Products Of Lebesgue Spaces Research Articles

Multilinear $$\varvec{\tau } $$-Wigner transform

In this work, we introduce a $$\varvec{\tau } $$ -dependent multilinear Wigner transform $$W_{\varvec{\tau }}$$ , $${\varvec{\tau }}\in [0,1]^d$$ . It also includes various types of multilinear time-frequency representations, among the others the classical multilinear Wigner and Rihaczek representations and some properties of multilinear $$\varvec{\tau } $$ -Wigner transform. We then prove the boundedness properties of multilinear $$\varvec{\tau } $$ -Wigner transform both on products of Lebesgue spaces and on modified version of modulation spaces.

Weighted estimates for bilinear square functions with non-smooth kernels and commutators

Under weaker conditions on the kernel functions, we discuss the boundedness of bilinear square functions associated with non-smooth kernels on the product of weighted Lebesgue spaces. Moreover, we investigate the weighted boundedness of the commutators of bilinear square functions (with symbols which are BMO functions and their weighted version, respectively) on the product of Lebesgue spaces. As an application, we deduce the corresponding boundedness of bilinear Marcinkiewicz integrals and bilinear Littlewood-Paley g-functions.

On compactness of commutators of multiplication and bilinear pseudodifferential operators and a new subspace of BMO

It is known that the compactness of the commutators of point-wise multiplication with bilinear homogeneous Calderón–Zygmund operators acting on product of Lebesgue spaces is characterized by the multiplying function being in the space CMO. This space is the closure in BMO of its subspace of smooth functions with compact support. It is shown in this work that for bilinear Calderón–Zygmund operators arising from smooth (inhomogeneous) bilinear Fourier multipliers or bilinear pseudodifferential operators, one can actually consider multiplying functions in a new subspace of BMO larger than CMO.

Characterization of compactness of commutators of bilinear singular integral operators

The commutators of bilinear Calderón–Zygmund operators and pointwise multiplication with a symbol in C M O \mathrm {CMO} are bilinear compact operators on products of Lebesgue spaces. We show that, for certain non-degenerate Calderón–Zygmund operators, the symbol being in C M O \mathrm {CMO} is not only sufficient but actually necessary for the compactness of the commutators.

Bounds of weighted multilinear Hardy-Cesàro operators in p-adic functional spaces

We introduce the p-adic weighted multilinear Hardy-Cesaro operator. We also obtain the necessary and sufficient conditions on weight functions to ensure the boundedness of that operator on the product of Lebesgue spaces, Morrey spaces, and central bounded mean oscillation spaces. In each case, we obtain the corresponding operator norms. We also characterize the good weights for the boundedness of the commutator of weighted multilinear Hardy-Cesaro operator on the product of central Morrey spaces with symbols in central bounded mean oscillation spaces.

L p $L^{p}$ and BMO bounds for weighted Hardy operators on the Heisenberg group

In the setting of the Heisenberg group $\mathbb{H}^{n}$ , we characterize those nonnegative functions w defined on $[0,1]$ for which the weighted Hardy operator $\mathsf{H}_{w}$ is bounded on $L^{p}(\mathbb{H}^{n})$ , $1\leq p\leq\infty$ , and on $\operatorname{BMO}(\mathbb{H}^{n})$ . Meanwhile, the corresponding operator norm in each case is derived. Furthermore, we introduce a type of weighted multilinear Hardy operators and obtain the characterizations of their weights for which the weighted multilinear Hardy operators are bounded on the product of Lebesgue spaces in terms of Heisenberg group. In addition, the corresponding norms are worked out.

New weighted multilinear operators and commutators of hardy-cesàro type

This paper deals with a general class of weighted multilinear Hardy-Cesàro operators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesáro operators (with symbols in central BMO spaces) are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.

Multilinear paraproducts revisited

We prove that multilinear paraproducts are bounded from products of Lebesgue spaces L^{p_1}\!\times \cdots \times L^{p_{m+1}} to L^{p,\infty} , when 1\le\! p_1, \dots , p_m , p_{m+1}<\infty , 1/p_1+\cdots +1/p_{m+1}=1/p . We focus on the endpoint case when some indices p_j are equal to 1 , in particular we obtain a new proof of the estimate L^1\times \cdots \times L^1\to L^{1/(m+1),\infty} .

Characterization of Compactness of the Commutators of Bilinear Fractional Integral Operators

The compactness of the commutators of bilinear fractional integral operators and point-wise multiplication, acting on products of Lebesgue spaces, is characterized in terms of appropriate mean oscillation properties of their symbols. The compactness of the commutators when acting on product of weighted Lebesgue spaces is also studied.

Estimates for the maximal bilinear singular integral operators

In this paper, the behavior on the product of Lebesgue spaces is considered for the maximal operators associated with the bilinear singular integral operators whose kernels satisfy certain minimal regularity conditions.

