A generalization of Gallagher's lemma for exponential sums

First we generalize a famous lemma of Gallagher on the mean square estimate for exponential sums by plugging a weight in the right hand side of Gallagher's original inequality. Then we apply it in the special case of the Ces`aro weight in order to establish some results mainly concerning the classical Dirichlet polynomials and the Selberg integrals of an arithmetic function {piccolosc f}, that are tools for studying the distribution of {piccolosc f} in short intervals. Furthermore, we describe the smoothing process via self-convolutions of a weight that is involved into our Gallagher type inequalities, and compare it with the analogous process via the so-called correlations. Finally, we discuss a comparison argument in view of refinements on the Gallagher weighted inequalities according to different instances of the weight.