The Implicit Function Theorem

Krantz, Steven G.; Parks, Harold R.

doi:10.1007/978-1-4612-0059-8

Cited by 148 publications

(24 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…combined with (4) and the formula for the Jacobian of the inverse function, that Hadamard's global inverse function theorem [25,Theorem 6.2.4] applies to the maps x −→ ∇ η (x, η) and η −→ ∇ x (x, η), which are therefore globally invertible. Moreover, these maps have bounded Jacobians uniformly with respect to η and x, respectively, so that they are globally Lipschitz continuous, uniformly with respect to η and x, respectively.…”

Section: Sg Fourier Integral Operatorsmentioning

confidence: 99%

On the global boundedness of Fourier integral operators

2010

View full text Add to dashboard Cite

We consider a class of Fourier integral operators, globally defined on R d , with symbols and phases satisfying product type estimates (the so-called SG or scattering classes). We prove a sharp continuity result for such operators when acting on the modulation spaces M p . The minimal loss of derivatives is shown to be d|1/2 − 1/ p|. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on L p spaces are presented.

show abstract

Section: Sg Fourier Integral Operatorsmentioning

confidence: 99%

On the global boundedness of Fourier integral operators

2010

View full text Add to dashboard Cite

show abstract

“…We then obtain the following corollary due to the real analytic inverse function theorem (see [20]) coupled with a counting argument. Proof of Proposition 4.1 First, note that F is not orthodecomposable since existence of a Hamiltonian path implies that γ (F ) is connected.…”

Section: The Case N = Dmentioning

confidence: 94%

Finite Frame Varieties: Nonsingular Points, Tangent Spaces, and Explicit Local Parameterizations

Strawn

2010

J Fourier Anal Appl

View full text Add to dashboard Cite

The (μ, S)-frames are frames with lengths in [μ 1 · · · μ N ] and with frame operator S, or the F = [f 1 · · · f N ] ∈ M d×N (E) with column lengths listed by μ and which satisfy F F * = S. In this paper, we characterize the nonsingular points of real and complex finite (μ, S)-frame varieties by determining where generalized tori and distorted Stiefel manifolds intersect transversally in Hilbert-Schmidt spheres. This provides us with a characterization of the tangent space for each nonsingular point of the (μ, S)-frame varieties, and we leverage this characterization to demonstrate the existence of structured, locally well defined analytic coordinate patches. We conclude by deriving explicit expressions for these coordinates.

show abstract

“…Remark 3.1 It follows from the assumptions (i), (ii) and (iii) and Hadamard's global inversion function theorem (see, e.g., [21]) that the mappings…”

Section: Boundedness Of Fios On M Pqmentioning

confidence: 99%

Boundedness of Schrödinger Type Propagators on Modulation Spaces

Cordero

Nicola

2009

J Fourier Anal Appl

View full text Add to dashboard Cite

It is known that Fourier integral operators arising when solving Schrö-dinger-type operators are bounded on the modulation spaces M p,q , for 1 ≤ p = q ≤ ∞, provided their symbols belong to the Sjöstrand class M ∞,1 . However, they generally fail to be bounded on M p,q for p = q. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on M p,q for p = q, and between M p,q → M q,p , 1 ≤ q < p ≤ ∞. We also study similar problems for operators acting on Wiener amalgam spaces, recapturing, in particular, some recent results for metaplectic operators. Our arguments make heavily use of the uncertainty principle.

show abstract

The Implicit Function Theorem

Cited by 148 publications

References 0 publications

On the global boundedness of Fourier integral operators

On the global boundedness of Fourier integral operators

Finite Frame Varieties: Nonsingular Points, Tangent Spaces, and Explicit Local Parameterizations

Boundedness of Schrödinger Type Propagators on Modulation Spaces

Contact Info

Product

Resources

About