Let $f:\mathbb{R}\times \mathbb{R}^N \to \mathbb{R}^N$ be an $L^1$ function. Assume that $$\mathcal M f(x,y) = \sup_{r< \bar r}\frac{1}{B_r(y)} \int_{B_r(y)} f(x,z)dz \to 0 $$ as $\bar r \to 0$ for a.e. $y \in \mathbb{R}^d$.

Is it true that for a.e. $y \in \mathbb{R}^d$ $$\int_I \mathcal M f(x,y)dx \to 0,$$ where $I \subset \mathbb{R}$ is an interval?

If not, what counterexample shows that?

At least, if we fix $R\gg 1$, does it hold for a.e. $x \in B_R(0) \setminus E$, where $E$ is a set of arbitrarily small (non zero) measure?


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