This is Theorem 4.3 page 106 in "Analysis" of Lieb and Loss.
There is a step in the proof that I cannot understand: (I am writing in a less elegant way for the sake of clarity) $$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\chi_{\{x:f(x)>a\}}{(x)}\chi_{\{x:h(x)>b\}}{(y)}\chi_{\{x:|x|<c\}}{(x-y)}\,dxdy$$ $$\leq \frac{\int_{\mathbb{R}^{n}}\chi_{\{x:f(x)>a\}}{(x)}dx\,\int_{\mathbb{R}^{n}}\chi_{\{x:h(x)>b\}}{(x)}\,dx\, C_n\, c^n}{\max\{\int_{\mathbb{R}^{n}}\chi_{\{x:f(x)>a\}}{(x)}dx,\,\int_{\mathbb{R}^{n}}\chi_{\{x:h(x)>b\}}{(x)}\,dx,\, C_n\, c^n\}} $$ It says in the book, the integrals in $x,y$ can be estimated from above by replacing one of the characteristic functions by 1.
My question is: why can we bound
$$\int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}} f(x)g(y) \chi_{\{x:|x|<R\}}{(x-y)}\, dx\,dy= \int_{|x-y|<R}f(x)g(y) \, dx\,dy$$ by $$ \frac{\int_{\mathbb{R}^{n}} f(x)dx\,\int_{\mathbb{R}^{n}} g(x)dx\; |B_{R}(o)|}{\max\{\int_{\mathbb{R}^{n}} f(x)dx,\,\int_{\mathbb{R}^{n}} g(x)dx,\; |B_{R}(o)|\} }$$
You have to use $0 \le f(x),g(y) \le 1$.
We always have $\int_{|x-y|<R} f(x)g(y)dxdy \le \int_{(\mathbb{R}^n)^2} f(x)g(y)dxdy = \int_{\mathbb{R}^n} f(x)dx\int_{\mathbb{R}^n} g(y)dy$, so we're done if the maximum is $|B_R(0)|$. We also have $\int_{|x-y| < R} f(x)g(y)dxdy \le \int_{|x-y|<R} g(y)dxdy = |B_R(0)|\int_{\mathbb{R}^n}g(y)dy$. Same thing with $f$.