The Titchmarsh problem with integers having a given number of prime divisors

The asymptotics for the number of representations of $N$ as $N\to\infty$ is expressed as the sum of a number having $k$ prime divisors and a product of two natural numbers. The asymptotics is found for $k\le(2-\varepsilon)\ln\ln N$ and $(2+\varepsilon)\ln\ln N\le k\le b\ln\ln N$, where $\varepsilon>0$. The results obtained are uniform with respect to $k$.