In the case where either r = 2, k = 1 or r = 3, k = 1, 2, for any q, p ≥ 1, β ∈ [0, 2π), and a Lebesgue-measurable set B ⊂ I2π ≔ [−π/2, 3π/2], μB ≤ β, we prove a sharp Kolmogorov–Remez-type inequality $$ {\left\Vert {f}^{(k)}\right\Vert}_q\le \frac{\left\Vert \varphi r-k\right\Vert q}{E_0{{\left(\varphi r\right)}_L^{\alpha}}_p\left({I}_{2\uppi}/{B}_{2m}\right)}{\left\Vert f\right\Vert}_{L_p}^{\alpha}\left({I}_{2\uppi}/B\right){\left\Vert {F}^{(r)}\right\Vert}_{\infty}^{1-\alpha },\kern1em f\in {L}_{\infty}^r, $$ with 𝛼 = min {1 − k/r, (r − k + 1/q)/(r + 1/p)}, where 𝜑r is the perfect Euler spline of order r, $$ {E}_0{(f)}_{L_p(G)} $$ is the best approximation of f by constants in Lp(G), $$ {B}_{2m}=\left[\frac{\pi -2m}{2},\frac{\pi +2m}{2}\right] $$ , and m = m(β) 𝜖 [0, 𝜋) is uniquely defined by β. We also establish a sharp Kolmogorov–Remez-type inequality, which takes into account the number of sign changes of the derivatives.

中文翻译：

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低平滑度周期函数的 Sharp Kolmogorov-Remez 型不等式

在 r = 2, k = 1 或 r = 3, k = 1, 2 的情况下，对于任何 q, p ≥ 1, β ∈ [0, 2π) 和 Lebesgue-可测集 B ⊂ I2π ≔ [ −π/2, 3π/2], μB ≤ β，我们证明了一个尖锐的 Kolmogorov-Remez 型不等式 $$ {\left\Vert {f}^{(k)}\right\Vert}_q\le \frac {\left\Vert \varphi rk\right\Vert q}{E_0{{\left(\varphi r\right)}_L^{\alpha}}_p\left({I}_{2\uppi}/{ B}_{2m}\right)}{\left\Vert f\right\Vert}_{L_p}^{\alpha}\left({I}_{2\uppi}/B\right){\left \Vert {F}^{(r)}\right\Vert}_{\infty}^{1-\alpha },\kern1em f\in {L}_{\infty}^r, $$ with 𝛼 = min {1 − k/r, (r − k + 1/q)/(r + 1/p)}，其中 𝜑r 是完美的 r 阶欧拉样条，$$ {E}_0{(f)}_ {L_p(G)} $$ 是 Lp(G) 中常数对 f 的最佳近似， $$ {B}_{2m}=\left[\frac{\pi -2m}{2},\frac{ \pi +2m}{2}\right] $$ ，并且 m = m(β) 𝜖 [0, 𝜋) 由 β 唯一定义。我们还建立了一个尖锐的 Kolmogorov-Remez 型不等式，它考虑了导数符号变化的次数。