On the Wave Equation Associated to the Hermite and the Twisted Laplacian

D’Ancona, Piero; Pierfelice, Vittoria; Ricci, Fulvio

doi:10.1007/s00041-009-9104-y

Cited by 12 publications

(11 citation statements)

References 17 publications

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“…Since the integral on the right hand side of the above gets larger when N gets smaller, (30) holds true for all N > 0 and all x ∈ X. It follows that (31)…”

Section: A New Class Of Distributionsmentioning

confidence: 95%

Weighted Besov and Triebel–lizorkin Spaces Associated With Operators and Applications

Bui

Duong

2020

Forum of Mathematics, Sigma

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Let X be a space of homogeneous type and L be a nonnegative self-adjoint operator on L 2 (X) satisfying Gaussian upper bounds on its heat kernels. In this paper we develop the theory of weighted Besov spacesḂ α,L p,q,w (X) and weighted Triebel-Lizorkin spacesḞ α,L p,q,w (X) associated to the operator L for the full range 0 < p, q ≤ ∞, α ∈ R and w being in the Muckenhoupt weight class A∞. Similarly to the classical case in the Euclidean setting, we prove that our new spaces satisfy important features such as continuous charaterizations in terms of square functions, atomic decompositions and the identifications with some well known function spaces such as Hardy type spaces and Sobolev type spaces. Moreover, with extra assumptions on the operator L, we prove that the new function spaces associated to L coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of L and the spectral multiplier of L in our new function spaces.

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“…Since the integral on the right hand side of the above gets larger when N gets smaller, (30) holds true for all N > 0 and all x ∈ X. It follows that (31)…”

Section: A New Class Of Distributionsmentioning

confidence: 95%

Weighted Besov and Triebel–lizorkin Spaces Associated With Operators and Applications

Bui

Duong

2020

Forum of Mathematics, Sigma

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“…(II) Thanks to Proposition 18, global regularity is equivalent to (77) e iΣ± ∈ S ∪ (C ∞ \ S ′ ). (5) Observe that when x < 0 we have x · x 1 2 = −e i π 2 (−x) (III) Then, if B is globally regular, there are only three possible behaviors of e iΣ± :…”

Section: Proof Of Theorem 17mentioning

confidence: 99%

“…We prove in Theorem 14 that (2) is globally regular in the sense of (1) if and only if (5) is globally regular and one-to-one as an operator from S ′ (R) into S ′ (R). In Proposition 18 we give a complete characterization of all operators (5) that are globally regular, in terms of the behavior of the complex roots of its Weyl symbol. In particular we avoid the additional hypotheses required in [13].…”

Section: Introductionmentioning

confidence: 99%

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Global regularity of second order twisted differential operators

Buzano

Oliaro

2020

Journal of Differential Equations

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In this paper we characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension 2. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree 2. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential operators, which can be completely characterized in terms of the asymptotic behavior of the Weyl symbol. In conclusion we observe that we have obtained a new class of globally regular partial differential operators which is disjoint from the class of hypo-elliptic operators in the sense of Shubin.

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“…For instance, if we choose φ(L) = √ L, we are asking if a dispersive estimate for the wave flow e it √ L can be deduced directly from a corresponding estimate for the Schrödinger flow e it L . The possibility of such a reduction is suggested by a suitable subordination formula connecting the two flows (see [33]); the formula was applied in [15] to the Hermite operator and the twisted Laplacian. Our main goal here is to extend this idea to more general functions φ(L) with power-like behaviour near 0 and ∞, and to prove sharper estimates in terms of Sobolev, Hardy or Besov norms, appropriately defined.…”

Section: Introductionmentioning

confidence: 99%

On the flows associated to selfadjoint operators on metric measure spaces

et al. 2019

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On the Wave Equation Associated to the Hermite and the Twisted Laplacian

Cited by 12 publications

References 17 publications

Weighted Besov and Triebel–lizorkin Spaces Associated With Operators and Applications

Weighted Besov and Triebel–lizorkin Spaces Associated With Operators and Applications

Global regularity of second order twisted differential operators

On the flows associated to selfadjoint operators on metric measure spaces

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