Generalisation of Lebesgue differentiation theorem to Orlicz spaces

Question

If $f\in L_{p}^{\rm loc}(\mathbb{R}^{n})$ and $1\leq p<\infty$, then a stronger version of Lebesgue differentiation theorem holds: $$\lim\limits_{r\rightarrow 0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}=|f(x)|$$ for almost all $x\in\mathbb{R}^{n}$. See, for example, "Grafakos, Classical Fourier Analysis, Third Edition, Page 101-102". Here $\chi_{B(x,r)}$ denotes the characteristic function of the open ball $B(x,r)$. I wonder that is there an analogue of this property in Orlicz spaces, that is, $$\lim\limits_{r\rightarrow0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{\Phi}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{\Phi}(\mathbb{R}^{n})}}=|f(x)|~~~~~~~~~~~~~~~~~~(\ast)$$ for almost all $x\in\mathbb{R}^{n}$ ?

Where $\Phi:[0,\infty)\to [0,\infty)$ is an increasing, continuous, convex function with $\Phi(0)=0$ and $$ \|f\|_{L_{\Phi}(\mathbb{R}^{n})}:=\inf\{\lambda>0:\int_{\mathbb{R}^{n}}\Phi\left(\frac{|f(x)|}{\lambda}\right)dx\leq 1\}. $$ It is a generalization of $L_p$ norm. Indeed, if we take $\Phi(t)=t^p,\,1\leq p< \infty$ we get $\|f\|_{L_{\Phi}}=\|f\|_{L_{p}}$.

I think we should define a maximal function such that $$ M^{\Phi}f(x)=\sup_{r>0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{\Phi}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{\Phi}(\mathbb{R}^{n})}} $$ and obtain a weak type inequality for this operator. As an application of this we get the desired equation. Unfortunately, i couldn't do any of these operations. I need your helps.

Answer 1

I found the answer of my own question. In the paper "Hencl, Stanislav; Kleprlík, Luděk; Composition of q-quasiconformal mappings and functions in Orlicz-Sobolev spaces. Illinois J. Math. 56 (2012), no. 3, 931–955." The authors proved that the equation ($\ast$) holds provided that $\Phi\in\Delta^{\prime}$, i.e. there exist $C>0$ such that $$\Phi(x y)\leq C \Phi(x)\Phi(y)\text{ for every }x,y\geq 0.$$

They also add a remark such that:

"Surprisingly we cannot have ($\ast$) for general Young functions, because the term $\|\chi_{B(x,r)}\|$ does not necessarily scale well for small $r$. For $\alpha<0$ we construct a function $f\in L^q\log^{\alpha}L$ such that the limit in ($\ast$) is infinite everywhere. "