Consider the strictly monotone continuous function $d:\mathbb{R^+}\to\mathbb{R^+}$, and denote by $\mathcal{D}$ the space of all measurable functions $f:[0,1]\to\mathbb{R}$ such that:
$$\int_0^1 d(|f(x)|) dx<\infty$$
I am wondering if there is any result on the (dense and continuous) embedding of this function space in the $L_p$ spaces?