An estimate for the Fourier coefficients of functions from Orlicz classes (Q1605553)

Let \(\Phi(t)\) be an \(N\)-function which generates an Orlicz space \(\Phi(L)\) of measurable functions \(f(x),x\in[0,1],\) such that \(\int_0^1\Phi(f(x))dx<\infty\). Let \(\{\phi_n(x)\}\) be an orthonormal system of functions on \([0,1]\), let \(\|\phi_n\|_s=M_n\), \(B_n=\sum_{k=1}^nM_n^2\), \(n=1,2\dots\), for \(s\in (2,+\infty]\), and let \(c_k\) be the Fourier coefficients of the function \(f\). The author proposes the following estimate \[ \sum_{k=1}^{\infty}\Psi\left(\frac{|c_k|}{M_k} B_k^{\frac{s-1}{s-2}}\right) \frac{M_k^2}{B_k^{2\frac{s-1}{s-2}}}\leq C_1\int_0^1\Phi(C_2f^{(s')}(t))dt,\;\frac{1}{s}+\frac{1}{s'}=1, \] under the condition that \(\varphi(s)=\Phi(\sqrt{s})\) is a concave function.