Multilinear Pseudo-differential Operators Research Articles

Multilinear Pseudo-differential Operators with $$S_{0,0}$$ Class Symbols of Limited Smoothness

Multilinear Pseudo-differential Operators with $$S_{0,0}$$ Class Symbols of Limited Smoothness

Boundedness of multilinear pseudo-differential operators with symbols in the Hörmander class S0,0

The multilinear pseudo-differential operators with symbols in the multilinear Hörmander class S0,0 are considered. A complete identification of the cases where those operators define bounded operators between local Hardy spaces is given. Some results for the boundedness between Wiener amalgam spaces are also given. These are extensions and improvements of the results known in the bilinear case.

Weighted boundedness of multilinear pseudo-differential operators

<abstract><p>In this paper, the weighted boundedness for some multilinear operators generated by the pseudo-differential operators and the weighted Lipschitz functions are obtained.</p></abstract>

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.

Weak and strong type estimates for the multilinear pseudo-differential operators

In this paper, we investigate the boundedness of the multilinear pseudo-differential operator Tσ. First, we establish the local exponential decay estimates for Tσ. In terms of the corresponding commutators Tσ,Σb, we obtain the local subexponential decay estimates. Secondly, we derive the weighted mixed weak type inequality for Tσ, which parallels Sawyer's conjecture for Calderón-Zygmund operators and covers the endpoint weighted inequalities. Last but not least, we present the sharp weighted estimates for Tσ and Tσ,Σb. It is worth mentioning that our results are totally new even in the linear case.

$$L^p$$-Boundedness and $$L^p$$-Nuclearity of Multilinear Pseudo-differential Operators on $${\mathbb {Z}}^n$$ and the Torus $${\mathbb {T}}^n$$

In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential operators on $L^p$-spaces. First, we prove analogues of known multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. Later, we investigate the $s$-nuclearity, $0<s \leq 1,$ of periodic and discrete pseudo-differential operators. To accomplish this, we classify those $s$-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on $\sigma$-finite measures spaces. We also study similar properties for periodic Fourier integral operators. Finally, we present some applications of our study to deduce the periodic Kato-Ponce inequality and to examine the $s$-nuclearity of multilinear Bessel potentials as well as the $s$-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.

Inglés

In this note we announce our investigation on the Lp properties for periodic and discrete multilinear pseudo-differential operators. First, we review the periodic analysis of multilinear pseudo-differential operators byshowing classical multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Tomita, Miyachi, Fujita, Grafakos, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. The s-nuclearity, 0 < s ≤ 1, for the discrete and periodic multilinear pseudo-differential operators will be investigated. To do so, we classify those s-nuclear, 0 < s ≤ 1, multilinear integral operators on arbitrary Lebesgue spaces defined on σ-finite measures spaces. Finally, we present some applications of our analysis to deduce the periodic Kato-Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentialsas well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.

Multilinear pseudo-differential operators on product of Local Hardy spaces with variable exponents

In this paper, we establish a new atomic decomposition theory for Local Hardy spaces with variable exponents via local grand maximal characterization. By applying the refined atomic decomposition result, we prove that multilinear pseudo-differential operators are bounded on product of local Hardy spaces with variable exponents.

Multilinear Stockwell transforms

The main aim of this paper is to introduce multilinear versions of the Stockwell transforms (also named S-transforms) by using the fact that S-transforms can be written as convolution products. Further on we extend the multilinear S-transforms from the Schwartz class of rapidly decreasing functions to the space of tempered distributions. In the sequel we give a relation between multilinear S-transforms and multilinear pseudo-differential operators. We also state and prove some boundedness results regarding multilinear S-transforms on the Lebegue’s spaces Lp(Rn) and also on the Hörmander’s spaces Bp,k(Rn), where p ≥ 1 and k is a temperate weight function. In the end, a weak uncertainty principle for multilinear S-transforms and for its adjoint is also given.

$L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators

In this paper, we establish the $L^p$ estimates for the maximal functions associated with the multilinear pseudo-differential operators. Our main result is Theorem 1.2. There are several major different ingredients and extra difficulties in our proof from those in Grafakos, Honzik and Seeger [ 15 ] and Honzik [ 22 ] for maximal functions generated by multipliers. First, in order to eliminate the variable $x$ in the symbols, we adapt a non-smooth modification of the smooth localization method developed by Muscalu in [ 26 , 30 ]. Then, by applying the inhomogeneous Littlewood-Paley dyadic decomposition and a discretization procedure, we can reduce the proof of Theorem 1.2 into proving the localized estimates for localized maximal functions generated by discrete paraproducts. The non-smooth cut-off functions in the localization procedure will be essential in establishing localized estimates. Finally, by proving a key localized square function estimate (Lemma 4.3) and applying the good-$\lambda$ inequality, we can derive the desired localized estimates.

