Bernstein type inequalities for quasipolynomials (Q5958885)

A quasipolynomial is defined as a finite sum \(q=\sum_{i=1}^kp_ie^{f_i},\) where \(p_i\in \mathbb C[z_1,\dots,z_n]\) are holomorphic polynomials and \(f_1,\dots,f_k\in(\mathbb C^n)^*\) are pairwise different complex linear functionals. The expression \(m(q):=\sum_{i=1}^k(1+\deg p_i)\) is said to be the \textit{degree} of \(q\) and the number \(\varepsilon(q):=\max_{1\leq i\leq k}\max_{B_c(0,1)}|f_i|\) is called the \textit{exponential type} of \(q\). Here \(B_c(z,t)\) stands for the complex Euclidean ball of radius \(t\) centered at \(z=(z_1,\dots,z_n)\in\mathbb C^n\). The main result of the paper is the following Bernstein type inequality: NEWLINE\[NEWLINE\max_{B_c(z,R)}|q|\leq C(\max\{1,\varepsilon(q)\})^{m-1}R^{m-1}e^{\varepsilon(q)R}\max_{B_c(z,1)}|q|\quad (R>1)NEWLINE\]NEWLINE which holds for any quasipolynomial \(q\) of degree \(m\) with a constant \(C=C(k,m)\), where the exponents \(m-1\) and \(\varepsilon(q)R\) are sharp. Using this inequality the author proves a Brudnyi-Ganzburg type inequality: NEWLINE\[NEWLINE\max_{V}|q|\leq\left(\frac{cn|V|}{\omega|}\right)^{\alpha}\max_{\omega}|q|, NEWLINE\]NEWLINE where \(V\subset\mathbb R^n\) is a convex body, \(\omega\) is a measurable subset of \(V\), \(c\) is an absolute constant and \(\alpha\) depends on the constant \(C\), the diameter of \(V\), the degree \(m(q)\) and the exponential type \(\varepsilon(q)\). Other applications of the key inequality are: the estimate of the BMO-norm for a quasipolynomial in terms of its degree and exponential type and the reverse Hölder inequality with a dimensionless constant.