Goldbach's problem with primes in arithmetic progressions and in short intervals

Abstract: Some mean value theorems in the style of Bombieri-Vinogradov's theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd n\not\equiv 1 (6), Goldbach's ternary problem n=p_1+p_2+p_3 is solvable with primes p_1,p_2 in short intervals p_i \in [X_i,X_i+Y] with X_{i}^{\theta_i}=Y, i=1,2, and \theta_1,\theta_2\geq 0.933 such that (p_1+2)(p_2+2) has at most 9 prime factors.