There are two well-known mapping reducibilities: polytime and logspace. In both cases, the length of the output string can be at most polynomial in the length of a given input string. If we allow to use more space (super-logspace), then the length of the output string can be super-polynomial.
Therefore, it seems interesting to me to ask whether this blow-up helps us to reduce some difficult problems to some easier problems when using super-logspace $ \left(\omega(\log n) \right) $. I am not interested in the "cheap" tricks such as padding.
More generally, is there any literature on super-logspace (or super-polynomial) mapping reducibility?
DEFINITION
Language $ \mathtt{A} $ is $s(n)$-space mapping reducible to language $ \mathtt{B} $, shown as $ \mathtt{A} \le_{m}^{s} \mathtt{B} $, if there exists a $ s(n) $-space DTM $ \mathcal{D} $ outputting $ f_{\mathcal{D}} (w) $ for a given input $ w $ such that
\begin{equation*} w \in \mathtt{A} \mbox{ if and only if } f_{\mathcal{D}} (w) \in \mathtt{B}, \end{equation*}
where $ \mathcal{D} $ has a two-way read-only input tape, a two-way read/write work tape, and a one-way write-only output tape.
UPDATE: I have found an answer to my own question. But, I am still interested in further results.