by $$\langle |f |\rangle_{A, Q}= \inf\{\lambda>0: \frac 1{|Q|}\int_Q A(|f|/\lambda)\le 1\}.$$

We say that $f\in L^A_{\rm{loc}}$ if $\langle |f |\rangle_{A, Q}<\infty$ for all cubes $Q$. The relative sizes of Young functions $A,B$ are compared with the symbol $\succeq;$ we say that $B\succeq A,$ if there exist constants $C,t_0>0$ such that $A(t)\leq CB(t),$ when $t>t_0.$ Finally, we define the $B_p$ class: a Young function $A\in B_p$ for $p>1$ if $$\int_1^\infty \frac{A(t)}{t^p}\frac{dt}{t}<\infty.$$ We record the following properties, which can be found at least in [@CMP10 Chapter 5] (see also [@P95]):

\[prop:young\] Given a Young function $A$, it holds that

i)  for any $t\ge 0$, $t \le A^{-1}(t) \bar A^{-1}(t)\le 2t$,

ii) for any cube $Q$, $$\label{eq:generalholder}
                \langle |f g| \rangle_Q\le 2\langle |f| \rangle_{A, Q} \langle |g| \rangle_{\bar A, Q}.$$ More generally, if $A$, $B$, and $C$ are Young functions such that for all $ t\ge t_0>0,$ $$B^{-1}(t) C^{-1}(t)\le c A^{-1}(t),$$ then $$\langle |fg|\rangle_{A,Q}\lesssim \langle |f|\rangle_{B,Q}\langle |g|\rangle_{C,Q},$$

iii) if $B\succeq A,$ then $\langle{|f|}\rangle_{A,Q}\lesssim \langle{|f|}\rangle_{B,Q}$ and $M_A \lesssim M_B,$

iv) if $\bar A\in B_{p'}$, then $A(t)\succeq t^p$ and $ \langle |f|^p\rangle_Q^{\frac 1p}\lesssim \langle|f|\rangle_{A, Q}$.

[@P95]\[PE95\] $M_A:L^p\to L^p$ boundedly if and only if $A\in B_p$.

Now we are ready to give the following definition:

\[sufdef\] Given 