Reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension

The paper is devoted to the exposition of results announced in [1]. We construct a reduction (following an idea of S. P. Novikov) of the calculus of pseudodifferential operators on Euclidean space ? ⁿ to a similar calculus in the space of sections of a one-dimensional fiber bundle ξ on the 2n-dimensional torus $\mathbb{T}^{2n} $ . This reduction enables us to identify the Schwartz space on ? ⁿ with the space of smooth sections Γ^∞(T ²ⁿ , ξ), compare the Sobolev norms on the corresponding spaces and pseudodifferential operators in them, and describe the class of elliptic operators that reduce to Fredholm operators in Sobolev norms. Thus, for a natural class of elliptic pseudodifferential operators on a noncompact manifold of ? ⁿ , we construct an index formula in accordance with the classical Atya-Singer formula.     

Fredholmness of pseudodifference operators in weighted Spaces

The aim of the paper is to present some results concerning pseudodifference operators on ?^N, which are a discrete analog of standard pseudodifferential operators on ?^N. We study the Fredholm property of pseudodifference operators acting in weighted l ^p spaces on ?^N and the Phragmen-Lindelöf principle for solutions of pseudodifference equations and give applications of these results to discrete Schrö dinger operators on ?^N.     

A boundedness result for twisted convolution

We consider twisted convolution operators with kernels having singularities on a sphere and having as Fourier transform the oscillatory symbol m_α(|ξ|) = |ξ|^–αe^i|ξ|, 0 ≤ ??α < 2n. We give integral representations for such operators and, as a principal result, we study L_p–L_q estimates for them.     

Rellich type inequalities related to Grushin type operator and Greiner type operator

We prove some sharp Hardy inequality associated with the gradient ? _?? = (? _x ,|x| ^?? ? _y ) by a direct and simple approach. Moreover, similar method is applied to obtain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L ^p . We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.     

Lp-Bounded Pseudodifferential Operators and Regularity for Multi-quasi-elliptic Equations

Using a classical result of Marcinkiewicz and Lizorkin about the L^p-continuity for Fourier multipliers, the authors study the action of a class of pseudodifferential operators with weighted smooth symbol on a family of weighted Sobolev spaces. Results about L^p-regularity for multi-quasi-elliptic pseudodifferential operators are also given.     

Pseudodifferential operators on Bochner spaces and an application

We consider pseudodifferential operators with operator valued symbols a(x,ξ) acting on a UMD Banach space X. Assuming some regularity of Hölder type in x and Mihlin type in ξ we prove L _p (? ⁿ ;X) boundedness of such operators. This result is then applied to the study of L _p -maximal regularity for nonautonomous parabolic evolution equations.     

Unbounded pseudodifferential calculus on Lie groupoids

We develop an abstract theory of unbounded longitudinal pseudodifferential calculus on smooth groupoids (also called Lie groupoids) with compact basis. We analyze these operators as unbounded operators acting on Hilbert modules over C^∗(G), and we show in particular that elliptic operators are regular. We construct a scale of Sobolev modules which are the abstract analogues of the ordinary Sobolev spaces, and analyze their properties. Furthermore, we show that complex powers of positive elliptic pseudodifferential operators are still pseudodifferential operators in a generalized sense.     

Weighted inequalities for multilinear commutators of some sublinear operators

We establish sharp estimates for some multilinear commutators related to the Littlewood-Paley and Marcinkiewicz operators. As an application, we obtain the weighted norm inequalities and L log L type estimate for the multilinear commutators.      

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators

This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For L in some class of elliptic operators, we study weighted norm Lp inequalities for singular “non-integral” operators arising from L; those are the operators φ(L) for bounded holomorphic functions φ, the Riesz transforms ∇L−1/2 (or (−Δ)^1/2L−1/2) and its inverse L1/2(−Δ)^−1/2, some quadratic functionals gL and GL of Littlewood-Paley-Stein type and also some vector-valued inequalities such as the ones involved for maximal Lp-regularity. For each, we obtain sharp or nearly sharp ranges of p using the general theory for boundedness of Part I and the off-diagonal estimates of Part II. We also obtain commutator results with BMO functions.     

Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators

We consider here pseudo-differential operators whose symbol σ(x,ξ) is not infinitely smooth with respect to x. Decomposing such symbols into four—sometimes five—components and using tools of paradifferential calculus, we derive sharp estimates on the action of such pseudo-differential operators on Sobolev spaces and give explicit expressions for their operator norm in terms of the symbol σ(x,ξ). We also study commutator estimates involving such operators, and generalize or improve the so-called Kato-Ponce and Calderon-Coifman-Meyer estimates in various ways.     