Boundedness of Multilinear Pseudo-differential Operators on Modulation Spaces

Boundedness results for multilinear pseudodifferential operators on products of modulation spaces are derived based on ordered integrability conditions on the short-time Fourier transform of the operators’ symbols. The flexibility and strength of the introduced methods is demonstrated by their application to the bilinear and trilinear Hilbert transform.

Symbolic Calculus and Boundedness of Multi-parameter and Multi-linear Pseudo-differential Operators

Abstract Since the work of Hörmander on linear pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, harmonic analysis, theory of several complex variables and other branches of modern analysis (e.g., they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂̅ problem, etc.). The work of Coifman and Meyer on multi-linear Fourier multipliers and pseudo-differential operators has stimulated further such applications. In [2], the authors developed a fairly satisfactory theory of symbolic calculus for multi-linear pseudo-differential operators. Motivated by this work [2] and Lp estimates of [34, 35] on multi-parameter and multi-linear Fourier multipliers and of [12] on multi-parameter and multi-linear pseudo-differential operators, we study and carry out the theory of symbolic calculus for multi-parameter and multi-linear pseudo-differential operators. Our results include the symbol estimates of the adjoints, asymptotic behavior, kernel estimates and boundedness properties and extend those in [2] to the multi-parameter and multi-linear setting. The estimates of the distributional kernel of associated multi-parameter and multilinear pseudo-differential operators can be found useful in establishing the boundedness of such multi-parameter and multi-linear pseudo-differential operators.

Weighted estimates for multilinear pseudodifferential operators

In this paper, we study the weighted estimates for multilinear pseudodifferential operators. We show that a multilinear pseudodifferential operator is bounded with respect to multiple weights whenever its symbol satisfies some smoothness and decay conditions. Our result generalizes similar ones from the classical Ap weights to multiple weights.

Multilinear pseudodifferential operators beyond Calderón–Zygmund theory

We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hörmander Sρ,δm classes. These results are new in the case ρ<1, that is, outwith the scope of multilinear Calderón–Zygmund theory.

Lp-boundedness of Multilinear Pseudo-differential Operators on Znand Tn

The aim of this paper is to introduce and study multilinear pseudo-differential operators on Zn and Tn = (Rn/ 2πZn) the n-torus. More precisely, we give sufficient conditions and sometimes necessary conditions for Lp-boundedness of these classes of operators. L2-boundedness results for multilinear pseudo-differential operators on Zn and Tn with L2-symbols are stated. The proofs of these results are based on elementary estimates on the multilinear Rihaczek transforms for functions in L2(Zn) respectively L2(Tn) which are also introduced. We study the weak continuity of multilinear operators on the m-fold product of Lebesgue spaces Lpj(Zn), j = 1,...,m and the link with the continuity of multilinear pseudo-differential operators on Zn. Necessary and sufficient conditions for multilinear pseudo-differential operators on Zn or Tn to be a Hilbert-Schmidt operators are also given. We give a necessary condition for a multilinear pseudo-differential operators on Zn to be compact. A sufficient condition for compactness is also given.

New class of multiple weights and new weighted inequalities for multilinear operators

Abstract In this paper, we first introduce the new class of multiple weights A p → ∞ ${A^\infty _{\vec{p}}}$ which is larger than the class of multiple weights in [Adv. Math. 220 (2009), 1222–1264]. Then, using this class of weights, we study the weighted norm inequalities for certain classes of multilinear operators and their commutators with new BMO functions introduced in [J. Fourier Anal. Appl. 17 (2011), 115–134]. Finally, we show that some multilinear pseudodifferential operators fall within the scope of the theory obtained in this paper.

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.

Multilinear localization operators

In this paper we introduce a notion of multilinear localization operators. By reinterpreting these operators as multilinear Kohn–Nirenberg pseudodifferential operators, we give sufficient conditions for their boundedness on products of modulation spaces. Moreover, we prove that these conditions are also necessary for the boundedness of the operators. Finally, as application, we construct various examples of bounded multilinear pseudodifferential operators.

Modulation space estimates for multilinear pseudodifferential operators

We prove that for symbols in the modulation spaces M, p ≥ q, the associated multilinear pseudodifferential operators are bounded on products of appropriate modulation spaces. In particular, the symbols we study here are defined without any reference to smoothness, but rather in terms of their time-frequency behavior.

On multilinear singular integrals of Calderón-Zygmund type

A variety of results regarding multilinear singular Calderon-Zygmund integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discrete decompositions, a multilinear version of Schur's test, and a multilinear version of the T1 Theorem suitable for the study of multilinear pseudodifferential and translation invariant operators. A maximal operator associated with multilinear singular integrals is also introduced and employed to obtain wiegted norm inequalities. [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].