  Let $X$ be a rearrangement-invariant space over a non-atomic $\sigma$-finite
measure space $(\mathscr{R},\mu)$ and let $\alpha\in(0,\infty)$. We define the
functional \begin{equation*}
  \|f\|_{X^{\langle \alpha \rangle}} =
\|((|f|^\alpha)^{**})^{\frac{1}{\alpha}}\|_{\overline{X}(0,\mu(\mathscr{R}))},
\end{equation*} in which $f$ is a $\mu$-measurable scalar function defined on
$(\mathscr{R},\mu)$ and $\overline{X}(0,\mu(\mathscr{R}))$ is the
representation space of $X$. We denote by $X^{\langle \alpha \rangle}$ the
collection of all almost everywhere finite functions $f$ such that
$\|f\|_{X^{\langle \alpha \rangle}}$ is finite. These spaces recently surfaced
in connection of optimality of target function spaces in general Sobolev
embeddings involving upper Ahlfors regular measures.
  We present a variety of results on these spaces including their basic
functional properties, their relations to customary function spaces and mutual
embeddings and, in a particular situation, a characterization of their
associate structures. We discover a new one-parameter path of function spaces
leading from a Lebesgue space to a Zygmund class and we compare it to the
classical one.
