  A generalization of the theory of Y. Brudnyi \cite{yuri}, and A. and Y.
Brudnyi \cite{BB20a}, \cite{BB20b}, is presented. Our construction connects
Brudnyi's theory, which relies on local polynomial approximation, with new
results on sparse domination. In particular, we find an analogue of the maximal
theorem for the fractional maximal function, solving a problem proposed by
Kruglyak--Kuznetsov. Our spaces shed light on the structure of the
John--Nirenberg spaces. We show that $SJN_{p}$ (sparse John--Nirenberg space)
coincides with $L^{p},1<p<\infty.$ This characterization yields the
John--Nirenberg inequality by extrapolation and is useful in the theory of
commutators.
