Integral version of the lebesgue differentation theorem

Question

Given $f\in L^p(B_1)$, where $B_1$ is the unit ball centered at $0$ in $\mathbb{R}^d$, we know that the point $a\in B$ such that $$ \lim_{r\to 0} \frac{1}{|B(a,r)|} \int_{B(a,r)} |f-(f)_{B(a,r)}|^p =0 $$ have full measure in $B$, where $(f)_{B(a,r)}$ is the average of $f$ on the ball $B(a,r)$ of center $a$ with radius $r$.
I wounder if there is an integral version of the kind : for $\frac{1}{2}>r>0$ consider $$\phi_r : a \in B(0,\frac{1}{2}) \mapsto \int_{B(a,r)} |f-(f)_{B(a,r)}|^p\ \ \ \text{and}\ \ \ \psi_r : a\in B(0,\frac{1}{2})\mapsto \int_{B(a,r)} |f|^p $$ Is there a $q\geq 1$, such that $$ \Vert \phi_r \|_{L^q(B(0,r))} \underset{r\to 0}{=} o \left( \Vert \psi_r\Vert_{L^q(B(0,r))} \right) $$ Or is there a weaker space $X$, such as Lorentz space, for which we can exchange the $L^q$ norm with the norm on $X$ ?