Basic functional properties of certain scale of rearrangement-invariant spaces

Abstract

Let $X$ be a rearrangement-invariant space over a non-atomic $\sigma$-finite measure space $(\mathcal{R},\mu )$ and let $\alpha \in (0,\infty )$. We define the functional \begin{equation*} \|f\|_{X^{\langle α\rangle}} = \|((|f|^α)^{**})^{\frac{1}α}\|_{\overline{X}(0,μ(\mathscr{R}))}, \end{equation*} in which $f$ is a $\mu$-measurable scalar function defined on $(\mathcal{R},\mu )$ and $\overline{X}(0,\mu (\mathcal{R}))$ is the representation space of $X$. We denote by $X^{⟨\alpha ⟩}$ the collection of all almost everywhere finite functions $f$ such that $\Vert f{\Vert }_{X^{⟨\alpha ⟩}}$ 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.

22 pages, 1 figure

References in corpus (1)

• Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

Cited by in corpus (3)

• Weighted inequalities for a superposition of the Copson operator and the Hardy operator
• Discretization and antidiscretization of Lorentz norms with no restrictions on weights
• Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces