Bilinear Pseudodifferential Operators Research Articles

Bi-parameter and bilinear Calderón–Vaillancourt theorem with critical order

Abstract In this paper, we establish the sharp Calderón–Vaillancourt theorem on L p L^{p} spaces for bi-parameter and bilinear pseudo-differential operators with symbols of critical order by deriving a sufficient and necessary condition on its symbol. This sharpens the result of [G. Lu and L. Zhang, Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order, Forum Math. 28 2016, 6, 1087–1094] which was only proved for symbols of subcritical order.

Multilinear Fourier integral operators on modulation spaces

Abstract 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 𝐒𝐆 {\mathbf{SG}} -class.

A remark on the condition $\bm{1/p = 1/p_1 + 1/p_2}$ for boundedness of bilinear pseudo-differential operators with exotic symbols

A remark on the condition $\bm{1/p = 1/p_1 + 1/p_2}$ for boundedness of bilinear pseudo-differential operators with exotic symbols

Bilinear pseudodifferential operators with symbol in BS1,1m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$BS_{1,1}^m$$\\end{document} on Triebel–Lizorkin spaces with critical Sobolev index

In this paper we obtain new estimates for bilinear pseudodifferential operators with symbol in the class BS1,1m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$BS_{1,1}^m$$\\end{document}, when both arguments belong to Triebel-Lizorkin spaces of the type Fp,qn/p(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{p,q}^{n/p}({\\mathbb {R}}^n)$$\\end{document}. The inequalities are obtained as a consequence of a refinement of the classical Sobolev embedding Fp,qn/p(Rn)↪bmo(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F^{n/p}_{p,q}({\\mathbb {R}}^n)\\hookrightarrow \ extrm{bmo}({\\mathbb {R}}^n)$$\\end{document}, where we replace bmo(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{bmo}({\\mathbb {R}}^n)$$\\end{document} by an appropriate subspace which contains L∞(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty ({\\mathbb {R}}^n)$$\\end{document}. As an application, we study the product of functions on Fp,qn/p(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{p,q}^{n/p}({\\mathbb {R}}^n)$$\\end{document} when 1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty $$\\end{document}, where those spaces fail to be multiplicative algebras.

Discrete Bilinear Operators and Commutators

We discuss boundedness properties of certain classes of discrete bilinear operators that are similar to those of the continuous bilinear pseudodifferential operators with symbols in the Hörmander classes $$BS^{\omega }_{1, 0}$$ . In particular, we investigate their relation to discrete analogs of the bilinear Calderón–Zygmund singular integral operators and show compactness of their commutators.

Some estimates for commutators of bilinear pseudo-differential operators

Abstract We obtain a class of commutators of bilinear pseudo-differential operators on products of Hardy spaces by applying the accurate estimates of the Hörmander class. And we also prove another version of these types of commutators on Herz-type spaces.

Endpoint estimates for bilinear pseudodifferential operators with symbol in [formula omitted

In this paper we establish some endpoint estimates for bilinear pseudodifferential operators with symbol in the class BS1,1m, involving the space of functions with local bounded mean oscillation bmo(Rn). As a consequence we also obtain an endpoint estimate of Kato-Ponce type.

Boundedness of bilinear pseudo-differential operators of S0,0-type in Wiener amalgam spaces and in Lebesgue spaces

We extend and improve the known results about the boundedness of the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class BS0,0m(Rn). We consider wider classes of symbols and improve estimates for the corresponding operators. A key idea is to consider the operators in Wiener amalgam spaces.

Weighted and endpoint estimates for commutators of bilinear pseudo-differential operators

&lt;abstract&gt;&lt;p&gt;In this paper, by applying the accurate estimates of the Hörmander class, the authors consider the commutators of bilinear pseudo-differential operators and the operation of multiplication by a Lipschitz function. By establishing the pointwise estimates of the corresponding sharp maximal function, the boundedness of the commutators is obtained respectively on the products of weighted Lebesgue spaces and variable exponent Lebesgue spaces with $ \sigma \in\mathcal{B}BS_{1, 1}^{1} $. Moreover, the endpoint estimate of the commutators is also established on $ L^{\infty}\times L^{\infty} $.&lt;/p&gt;&lt;/abstract&gt;

Bilinear Pseudodifferential Operators and the Hörmander Classes

Bilinear Pseudodifferential Operators and the Hörmander Classes

Bilinear Pseudo-Differential Operators with Exotic Class Symbols of Limited Smoothness

We consider the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class \(BS_{\rho , \rho }^m ({\mathbb {R}}^n) \) for \(m \in {\mathbb {R}}\) and \(0 \le \rho < 1\). The aim of this paper is to discuss smoothness conditions for symbols to assure the boundedness from \(h^p \times h^q\) to \(h^r\) for \(1 \le r \le 2 \le p,q \le \infty \) satisfying \(1/p+1/q=1/r\), where \(h^\infty \) is replaced by the space bmo.

