# Application of Lebesgue differentiation theorem

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1. Mar 21, 2015

### fderingoz

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, $$\lim\limits_{r\rightarrow0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{\Phi}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{\Phi}(\mathbb{R}^{n})}}=|f(x)|,$$ 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 $\|f\|_{L_{\Phi}}=\|f\|_{L_{p}}$.

2. Mar 26, 2015