I was reading the book "Partial Differential Equation" written by Lawrence C. Evans, coming up with a question.
On page 718, Evans wrote
$$\int_V \left(\int_{B(x,\epsilon)} \eta_{\epsilon}(x-y) |f(y)|^p dy\right) dx \\ \leq \int_W |f(y)|^p \left(\int_{B(y,\epsilon)} \eta_{\epsilon}(x-y) dx\right) dy$$
Where $V,W,U$ are open sets, $V\subset\subset W\subset \subset U$, and $\eta_{\epsilon}(x) := \frac{1}{\epsilon^n} \eta(\frac{x}{\epsilon})$, $\eta$ is the standard mollifier, $f \in L_{loc}^p(U)$
I want to ask why the inequality holds. I was trying to prove it so hard but I have no ideas.
Best Answer
This is an application of Fubini's theorem, followed by the fact that the integrand is positive and $V$ is a subset of $W$.