XMO and Weighted Compact Bilinear Commutators

To study the compactness of bilinear commutators of certain bilinear Calderón–Zygmund operators which include (inhomogeneous) Coifman–Meyer bilinear Fourier multipliers and bilinear pseudodifferential operators as special examples, Torres and Xue (Rev Mat Iberoam 36:939–956, 2020) introduced a new subspace of BMO\(\,(\mathbb {R}^n)\), denoted by XMO\(\,(\mathbb {R}^n)\), and conjectured that it is just the space VMO\(\,(\mathbb {R}^n)\) introduced by D. Sarason. In this article, the authors give a negative answer to this conjecture by establishing an equivalent characterization of XMO\(\,(\mathbb {R}^n)\), which further clarifies that XMO\(\,(\mathbb {R}^n)\) is a proper subspace of VMO\(\,(\mathbb {R}^n)\). This equivalent characterization of XMO\(\,(\mathbb {R}^n)\) is formally similar to the corresponding one of CMO\(\,(\mathbb {R}^n)\) obtained by A. Uchiyama, but its proof needs some essential new techniques on dyadic cubes as well as some exquisite geometrical observations. As an application, the authors also obtain a weighted compactness result on such bilinear commutators, which optimizes the corresponding result in the unweighted setting.

Kato-Ponce type inequality for bilinear pseudo-differential operators of $S_{0, 0}$-type in the scale of Besov spaces

We consider the Kato-Ponce type inequality for bilinear pseudo-differential operators with $S_{0,0}$-type symbols in the scale of Besov spaces. In particular, the borderline whether the boundedness of those operators holds or not is discussed.

Bilinear pseudo-differential operators with exotic symbols

The boundedness from $H^p \times L^2$ to $L^r$, $1/p+1/2=1/r$, and from $H^p \times L^{\infty}$ to $L^p$ of bilinear pseudo-differential operators is proved under the assumption that their symbols are in the bilinear H\ormander class $BS^m_{\rho,\rho}$, $0 \le \rho <1$, of critical order $m$, where $H^p$ is the Hardy space. This combined with the previous results of the same authors establishes the sharp boundedness from $H^p \times H^q$ to $L^r$, $1/p+1/q=1/r$, of those operators in the full range $0< p, q \le \infty$, where $L^r$ is replaced by $BMO$ if $r=\infty$.

Boundedness of bilinear pseudo-differential operators of $$S_{0,0}$$-type on $$L^2 \times L^2$$

We extend the known result that the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class $$BS^{-n/2}_{0,0}(\mathbb {R}^n)$$ are bounded from $$L^2 \times L^2$$ to $$h^1$$ . We show that those operators are also bounded from $$L^2 \times L^2$$ to $$L^r $$ for every $$1< r\le 2$$ . Moreover we give similar results for symbol classes wider than $$BS^{-n/2}_{0,0}(\mathbb {R}^n)$$ . We also give results for symbols of limited smoothness.

Equivalence of (quasi-)norms on a vector-valued function space and its applications to multilinear operators

In this paper we present (quasi-)norm equivalence on a vector-valued function space $L^p_A(l^q)$ and extend the equivalence to $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin space, motivated by Fraizer-Jawerth. By applying the results, we improve the multilinear Hormander's multiplier theorem of Tomita, that of Grafakos-Si, and the boundedness results for bilinear pseudo-differential operators, given by Koezuka-Tomita.

On the ranges of bilinear pseudo-differential operators of S0,0-type on L2 × L2

In this paper, the ranges of bilinear pseudo-differential operators of S0,0-type on L2×L2 are determined in the framework of Besov spaces. Our result improves the L2×L2→L1 boundedness of those operators with symbols in the bilinear Hörmander class BS0,0m.

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.

Bilinear Pseudo-differential Operators with Gevrey–Hörmander Symbols

We consider bilinear pseudo-differential operators whose symbols may have a sub-exponential growth at infinity, together with all their derivatives. It is proved that those symbol classes can be described by the means of the short-time Fourier transform and modulation spaces. Our first main result is the invariance property of the corresponding bilinear operators. Furthermore, we prove the continuity of such operators when acting on modulation spaces. As a consequence, we derive their continuity on anisotropic Gelfand–Shilov type spaces.

Boundedness of multilinear pseudo-differential operators of $S_{0,0}$-type in $L^2$-based amalgam spaces

We consider the multilinear pseudo-differential operators with symbols in a generalized $S_{0,0}$-type class and prove the boundedness of the operators from $(L^2, \ell^{q_1}) \times \cdots \times (L^2, \ell^{q_{N}})$ to $(L^2, \ell^{r})$, where $(L^2, \ell^{q})$ denotes the $L^2$-based amalgam space. This extends the previous result by the same authors, which treated the bilinear pseudo-differential operators and gave the $L^2 \times L^2$ to $(L^2, \ell^{1})$ boundedness.