Bilinear Embedding Theorems for Differential Operators in ℝ2

Stolyarov, Dmitriy

doi:10.1007/s10958-015-2527-x

Cited by 7 publications

(13 citation statements)

References 13 publications

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“…with Fourier support in the unit ball, there exists a functiong, also with Fourier support in the unit ball, such that (24) (…”

Section: Corollary 115 Let M Be An Integermentioning

confidence: 99%

“…Note that this curve is convex outside the origin. Application of the Littlewood-Paley inequality and homogeneity considerations (see [24]) reduce (35) to the case where the spectrum of f lies in a small neighborhood of a point on Γ k,l . So, by the results of [2], the inequality 35 Note that the Fourier transform of the function (∂ k 1 − σ∂ l 2 )f vanishes on Γ k,l , which is a smooth convex curve in the plane (with, possibly, a singularity at zero).…”

Section: Precursors To the Current Workmentioning

confidence: 99%

“…They appeared in [12] (see Proposition 12 in that paper) and [11] where the authors investigated the action of Bochner-Riesz operators of negative order on these spaces. They arose in [24] in connection with Sobolev-type embedding theorems. We describe this development in Section 2.…”

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confidence: 99%

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Restrictions of higher derivatives of the Fourier transform

Goldberg

Stolyarov

2020

Trans. Amer. Math. Soc. Ser. B

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We consider several problems related to the restriction of (∇ k)f to a surface Σ ⊂ R d with nonvanishing Gauss curvature. While such restrictions clearly exist if f is a Schwartz function, there are few bounds available that enable one to take limits with respect to the L p (R d) norm of f. We establish three scenarios where it is possible to do so: • When the restriction is measured according to a Sobolev space H −s (Σ) of negative index, we determine the complete range of indices (k, s, p) for which such a bound exists. • Among functions wheref vanishes on Σ to order k − 1, the restriction of (∇ k)f defines a bounded operator from (this subspace of) L p (R d) to L 2 (Σ) provided 1 ≤ p ≤ 2d+2 d+3+4k. • When there is a priori control off | Σ in a space H (Σ), > 0, this implies improved regularity for the restrictions of (∇ k)f. If is large enough, then even ∇f L 2 (Σ) can be controlled in terms of f H (Σ) and f L p (R d) alone. The proofs are based on three main tools: the spectral synthesis work of Y. Domar, which provides a mechanism for L p approximation by "convolving along surfaces in spectrum", a new bilinear oscillatory integral estimate valid for ordinary L p functions, and a convexity-type property of the quantity (∇ k)f H −s (Σ) as a function of k and s that allows one to employ the control of f H (Σ). Contents 65 5. Robust estimates 72 6. Sharpness 85 7. Additional lemmas and supplementary material 91 Acknowledgment 95 References 95

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“…with Fourier support in the unit ball, there exists a functiong, also with Fourier support in the unit ball, such that (24) (…”

Section: Corollary 115 Let M Be An Integermentioning

confidence: 99%

Section: Precursors To the Current Workmentioning

confidence: 99%

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Restrictions of higher derivatives of the Fourier transform

Goldberg

Stolyarov

2020

Trans. Amer. Math. Soc. Ser. B

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“…We also note that the isotropic nature of homogeneity is not of crucial importance here. See [14] and [29] for anisotropic inequalities in the style of Theorem 1.1. The authors suppose that the nature of the effect discussed above has nothing to do with Euclidean space instruments such as the Newton-Leibniz, Stokes, or coarea formulas, but has a purely harmonic analytic explanation.…”

Section: Introductionmentioning

confidence: 99%

Martingale approach to Sobolev embedding theorems

Ayoush¹,

Stolyarov²,

Wojciechowski³

2018

Preprint

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“…They appeared in [12] (see Proposition 12 in that paper) and [11] where the authors investigated the action of Bochner-Riesz operators of negative order on these spaces. They arose in [24] in connection with Sobolev type embedding theorems. We describe this development in Subsection 1.4.…”

mentioning

confidence: 99%

Restrictions of higher derivatives of the Fourier transform

Goldberg¹,

Stolyarov²

2018

Preprint

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We consider several problems related to the restriction of (∇ k ) f to a surface Σ ⊂ R d with nonvanishing Gauss curvature. While such restrictions clearly exist if f is a Schwartz function, there are few bounds available that enable one to take limits with respect to the Lp(R d ) norm of f . We establish three scenarios where it is possible to do so:• When the restriction is measured according to a Sobolev space H −s (Σ) of negative index. We determine the complete range of indices (k, s, p) for which such a bound exists. • Among functions where f vanishes on Σ to order k − 1, the restriction of (∇ k ) f defines a bounded operator from (this subspace of)• When there is a priori control of f | Σ in a space H (Σ), > 0, this implies improved regularity for the restrictions of (∇ k ) f . If is large enough then even ∇ f L 2 (Σ) can be controlled in terms of f H (Σ) and f Lp(R d ) alone. The techniques underlying these results are inspired by the spectral synthesis work of Y. Domar, which provides a mechanism for Lp approximation by "convolving along surfaces", and the Stein-Tomas restriction theorem. Our main inequality is a bilinear form bound with similar structure to the Stein-Tomas T * T operator, generalized to accommodate smoothing along Σ and derivatives transverse to it. It is used both to establish basic H −s (Σ) bounds for derivatives of f and to bootstrap from surface regularity of f to regularity of its higher derivatives. p 2.1. Description of the annihilator and Domar's theory 2.2. Coincedence of Σ L k p and the spaces defined as kernels of restriction operators 2.3. Proofs of "if" part in Theorem 1.4 and Theorem 1.6 2.4. Proof of Corollary 1.14 3. Proof of Theorem 1.16 3.1. Pointwise estimates of the kernel 3.2. Interpolation 3.3. Strichartz estimates 4. Robust estimates 4.1. Introduction to "numerology" 4.2. Two simple examples

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Bilinear Embedding Theorems for Differential Operators in ℝ2

Cited by 7 publications

References 13 publications

Restrictions of higher derivatives of the Fourier transform

Restrictions of higher derivatives of the Fourier transform

Martingale approach to Sobolev embedding theorems

Restrictions of higher derivatives of the Fourier transform

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