Let $\Delta\geq 1$ be a constant. I need to prove that
$$ 2\int_1^T x\int_{T/2x^2}^{2T/x^2}\left(\frac{\sin(\frac{\Delta}{4}\log\frac{2\pi}{t})}{\frac{\Delta}{4}\log\frac{2\pi}{t}}\right)^2 dt\,dx = \int_{T/2}^{2T}\left(\int_{1/t}^{T/t}\left(\frac{\sin (\frac{\Delta}{4}\log 2\pi y)}{\frac{\Delta}{4}\log 2\pi y}\right)^2\frac{dy}{y}\right)dt $$ I guess I should manipulate the LHS by some kind of change of variable to obtain the RHS but everything I tried doesn't work. Do you have any suggestion? Thanks!