Analogs of the Jackson-Nikol'skij inequalities for trigonometric polynomials in spaces with nonsymmetric norm (Q1274086)

The author deals with the evaluation of \[ \sup_{\substack{ t_n\in T_n\\ t_n\not\equiv 0}} {\| t_n\|_{q_1,q_2}\over\| t_n\|_{p_1,p_2}}, \] where \(\| f\|_{p,q}:= \| f^+\|_p+\| f^-\|_q\) denotes the nonsymmetric norm of \(f\). E.g., he proves that if \(1\leq p\leq\infty\), then \[ \sup_{\substack{ t_n\in T_n\\ t_n\not\equiv 0}} {\| t_n\|_{1,\infty}\over \| t_n\|_{1,p}}\asymp n^{3/(2p+ 1)}. \] {}.