Bilinear forms, Schur multipliers, complete boundedness and duality
Erik Christensen
备注：
A part of this was previously announced on the arXiv as the first half of the article "Norm optimal factorizations of scalar and block matrices", arXiv:2211.00591. Later results have forced a division of the article. The part on block matrices will appear separately
Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m x n matrices. Based on the theory of operator spaces and completely bounded mappings we present norm optimal versions of these results and two norm optimal factorization results related to the Schur product. We show that the spaces of respectively bilinear forms and Schur multipliers are conjugate duals to each other with respect to their completely bounded norms.
In this article, we investigate the boundedness of the bilinear Bochner-Riesz means $S^{α}$ in the non-Banach triangle case. We improve the corresponding results in [Bern] in two aspects: Our partition of the non-Banach triangle is simpler and we obtain lower smoothness indices $α(p_{1},p_{2})$ for various cases apart from $1 \leq p_1=p_2 <2$.
Let $\mathcal{M}$ be the bilinear Hardy-Littlewood maximal function and $\vec{b}=(b,b)$ be a collection of locally integrable functions. In this paper, the authors establish characterizations of the weighted {\rm BMO} space in terms of several different commutators of bilinear Hardy-Littlewood maximal function, respectively; these commutators include the maximal iterated commutator $\mathcal{M}_{Π\vec{b}}$, the maximal linear commutator $\mathcal{M}_{Σ\vec{b}}$, the iterated commutator $[Π\vec{b},\mathcal{M}]$ and the linear commutator $[Σ\vec{b},\mathcal{M}]$.
We investigate nontrigonometric forms of Riesz transforms in the context of Schur multipliers. This refines Grothendieck-Haagerup's endpoint criterion with a new condition for the Schatten p-boundedness of Schur multipliers and strengthens Potapov/Sukochev's solution of Arazy's conjecture. We recover as well dimension-free estimates for trigonometric Riesz transforms. Our discrete approach is much simpler than previous harmonic analysis and probabilistic approaches. As an application, we find a very simple proof of recent criteria for Schur multipliers of Hörmander-Mikhlin and Marcinkiewicz ty…
The boundedness of the bilinear fractional integral operator is investigated. This bilinear fractional integral operator goes back to Kenig and Stein. This paper is oriented to the boundedness of this operator on products of Morrey spaces. Compared to the earlier work by He and Yan, the local integrability condition of the domain is expanded. The local integrability condition can be relaxed with the help of the averaging technique.
We obtain descriptions of central operator-valued Schur and Herz-Schur multipliers, akin to a classical characterisation due to Grothendieck, that reveals a close link between central (linear) multipliers and bilinear multipliers into the trace class. Restricting to dynamical systems where a locally compact group acts on itself by translation, we identify their convolution multipliers as the right completely bounded multipliers, in the sense of Junge-Neufang-Ruan, of a canonical quantum group associated with the underlying group. We provide characterisations of contractive idempotent operator-…
In this paper, we investigate the $L^{p}$-boundedness of the Riesz means and the $L^{p_{1}}\times L^{p_{2}}\rightarrow L^{p}$ boundedness of the bilinear Riesz means on Métivier groups. Métivier groups are generalization of Heisenberg groups and general H-type groups. Because general Métivier groups only satisfy the non-degeneracy condition and have high-dimensional centre, we have to use different methods and techniques from those on Heisenberg groups and H-type groups.
We study the mapping property of the commutator of bilinear Hardy-Littlewood maximal operator in homogeneous Triebel-Lizorkin space. We also show that the commutator of bilinear Hardy-Littlewood maximal operator is a compact operator acting on product of Lebesgue spaces. The results are new even in the linear case.
We study new weighted estimates for the 2-fold product of Hardy-Littlewood maximal operators defined by $M^{\otimes}(f,g):= MfMg$. This operator appears very naturally in the theory of bilinear operators such as the bilinear Calderón-Zygmund operators, the bilinear Hardy-Littlewood maximal operator introduced by Calderón or in the study of pseudodifferential operators. To this end, we need to study Hölder's inequality for Lorentz spaces with change of measures $$ \Vert fg \Vert_{L^{p,\infty}\left(w_1^{p/p_1} w_2^{p/p_2}\right)} \le C \Vert f \Vert_{L^{p_1,\infty}(w_1 )} \Vert g \Vert_{L^{p_2,\…
In this paper, we give the definability of bilinear singular and fractional integral operators on Morrey-Banach space, as well as their commutators and we prove the boundedness of such operators on Morrey-Banach spaces. Moreover, the necessary condition for BMO via the bounedness of bilinear commutators on Morrey-Banach space is also given. As a application of our main results, we get the necessary conditions for BMO via the bounedness of bilinear integral operators on weighted Morrey space and Morrey space with variable exponents. Finally, we obtain the boundedness of bilinear C-Z operator on…
In this paper, we characterize the multiple operator integrals mappings which are bounded on the Haagerup tensor product of spaces of compact operators. We show that such maps are automatically completely bounded and prove that this is equivalent to a certain factorization property of the symbol associated to the operator integral mapping. This generalizes a result by Juschenko-Todorov-Turowska on the boundedness of continuous multilinear Schur multipliers.
