Let $\mathcal{C}$ be the collection of function f such that f is a Lebesgue integrable
function on $[0, 1]^2$ such that it vanishes outside the set $\{ (x, y) \in [0, 1]^2 | x + y \leq 1 \}$, and below integrals are non zero for f.
$I(f) = {{\int}_0}^1{{\int}_0}^1 (f(t_1, t_2)^2 dt_1 dt_2 $,
$J_1(f) = {{\int}_0}^1 ({{\int}_0}^1 f(t_1, t_2)^2 dt_1 ) dt_2 $ and
$J_2(f) = {{\int}_0}^1 ({{\int}_0}^1 f(t_1, t_2)^2 dt_2 ) dt_1 $.
Then I've to show that M = $sup_{f \in \mathcal{C}} \frac{J_1(f) + J_2(f)}{I(f)} < \infty$
I came to this problem when I was reading James Maynard's "Small Gap Between Primes". I was trying to show that $(\int_0^1 f(t_1,t_2) dt_1)^2 \leq \int_0^1 f^2(t_1, t_2) dt_1$, then from Fubini Toneli Theorem will get $M \leq 2$. But I couldn't. Please help me.
Best Answer
Let $X$ be a finite measure space. Then, for any $1\leq p< q\leq +\infty$ $$L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m).$$ The proof follows from Hölder inequality. So here, you have $L^2([0,1]^2)\subset L^1([0,1]^2) $. Therefore, you can use Tonelli-Fubini and get that $M=2$.