Multilinear Calderón–Zygmund operators on Hardy spaces, II

In this note we explain a point left open in the literature of Hardy spaces, namely that for a sufficiently smooth m-linear Calderón–Zygmund operator bounded on a product of Lebesgue spaces we haveT(f1,…,fm)=∑i1⋯∑imλ1,i1⋯λm,imT(a1,i1,…,am,im)a.e., where aj,ij are Hpj atoms, λj,ij∈C, and fj=∑ijλj,ijaj,ij are Hpj distributions. In some particular cases the proof is new even when m=1.

Weighted multilinear Hardy operators and commutators

Abstract In this paper, we introduce a type of weighted multilinear Hardy operators and obtain their sharp bounds on the product of Lebesgue spaces and central Morrey spaces. In addition, we obtain sufficient and necessary conditions of the weight functions so that the commutators of the weighted multilinear Hardy operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Riemann–Liouville and Weyl.

End-point estimates for iterated commutators of multilinear singular integrals

Iterated commutators of multilinear Calderón–Zygmund operators and pointwise multiplication with functions in bounded mean oscillation are studied in products of Lebesgue spaces. Both strong type and weak end-point estimates are obtained, including weighted results involving the vectors weights of the multilinear Calderón–Zygmund theory recently introduced in the literature. Some better than expected estimates for certain multilinear operators are presented too.

Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus

The reappearance of a sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be present in a recently introduced class of bilinear pseudodifferential operators which can be seen as more general variable coefficient counterparts of the bilinear Hilbert transform and other singular bilinear multipliers operators. The unboundedness on product of Lebesgue spaces but the boundedness on spaces of smooth functions (which is the exotic behavior referred to) of such operators is obtained. In addition, by introducing a new way to approximate the product of two functions, estimates on a new paramultiplication are obtained.

Bilinear Fourier integral operators

We study the boundedness of bilinear Fourier integral operators on products of Lebesgue spaces. These operators are obtained from the class of bilinear pseudodifferential operators of Coifman and Meyer via the introduction of an oscillatory factor containing a real-valued phase of five variables $${\Phi(x,y_1,y_2,\xi_1,\xi_2)}$$ which is jointly homogeneous in the phase variables (ξ1, ξ2). For symbols of order zero supported away from the axes and the antidiagonal, we show that boundedness holds in the local-L 2 case. Stronger conclusions are obtained for more restricted classes of symbols and phases.

Translation-Invariant Bilinear Operators with Positive Kernels

We study L r (or L r, ∞) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on \({\mathbb {R}^n}\). We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on \({\mathbb R^{2n}}\), while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some sufficient conditions on nonnegative kernels yielding boundedness for the corresponding operators on products of Lebesgue spaces. We also prove that, unlike the linear case where boundedness from L 1 to L 1 and from L 1 to L 1, ∞ are equivalent properties, boundedness from L 1 × L 1 to L 1/2 and from L 1 × L 1 to L 1/2, ∞ may not be equivalent properties for bilinear translation-invariant operators with nonnegative kernels.

A Class of Commutators for Multilinear Fractional Integrals in Nonhomogeneous Spaces

Let be a nondoubling measure on . A class of commutators associated with multilinear fractional integrals and RBMO( ) functions are introduced and shown to be bounded on product of Lebesgue spaces with .

Multilinear Calderón–Zygmund operators on the product of Lebesgue spaces with non-doubling measures

Under the assumption that μ is a non-doubling measure on R d , the author proves that for the multilinear Calderón–Zygmund operator, its boundedness from the product of Hardy space H 1 ( μ ) × H 1 ( μ ) into L 1 / 2 ( μ ) implies its boundedness from the product of Lebesgue spaces L p 1 ( μ ) × L p 2 ( μ ) into L p ( μ ) with 1 < p 1 , p 2 < ∞ and p satisfying 1 / p = 1 / p 1 + 1 / p 2 .

Some remarks on bilinear Littlewood–Paley theory

In this work, some bilinear analogues of linear Littlewood–Paley theory are explored. Paraproducts with functions decomposed at different scales are shown to be bounded on certain products of Lebesgue spaces. Results concerning square functions associated with smooth and nonsmooth cutoffs are also obtained.

Extrapolation of Weighted norm inequalities for multivariable operators and applications

Two versions of Rubio de Francia’s extrapolation theorem for multivariable operators of functions are obtained. One version assumes an initial estimate with different weights in each space and implies boundedness on all products of Lebesgue spaces. Another version assumes an initial estimate with the same weight but yields boundedness on a product of Lebesgue spaces whose indices lie on a line. Applications are given in the context of multilinear Calderon-Zygmund operators. Vector-valued inequalities are automatically obtained for them without developing a multilinear Banach-valued theory. A multilinear extension of the Marcinkiewicz and Zygmund theorem on l2-valued extensions of bounded linear operators is also obtained.