Boyanov-Naydenov problem and Kolmogorov type inequalities for positive (negative) parts of functions

Abstract

We prove that Boyanov-Naidenov problem $$$\|x^{(k)}_{\pm}\|_{q,\, \mu} \to \sup$$$ on classes of functions $$$\Omega^r_p(A_0, A_r):=\{x\in L^r_{\infty}: \|x^{(r)}\|_{\infty}\le A_r, L(x)_p\le A_0 \}$$$, where $$$k= 0,1,...,r-1$$$, $$$q \ge 1$$$ for $$$k\ge 1$$$, $$$q \ge p$$$ for $$$k=0$$$, is equivalent to the problem of the sharp constant $$$C = C(\lambda)$$$, $$$\lambda > 0$$$, in the Kolmogorov type inequality
$$\|x^{(k)}_{\pm}\|_{q,\, \mu} \leq C L(x)_{p}^{\alpha} \left\|x^{\left( r\right) }\right\|_\infty^{1-\alpha}, \;\;\;x\in \Omega^{\,r}_{p, \lambda} \qquad \qquad \qquad \qquad (1)$$ where $$$\|x_{\pm}\|_{p, \mu}:=\sup \{ \|x_{\pm}\|_{L_p[a, b]}: a, b\in {\rm \bf R}, \mu \left( {\rm supp}_{[a, b]} x_{\pm} \right) \le \mu \}$$$, $$$\Omega^{\,r}_{p, \lambda}:= \{x\in L^r_{\infty}: L(x)_p = L(\varphi_{\lambda, r})_p \cdot \|x^{(r)}\|_{\infty} \}$$$, $$$\alpha=\frac{r-k+1/q}{r+1/p}$$$, $$$\varphi_{\lambda, r}$$$ is the contraction of the ideal Euler spline of order $$$r$$$,
$$L(x)_p:=\sup \left\{ \|x\|_{L_p[a,\, b]}: \; a, b \in {\rm \bf R},\; |x(t)|>0,\;t\in (a, b) \right\}$$
In particular, we obtain the sharp inequality of the form (1).
Similar results were obtained on asymmetric classes of functions and on the spaces of trigonometric polynomials and splines.