Sums over numbers with restricted prime factors

For k a non-negative integer, let P_k(n) denote the kth largest prime factor of n where P₀(n) = +∞ and if the number of prime factors of n is less than k, then P_k(n) = 1. We shall study the asymptotic behavior of the sum Ψ_k(x, y; g) = Σ_{1 ≤ n ≤ x, Pk(n) ≤ y}g(n), where g(n) is an arithmetic function satisfying certain general conditions regarding its behavior on primes. The special case where g(n) = μ(n), the Möbius function, is discussed as an application.