Pseudodifferential operators
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Gabor representations of evolution operators Open
We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultra-analytic regularity. As an application we show that Gabor frames, which provide optima…
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Microlocal analysis of forced waves Open
We use radial estimates for pseudodifferential operators to describe long\ntime evolution of solutions to $ i u_t - P u = f $ where $ P $ is a\nself-adjoint 0th order pseudodifferential operator satisfying hyperbolic\ndynamical assumptions…
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Schrödinger equations with rough Hamiltonians Open
We consider a class of linear Schr\\"odinger equations in R^d with rough\nHamiltonian, namely with certain derivatives in the Sj\\"ostrand class\n$M^{\\infty,1}$. We prove that the corresponding propagator is bounded on\nmodulation spaces.…
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Refined and microlocal Kakeya–Nikodym bounds for eigenfunctions in two dimensions Open
We obtain some improved essentially sharp Kakeya–Nikodym estimates for eigenfunctions in two dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to t…
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Two-dimensional Schrödinger operators with point interactions: Threshold expansions, zero modes and Lp-boundedness of wave operators Open
We study the threshold behavior of two-dimensional Schrödinger operators with finitely many local point interactions. We show that the resolvent can either be continuously extended up to the threshold, in which case we say that the operato…
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Remarks on $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities Open
We consider the continuity property in Lebesgue spaces L^{p}(\mathbb R^{m}) of the wave operators W_\pm of scattering theory for Schrödinger operators H=-\Delta + V on \mathbb R^m , |V(x)|\le C\langle x \rangle^{-\delta} for some \delta>2 …
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Deep neural networks for inverse problems with pseudodifferential\n operators: an application to limited-angle tomography Open
We propose a novel convolutional neural network (CNN), called $\\Psi$DONet,\ndesigned for learning pseudodifferential operators ($\\Psi$DOs) in the context\nof linear inverse problems. Our starting point is the Iterative Soft\nThresholding…
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Pseudodifferential equations, wave factorization, and related problems Open
We consider a model elliptic pseudodifferential equation in a special canonical domains of a multidimensional space. Using a special representation for an elliptic symbol, we give the formula for a general solution of such an equation and …
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Singular integrals in quantum Euclidean spaces Open
We shall establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes’ pseudodifferential c…
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Boundedness and continuity of maximal operators associated to polynomial compound curves on Triebel-Lizorkin spaces Open
In this paper we study the Triebel-Lizorkin space boundedness and continuity of maximal operators related to rough singular integrals associated to polynomial compound curves.We prove that the above operators are bounded and continuous on …
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Growth of Sobolev norms for linear Schrödinger operators Open
We give an example of a linear, time-dependent, Schrödinger operator with optimal growth of Sobolev norms. The construction is explicit, and relies on a comprehensive study of the linear Lowest Landau Level equation with a time-dependent p…
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Elliptic regularity for Dirac operators on families of noncompact manifolds Open
We develop elliptic regularity theory for Dirac operators in a very general framework: we consider Dirac operators linear over $C^*$-algebras, on noncompact manifolds, and in families which are not necessarily locally trivial fibre bundles.
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Besov continuity of pseudo-differential operators on compact Lie groups revisited Open
In this note we present some results on the action of global pseudo-differential operators on Besov spaces on compact Lie groups.
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Limited Data Problems for the Generalized Radon Transform in R<sup>n</sup> Open
We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in ${{\mathbb R}^n}$. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by ge…
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Bilinear Localization Operators on Modulation Spaces Open
We introduce a class of bilinear localization operators and show how to interpret them as bilinear Weyl pseudodifferential operators. Such interpretation is well known in linear case whereas in bilinear case it has not been considered so f…
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Index theory of uniform pseudodifferential operators Open
We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. This generalization will follow as a corollary from a local index theorem that is …
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Viscosity Limits for Zeroth‐Order Pseudodifferential Operators Open
Motivated by the work of Colin de Verdière and Saint‐Raymond on spectral theory for zeroth‐order pseudodifferential operators on tori, we consider viscosity limits in which zeroth‐order operators, P , are replaced by P + iν Δ, ν > 0 . By a…
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A groupoid approach to pseudodifferential operators Open
We give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural $\mathbb{R}^\times_+$-action. Specifically, we show that a properly support…
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Uncertainty principles for the Opdam–Cherednik transform on modulation spaces Open
In this paper, we establish the Cowling--Price's, Hardy's and Morgan's\nuncertainty principles for the Opdam-Cherednik transform on modulation spaces\nassociated with this transform. The proofs of the theorems are based on the\nproperties …
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Symbol calculus for operators of layer potential type on Lipschitz surfaces with VMO normals, and related pseudodifferential operator calculus Open
We show that operators of layer potential type on surfaces that are locally graphs of Lipschitz functions with gradients in vmo are equal, modulo compacts, to pseudodifferential operators (with rough symbols), for which a symbol calculus i…
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Born–Jordan Pseudodifferential Calculus, Bopp Operators and Deformation Quantization Open
There has recently been a resurgence of interest in Born–Jordan quantization, which historically preceded Weyl's prescription. Both mathematicians and physicists have found that this forgotten quantization scheme is actually not only of gr…
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Strichartz estimates for Schrödinger equations with slowly decaying potentials Open
For Schrödinger equations with a class of slowly decaying repulsive potentials, we show that the solution satisfies global-in-time Strichartz estimates for any admissible pairs. Our admissible class of potentials includes the positive homo…
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Microlocal theory of Legendrian links and cluster algebras Open
We show the existence of quasicluster Ꮽ-structures and cluster Poisson structures on moduli stacks of sheaves with singular support in the alternating strand diagram of grid plabic graphs by studying the microlocal parallel transport of sh…
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Regularity of Commutators of Maximal Operators with Lipschitz Symbols Open
This paper is devoted to studying Sobolev regularity properties of commutators of Hardy–Littlewood maximal operator and its fractional case with Lipschitz symbols, both in the global and local case. Some new pointwise estimates for the wea…
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Geometry of pseudodifferential algebra bundles and Fourier integral operators Open
We study the geometry and topology of (filtered) algebra bundles ψZ over a smooth manifold X with typical fiber ψZ (Z;V), the algebra of classical pseudodifferential operators acting on smooth sections of a vector bundle V over the compact…
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Strongly singular bilinear Calderón–Zygmund operators and a class of bilinear pseudodifferential operators Open
Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a Hörmander class of critical order, we investigate boundedness properties of strongly singular Calderón–Zygmund operators in the bilinear setting. …
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The Lifshits--Krein trace formula and operator Lipschitz functions Open
We describe the maximal class of functions $f$ on the real line, for which the Lifshitz--Krein trace formula $\operatorname{trace}(f(A)-f(B))=\int_{\Bbb R} f'(s)\boldsymbolξ(s)\,ds$ holds for arbitrary self-adjoint operators $A$ and $B$ wi…
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m‐Microlocal elliptic pseudodifferential operators acting on Open
In the first part of the paper we study the minimal and maximal extension of a class of weighted pseudodifferential operators in the Fréchet space . In the second one non homogeneous microlocal properties are introduced and propagation of …
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Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras Open
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Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals Open
This paper is the first part of a two-paper series whose aim is to give a thorough account on Connes' pseudodifferential calculus on noncommutative tori. This pseudodifferential calculus has been used in numerous recent papers, but a detai…