The purpose of this paper is to establish bilinear estimates in Besov spaces generated by the Dirichlet Laplacian on a domain of Euclidian spaces. These estimates are proved by using the gradient estimates for heat semigroup together with the Bony paraproduct formula and the boundedness of spectral multipliers.
We characterize the boundedness of Hankel bilinear forms on a product of generalized Fock-Sobolev spaces on ${\mathbb C}^n$ with respect to the weight $(1+|z|)^ρe^{-\fracα2|z|^{2\ell}}$, for $\ell\ge 1$, $α>0$ and $ρ\in{\mathbb R}$. We obtain a weak decomposition of the Bergman kernel with estimates and a Littlewood-Paley formula, which are key ingredients in the proof of our main results. As an application, we characterize the boundedness, compactness and the membership in the Schatten class of small Hankel operators on these spaces.
This paper is a follow up of [6]. We investigate the boundedness of the bilinear fractional integral operator introduced by Grafakos in [3]. When the local integrability index $s$ falls 1 with weights and $t$ exceeds 1, He and Yan obtained some results on this operator was worked on Morrey spaces earlier in [7]. Later in the paper [6], we considered the case $t=1$. This paper handles the remaining case $0<t<1$.
An RD-space $\mathcal{X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\mathcal{X}$. In this setting, the authors establish the boundedness of bilinear $θ$-type Calderón-Zygmund operator $T_θ$ and its commutator $[b_1,b_2,T_θ]$ generated by the function $b_1,b_2\in BMO(μ)$ and $T_θ$ on generalized weighted Morrey space $\mathcal{M}^{p,ϕ}(ω)$ and generalized weighted weak Morrey space $W\mathcal{M}^{p,ϕ}(ω)$ over RD-spaces.
Let $G$ be a locally compact unimodular group, and let $ϕ$ be some function of $n$ variables on $G$. To such a $ϕ$, one can associate a multilinear Fourier multiplier, which acts on some $n$-fold product of the non-commutative $L_p$-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an $n$-fold product of Schatten classes $S_p(L_2(G))$. We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called `multiplicatively bounded $(p_1,\ldots,p_n)$-norm' of a multilinear Sch…
Arazy's conjecture concerning Schur multipliers: revisited and resolved
Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）；School of Mathematics and Statistics, University of NSW（新南威尔士大学数学与统计学院）
In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely $\textbf{SG}$-class.
For an $m \times n$ complex matrix $X$ of rank $r$ with Schur multiplier $S_X$ we show that there exist an $ r \times m $ complex matrix $L$ and an $ r\times n $ complex matrix $R$ such that $X = L^*R$ and $\|S_X\|\, =\, \|\mathrm{diag} (L^*L) \|^{\frac{1}{2}} \| \mathrm{diag} (R^*R) \| ^{\frac{1}{2}},$ and the norm condition is optimal. Let the completely bounded norm of the bilinear form $B_X$ induced by $X$ on $(\mathbb{C}^m, \|.\|_\infty) \times (\mathbb{C}^n, \|.\|_\infty)$ be denoted $\|B_X\|_{cb},$ then $X$ has a factorization $ X = Δ(η)^* C Δ(ξ)$ with $η$ in $\mathbb{C}^m,$ $ξ$ in $\ma…
The boundedness of the small Hankel operator $h_f^ν(g)=P_ν(f\bar{g})$, induced by an analytic symbol $f$ and the Bergman projection $P_ν$ associated to $ν$, acting from the weighted Bergman space $A^p_\om$ to $A^q_ν$ is characterized on the full range $0<p,q<\infty$ when $ω,ν$ belong to the class $\mathcal{D}$ of radial weights admitting certain two-sided doubling conditions. Certain results obtained are equivalent to the boundedness of bilinear Hankel forms, which are in turn used to establish the weak factorization $A_η^{q}=A_ω^{p_{1}}\odot A_ν^{p_{2}}$, where $1<q,p_{1},p_{2}<\infty$ such t…
We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the $\mathcal R$-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a $T(1)$-typ…
