On “rough” numbers represented by the sum of two squares

An integer $n\geqslant 2$ is called as $y$-rough number if $n$ is free of prime divisors less than or equal to $y$. Let $\mathcal{B}(x;y)$ be the quantity of $y$-rough numbers not exceeding $x$, represented by the sum of two squares. In the talk, we discuss the estimate
$$ \mathcal{B}(x;y) \ll \dfrac{x}{\sqrt{\ln x\ln y}}, $$
which is valid for $x\to +\infty$ and for any $y$, $2\leqslant y \leqslant \sqrt{x}$ (the constant in the symbol $\ll$ is absolute).