Caffarelli–Kohn–Nirenberg inequalities for curl-free vector fields and second order derivatives

Cazacu, Cristian; Flynn, Joshua; Lam, Nguyen

doi:10.1007/s00526-023-02454-1

Cited by 6 publications

(1 citation statement)

References 37 publications

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“…Cazacu et al. [8], Do et al. [15] proved the sharp

L^2

and

L^p

inequalities and their quantitative stability, in which the sharp constants and explicit extremal functions of the stability inequality were identified.…”

Section: Introductionmentioning

confidence: 99%

Remainder terms of a nonlocal Sobolev inequality

Deng,

Tian,

Yang

et al. 2023

Mathematische Nachrichten

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In this note, we study a nonlocal version of the Sobolev inequality where is the best constant, * denotes the standard convolution and denotes the classical Sobolev space with respect to the norm . By using the nondegeneracy property of the extremal functions, we prove that the existence of the gradient type remainder term and a reminder term in the weak ‐norm of above inequality for all with .

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“…Cazacu et al. [8], Do et al. [15] proved the sharp

L^2

and

L^p

inequalities and their quantitative stability, in which the sharp constants and explicit extremal functions of the stability inequality were identified.…”

Section: Introductionmentioning

confidence: 99%

Remainder terms of a nonlocal Sobolev inequality

Deng,

Tian,

Yang

et al. 2023

Mathematische Nachrichten

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A note on the 1-D minimization problem related to solenoidal improvement of the uncertainty principle inequality

Hamamoto

2024

Arch. Math.

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This paper gives a second way to solve the one-dimensional minimization problem of the form : $$\begin{aligned} \min _{f\not \equiv 0}\frac{\displaystyle \int \limits _0^\infty \left( f''\right) ^2x^{\mu +1}dx\int \limits _0^\infty \left( {x}^2\left( f'\right) ^2 -\varepsilon f^2\right) {{x}}^{\mu -1}d{x}}{\displaystyle \left( \int \limits _0^\infty \left( f'\right) ^2 {{x}}^{\mu }d{x}\right) ^2} \end{aligned}$$ min f ≢ 0 ∫ 0 ∞ f ′ ′ 2 x μ + 1 d x ∫ 0 ∞ x 2 f ′ 2 - ε f 2 x μ - 1 d x ∫ 0 ∞ f ′ 2 x μ d x 2 for scalar-valued functions f on the half line, where $$f'$$ f ′ and $$f''$$ f ′ ′ are its derivatives and $$\varepsilon $$ ε and $$\mu $$ μ are positive parameters with $$\varepsilon < \frac{\mu ^2}{4}.$$ ε < μ 2 4 . This problem plays an essential part of the calculation of the best constant in Heisenberg’s uncertainty principle inequality for solenoidal vector fields. The above problem was originally solved by using (generalized) Laguerre polynomial expansion; however, the calculation was complicated and long. In the present paper, we give a simpler method to obtain the same solution, the essential part of which was communicated in Theorem 5.1 of the preprint (Hamamoto, arXiv:2104.02351v4, 2021).

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