Embedding of \(H_p^\omega\) in the class \(e^L\) (Q1956582)

Let \(1\leq p< +\infty\) and \(\omega_p(f,\delta)=\sup_{0\leq h\leq\delta}(\int_0^{1-h}|f(x+h)-f(x)|^pdx)^{1/p}\) for \(f\in L^p(0,1)\) and \(0\leq\delta\leq 1\). For an increasing function \(\omega\) on \([0,1]\) satisfying \(\omega(0)=0\) and \(\omega(\delta +\eta)\leq\omega(\delta)+\omega(\eta)\) for \(0\leq\delta\leq \delta +\eta\leq 1\), let \(H^\omega_p\) be the class of all functions of \(f\in L^p(0,1)\) such that \(\omega_p(f,\delta)\leq \omega(\delta)\) for \(0\leq\delta\leq 1\). The author studies criteria for the embedding \(H^\omega_p\subseteq e^L\), where \(e^L\) is the space measurable functions \(f:[0,1]\rightarrow \mathbb{R}\) such that \(\int_0^1e^{|f(x)|}dx<\infty\).