Local divisor correlations in almost all short intervals

Let $k,l\ge 2$ be natural numbers, and let $d_k,d_l$ denote the $k$-fold and $l$-fold divisor functions, respectively. We analyse the asymptotic behavior of the sum ${\sum }_{x<n\le x+H_1}{d}_k(n)d_l(n+h)$. More precisely, let $\epsilon >0$ be a small fixed number and let $\mathrm{\Phi }(x)$ be a positive function that tends to infinity arbitrarily slowly as $x\to \mathrm{\infty }$. We then show that whenever $H_1\ge (\log x{)}^{\mathrm{\Phi }(x)}$ and $(\log x{)}^{1000k\log k}\le H_2\le H_1^{1-\epsilon }$, the expected asymptotic formula holds for almost all $x\in [X,2X]$ and almost all $1\le h\le H_2$.