On the properties of sums of trigonometric series with monotone coefficients (Q1358548)

The authors, among others, extend a well-known theorem of Hardy-Littlewood to the case \(0<p\leq 1\). Let \(f(x)\) and \(g(x)\) denote the sums of the series \({a_0\over 2}+ \sum^\infty_{n=1} a_n\cos nx\) and \(\sum^\infty_{n=1} a_n\sin nx\), respectively, where \(a_n\to 0\) and \(\Delta_ka_n\geq 0\) for some \(k\geq 1\) and any \(n\). The new theorem reads as follows: a) If \(\Delta_2a_n\geq 0\), then for any \(p\in(0,\infty)\), \[ C_1\left(\sum^\infty_{n=0}(\Delta_1a_n)^p(n+ 1)^{2p-2}\right)^{1/p}\leq|f|_p\leq C_2\Biggl(\sum^\infty_{n=0}(\Delta_1 a_n)^p(n+ 1)^{2p-2}\Biggr)^{1/p}. \] b) If \(\Delta_1a_n\geq 0\), then for any \(p\in(0,\infty)\), \[ C_3\Biggl(\sum^\infty_{n=1} a^p_n n^{p-2}\Biggr)^{1/p}\leq|g|_p\leq C_4\Biggl(\sum^\infty_{n=1} a^p_n n^{p-2}\Biggr)^{1/p}, \] where the positive constants \(C_i\) do not depend on \(\{a_n\}\). Unfortunately in the English translation there are several upsetting misprints.