaces in relation to compactness is analyzed, and an one-sided compactness theorem is obtained for bilinear operators interpolated by the ρ interpolation method.

Banach Center Publications, 2019

A locally integrable function m(ξ, η) defined on R n × R n is said to be a bilinear multiplier on R n of type (p 1 , p 2 , p 3) if Bm(f, g)(x) = Z R n Z R nf (ξ)ĝ(η)m(ξ, η)e 2πi(ξ+η,x dξdη defines a bounded bilinear operator from L p 1 (R n) × L p 2 (R n) to L p 3 (R n). The study of the basic properties of such spaces is investigated and several methods of constructing examples of bilinear multipliers are provided. The special case where m(ξ, η) = M (ξ − η) for a given M defined on R n is also addressed. R n f (x)e −2πi x,ξ dx. We shall use the notation M p,q (R n) (respect.M p,q (R n)), for 1 ≤ p, q ≤ ∞, for the space of distributions u ∈ S ′ (R n) such that u * φ ∈ L q (R n) for all φ ∈ L p (R n) (respect. for the space of bounded functions m such that T m defines a bounded operator from L p (R n) to L q (R n) where T m (φ)(ξ) = m(ξ)f (ξ).) We endow the spaceM p,q (R n) with the "norm" of the operator T m , that is m p,q = T m. Let us start off by mentioning some well known properties of the space of linear multipliers (see [1, 14]): M p,q (R n) = {0} whenever q < p, M p,q (R n) = M q ′ ,p ′ (R n) for 1 < p ≤ q < ∞ and for 1 ≤ p ≤ 2, M 1,1 (R n) ⊂ M p,p (R n) ⊂ M 2,2 (R n). We also have the identificationsM

Indiana University Mathematics Journal

In this work we prove Hardy space estimates for bilinear Littlewood-Paley-Stein square function and Calderón-Zygmund operators. Sufficient Carleson measure type conditions are given for square functions to be bounded from H p 1 × H p 2 into L p for indices smaller than 1, and sufficient BMO type conditions are given for a bilinear Calderón-Zygmund operator to be bounded from H p 1 × H p 2 into H p for indices smaller than 1. Subtle difficulties arise in the bilinear nature of these problems that are related to frequency properties of products of functions. Moreover, three types of bilinear paraproducts are defined and shown to be bounded from H p 1 × H p 2 into H p for indices smaller than 1. The first is a bilinear Bony type paraproduct that was defined in [33]. The second is a paraproduct that resembles the product of two Hardy space functions. The third class of paraproducts are operators given by sums of molecules, which were introduced in [2].

Journal of Fourier Analysis and Applications, 2005

Illinois Journal of Mathematics, 1997

Dissertationes Mathematicae, 2014

Transactions of the American Mathematical Society, 2018

Working in the setting of quasi-Banach couples, we establish a formula for the measure of noncompactness of bilinear operators interpolated by the general real method. The result applies to the real method and to the real method with a function parameter. c

… , Integration and Related …, 2010

Mathematische Nachrichten, 2003

We study some properties of strongly and absolutely p−summing bilinear operators. We show that Hilbert-Schmidt bilinear mappings are both strongly and absolutely p−summing, for every p ≥ 1. Giving an example of a strongly 1−summing bilinear mapping which fails to be weakly compact, we answer a question posed in [6]. We prove that, as in the linear case, every bilinear operator from L∞−spaces to an L2−space is absolutely 2−summing.

2000

Let H p denote the Lebesgue space Lp for p&gt; 1 and the Hardy space H p for p ≤ 1. For 0 &lt;p , q, r &lt;∞, we study H p × H q → H r mapping properties of bilinear operators given by finite sums of products of Calderon-Zygmund operators on stratified homogeneous Lie groups. When r ≤ 1, we show that such mapping properties hold when a number of moments of the operator vanish. This hypothesis is natural and the conditions imposed are the minimal required for any operator of this type to map into the space H r. Our proofs employ both the maximal function and atomic characterization of H p. We also discuss some applications.

The Journal of Fourier Analysis and Applications, 2001

×ØÖ Øº This paper proves the L p -boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of Coifman-Meyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth. Using a Whitney decomposition in the Fourier plane a general bilinear operator is represented as infinite discrete sums of time-frequency paraproducts obtained by associating wave-packets with tiles in phase-plane. Boundedness for the general bilinear operator then follows once the corresponding L p -boundedness of time-frequency paraproducts has been established. The latter result is the main theorem proved in Part II, our subsequent paper [11], using phase-plane analysis.