Carleman estimates and boundedness of associated multiplier operators

Jeong, Eun-Hee; Kwon, Yehyun; Lee, Sanghyuk

doi:10.48550/arxiv.1803.03040

Cited by 3 publications

(6 citation statements)

References 14 publications

(36 reference statements)

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“…which is uniform in v ∈ R d . Later, the range of p, q on which the uniform Sobolev inequality (1.8) and Carleman inequality (1.9) hold was completely characterized in [13] and [14], respectively, where the authors proved that both (1.8) and (1.9) hold if and only if 2…”

Section: Introductionmentioning

confidence: 99%

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Strichartz and uniform Sobolev inequalities for the elastic wave equation

Kim¹,

Kwon²,

Lee³

et al. 2021

Preprint

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We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are deduced. For the purpose we diagonalize the symbols of the Lamé operator and its semigroup, which also gives an alternative and simpler proofs of the previous results on perturbed elastic wave equations. Furthermore, we obtain uniform Sobolev inequalities for the elastic wave operator.

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Section: Introductionmentioning

confidence: 99%

“…If we consider the Laplacian ∆ R d instead of , the ranges of p, q for the uniform Sobolev and Carleman inequalities do not coincide. We refer the interested readers to[14] for details.…”

mentioning

confidence: 99%

Strichartz and uniform Sobolev inequalities for the elastic wave equation

Kim¹,

Kwon²,

Lee³

et al. 2021

Preprint

Self Cite

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show abstract

“…In [18] it was shown that, for any d ≥ 3 and v ∈ R d \ {0}, the Carleman estimate (1.7) holds if and only if…”

Section: Introductionmentioning

confidence: 99%

“…Such difference in the boundedess is attributed to different size of sets which carry singularities of the relevant Fourier multipliers. See [18] for more details.…”

Section: Introductionmentioning

confidence: 99%

Resolvent estimates for the Lamé operator and failure of Carleman estimates

Kwon¹,

Lee²,

Seo³

2019

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In this paper, we consider the Lamé operator −∆ * and study resolvent estimate, uniform Sobolev estimate, and Carlman estimate for −∆ * . First, we obtain sharp L p -L q resolvent estimates for −∆ * for admissible p, q. This extends the particular case q = p p−1 due to Cossetti [6]. Secondly, we show failure of uniform Sobolev estimate and Carlman estimate for −∆ * . Our approach is different from that in [6] which relies heavily on the Helmholtz decomposition. Instead, we directly analyze the Fourier multiplier of the resolvent. This direct approach allows us to prove not only the upper bound but also the lower bound on the resolvent, so we get the sharp L p -L q bounds for the resolvent of −∆ * . Strikingly, the relevant uniform Sobolev and Carleman estimates turn out to be false for the Lamé operator −∆ * even though the uniform resolvent estimates for −∆ * are valid for certain range of p, q. This contrasts with the classical result regarding the Laplacian ∆ due to Kenig, Ruiz, and Sogge [21] in which the uniform resolvent estimate plays crucial role in proving the uniform Sobolev and Carlman estimates for ∆. We also describe locations of the L q -eigenvalues of −∆ * + V with complex potential V by making use of the sharp L p -L q resolvent estimates for −∆ * .

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“…We refer the reader forward to Section 2 for the definition of the spaces X estimate of the form e v•x ∇u L q ≤ C e v•x ∆u L p when d ≥ 3. See [22,3,46,21]. To get around the difficulty averaged estimates over O d and τ were considered ( [20,34,18]).…”

Section: Introductionmentioning

confidence: 99%

Uniqueness in the Calderón problem and bilinear restriction estimates

Ham¹,

Kwon²,

Lee³

2019

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Uniqueness in the Calderón problem in dimension bigger than two was usually studied under the assumption that conductivity has bounded gradient. For conductivities with unbounded gradients uniqueness results have not been known until recent years. The latest result due to Haberman basically relies on the optimal L 2 restriction estimate for hypersurface which is known as the Tomas-Stein restriction theorem. In the course of developments of the Fourier restriction problem bilinear and multilinear generalizations of the (adjoint) restriction estimates under suitable transversality condition between surfaces have played important roles. Since such advanced machineries usually provide strengthened estimates, it seems natural to attempt to utilize these estimates to improve the known results. In this paper, we make use of the sharp bilinear restriction estimates, which is due to Tao, and relax the regularity assumption on conductivity. We also consider the inverse problem for the Schrödinger operator with potentials contained in the Sobolev spaces of negative orders.

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Carleman estimates and boundedness of associated multiplier operators

Cited by 3 publications

References 14 publications

Strichartz and uniform Sobolev inequalities for the elastic wave equation

Strichartz and uniform Sobolev inequalities for the elastic wave equation

Resolvent estimates for the Lamé operator and failure of Carleman estimates

Uniqueness in the Calderón problem and bilinear restriction estimates

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