Quantitative weighted estimates for rough homogeneous singular integrals

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L ²(w), we obtain a bound that is quadratic in A ₂ constant \({\left[ w \right]_{{A_2}}}\). We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.     

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.     

Sharp function boundedness for vector-valued multilinear singular integral operators with non-smooth kernels

Sharp function inequalities for several vector-valued, multilinear singular integral operators with non-smooth kernels are obtained. As an application, some weighted L ^p (p > 1) norm inequalities for vector-valued multilinear operators are derived.     

Error estimates for Hermite interpolation on spheres

In this paper, we prove convergence rates for spherical spline Hermite interpolation on the sphere S^d−1 via an error estimate given in a technical report by Luo and Levesley. The functionals in the Hermite interpolation are either point evaluations of pseudodifferential operators or rotational differential operators, the desirable feature of these operators being that they map polynomials to polynomials. Convergence rates for certain derivatives are given in terms of maximum point separation.     

加权Morrey-Herz空间上的广义Hausdorff算子

引入了一种新的广义Hausdorff算子.它包括了许多著名的算子作为其特殊情形.我们得到了这些算子在加权Morrey-Herz空间上有界的充分必要条件,还得到了相应的新的算子范数不等式.它们是许多著名结果有意义的改进和推广.     

Mixed modulation spaces and their application to pseudodifferential operators

This paper uses frame techniques to characterize the Schatten class properties of integral operators. The main result shows that if the coefficients {〈k,Φm,n〉} of certain frame expansions of the kernel k of an integral operator are in ?2,p, then the operator is Schatten p-class. As a corollary, we conclude that if the kernel or Kohn-Nirenberg symbol of a pseudodifferential operator lies in a particular mixed modulation space, then the operator is Schatten p-class. Our corollary improves existing Schatten class results for pseudodifferential operators and the corollary is sharp in the sense that larger mixed modulation spaces yield operators that are not Schatten class.     

Extrapolation from A∞ weights and applications

We generalize the A_p extrapolation theorem of Rubio de Francia to A_∞ weights in the context of Muckenhoupt bases. Our result has several important features. First, it can be used to prove weak endpoint inequalities starting from strong-type inequalities, something which is impossible using the classical result. Second, it provides an alternative to the technique of good-λ inequalities for proving L^p norm inequalities relating operators. Third, it yields vector-valued inequalities without having to use the theory of Banach space valued operators. We give a number of applications to maximal functions, singular integrals, potential operators, commutators, multilinear Calderón-Zygmund operators, and multiparameter fractional integrals. In particular, we give new proofs, which completely avoid the good-λ inequalities, of Coifman's inequality relating singular integrals and the maximal operator, of the Fefferman-Stein inequality relating the maximal operator and the sharp maximal operator, and the Muckenhoupt-Wheeden inequality relating the fractional integral operator and the fractional maximal operator.     

Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions

We investigate weighted composition operators that attain their norm on weighted Banach spaces of holomorphic functions on the unit disc of type H ^∞. Applications for composition operators on weighted Bloch spaces are given.     

Non-commutative operator Bohr inequality

It is shown that if A, B, X are Hilbert space operators such that X?γI, for the positive real number γ, and p,q>1 with 1/p+1/q=1, then |A−B|²?p|A|²+q|B|² with equality if and only if (1−p)A=B and γ||||A−B|²|||?|||p|A|²X+qX|B|²||| for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert-Schmidt operator, then ‖δ_A,B²(X)‖₂?‖p|A|²X+qX|B|²‖₂ with equality if and only if (1−p)AX=XB.     

An algebra of pseudodifferential operators and the asymptotics of quantum mechanics

We discuss in detail the regularity properties of a class of pseudodifferential operators on $\text{R}$^l introduced by Grossmann, Loupias, and Stein, which are more regular and more symmetrical than usual pseudodifferential operators. Those operators that are self-adjoint form a suitable class of smooth observables for a nonrelativistic quantum theory. If their symbols are allowed to depend smoothly upon Planck's constant $\text{h}\text{?}$, those operators provide the framework for regular asymptotics expansions as $\text{h}\text{?}\text{→ 0}$ of quantum mechanics around classical mechanics.