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Application of Lebesgue differentiation theorem

  1. Mar 21, 2015 #1
    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. jcsd
  3. Mar 26, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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