Exponential sums over primes in short intervals and an application to the Waring-Goldbach problem

Let λ(n) be the von Mangoldt function, be real and 2 ≤ y ≤ x. This paper improves the estimate for the exponential sum over primes in short intervals S_k(x,y;α)=∑ λ(n)e(n^k α) x<n≤x+y when k ≥ 3 for in the minor arcs. When combined with the Hardy-Littlewood circle method, this enables us to investigate the Waring-Goldbach problem concerning the representation of a positive integer as the sum of th powers of almost equal prime numbers, and improve the results of Wei and Wooley [On sums of powers of almost equal primes. Proc. Lond. Math. Soc. (3) 111(5) (2015), 1130-1162].