Results 1  10
of
21
TIMEFREQUENCY ANALYSIS OF SJÖSTRAND’S CLASS
, 2004
"... We investigate the properties an exotic symbol class of pseudodifferential operators, Sjöstrand’s class, with methods of timefrequency analysis (phase space analysis). Compared to the classical treatment, the timefrequency approach leads to striklingly simple proofs of Sjöstrand’s fundamental resu ..."
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Cited by 30 (9 self)
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We investigate the properties an exotic symbol class of pseudodifferential operators, Sjöstrand’s class, with methods of timefrequency analysis (phase space analysis). Compared to the classical treatment, the timefrequency approach leads to striklingly simple proofs of Sjöstrand’s fundamental results and to farreaching generalizations.
UNIMODULAR FOURIER MULTIPLIERS FOR MODULATION SPACES
"... Abstract. We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol eiξα, where α ∈ [0, 2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual Lpspaces. As a consequence, the phases ..."
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Cited by 17 (2 self)
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Abstract. We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol eiξα, where α ∈ [0, 2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual Lpspaces. As a consequence, the phasespace concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers ξ  −δ sin(ξ  α) for 0 ≤ δ ≤ α. 1.
Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation
, 2007
"... We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixedtime estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with ..."
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Cited by 12 (8 self)
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We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixedtime estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with potentials V(x) = ±x².
Counterexamples for Boundedness of Pseudodifferential Operators
, 2002
"... The KohnNirenberg correspondence assigns to a symbol #(x, #) in the space of tempered distributions S ) the operator #(X, D) : ) defined by #(x, #) f(#) e 2#ix# d# . This is the classical version of pseudodi#erential operators that is used in the investigation of partial diff ..."
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Cited by 10 (5 self)
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The KohnNirenberg correspondence assigns to a symbol #(x, #) in the space of tempered distributions S ) the operator #(X, D) : ) defined by #(x, #) f(#) e 2#ix# d# . This is the classical version of pseudodi#erential operators that is used in the investigation of partial differential operators, cf. [21]. In the language of physics, the KohnNirenberg correspondence and its relatives such as the Weyl correspondence are methods of quantization. In the language of engineering, they are timevarying filters. The KohnNirenberg correspondence is usually analyzed using methods from hard analysis. The problems arising from the theory of partial differential equations suggest using the classical Hormander symbol classes S #,# (R ), which are defined in terms of di#erentiability conditions [21], [31]. On the other hand, if we introduce the timefrequency shifts M # T x f(t) = e 2#i#t f(t x) , (1) then we can write #(X, D) as a formal superposition o
Modulation spaces and a class of bounded multilinear pseudodifferential operators
 J. Operator Theory
"... Abstract. We show that multilinear pseudodifferential operators with symbols in the modulation space M ∞,1 are bounded on products of modulation spaces. In particular, M ∞,1 includes nonsmooth symbols. Several multilinear Calderón– Vaillancourttype theorems are then obtained by using certain embed ..."
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Cited by 9 (7 self)
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Abstract. We show that multilinear pseudodifferential operators with symbols in the modulation space M ∞,1 are bounded on products of modulation spaces. In particular, M ∞,1 includes nonsmooth symbols. Several multilinear Calderón– Vaillancourttype theorems are then obtained by using certain embeddings of classical function spaces into modulation spaces. 1.
A Pedestrian's Approach to Pseudodifferential Operators
, 2005
"... This article, in particular, owes much to my joint work and many stimulating discussions with Chris Heil ..."
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Cited by 9 (2 self)
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This article, in particular, owes much to my joint work and many stimulating discussions with Chris Heil
Timefrequency Analysis of Fourier Integral Operators
, 2007
"... Abstract. We use timefrequency methods for the study of Fourier Integral operators (FIOs). In this paper we shall show that Gabor frames provide very efficient representations for a large class of FIOs. Indeed, similarly to the case of shearlets and curvelets frames [6, 27], the matrix representati ..."
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Cited by 8 (7 self)
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Abstract. We use timefrequency methods for the study of Fourier Integral operators (FIOs). In this paper we shall show that Gabor frames provide very efficient representations for a large class of FIOs. Indeed, similarly to the case of shearlets and curvelets frames [6, 27], the matrix representation of a Fourier Integral Operator with respect to a Gabor frame is wellorganized. This is used as a powerful tool to study the boundedness of FIOs on modulation spaces. As special cases, we recapture boundedness results on modulation spaces for pseudodifferential operators with symbols in M ∞,1 [25], for some unimodular Fourier multipliers [2] and metaplectic operators [10, 23]. 1.
Modulation Spaces as Symbol Classes for Pseudodifferential Operators
, 2002
"... We investigate the Weyl calculus of pseudodifferential operators with the methods of timefrequency analysis. As symbol classes we use the modulation spaces, which are the function spaces associated to the shorttime Fourier transform and the Wigner distribution. We investigate the boundedness and S ..."
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Cited by 7 (4 self)
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We investigate the Weyl calculus of pseudodifferential operators with the methods of timefrequency analysis. As symbol classes we use the modulation spaces, which are the function spaces associated to the shorttime Fourier transform and the Wigner distribution. We investigate the boundedness and Schattenclass properties of pseudodifferential operators, and furthermore we study their mapping properties between modulation spaces.
SYMBOLIC CALCULUS AND FREDHOLM PROPERTY FOR LOCALIZATION OPERATORS
, 2005
"... We study the composition of timefrequency localization operators (wavepacket operators) and develop a symbolic calculus of such operators on modulation spaces. The use of timefrequency methods (phase space methods) allows the use of rough symbols of ultrarapid growth in place of smooth symbols in ..."
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Cited by 6 (3 self)
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We study the composition of timefrequency localization operators (wavepacket operators) and develop a symbolic calculus of such operators on modulation spaces. The use of timefrequency methods (phase space methods) allows the use of rough symbols of ultrarapid growth in place of smooth symbols in the standard classes. As the main application it is shown that, in general, a localization operators possesses the Fredholm property, and thus its range is closed in the target space.
Some new Strichartz estimates for the Schrödinger equation. Preprint (2007), available at http://arxiv.org
"... Abstract. We deal with fixedtime and Strichartz estimates for the Schrödinger propagator as an operator on Wiener amalgam spaces. We discuss the sharpness of the known estimates and we provide some new estimates which generalize the classical ones. As an application, we present a result on the well ..."
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Cited by 6 (5 self)
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Abstract. We deal with fixedtime and Strichartz estimates for the Schrödinger propagator as an operator on Wiener amalgam spaces. We discuss the sharpness of the known estimates and we provide some new estimates which generalize the classical ones. As an application, we present a result on the wellposedness of the linear Schrödinger equation with a rough time dependent potential.