On Lp integrability and convergence of trigonometric series

We first give a necessary and sufficient condition for x(-gamma)phi(x) is an element of L-p, 1 < p < infinity, 1/p - 1 < gamma < 1/p, where phi(x) is the sum of either Sigma(infinity)(k=1) a(k) cos kx or Sigma(infinity)(k=1) b(k) sin kx, under the condition that {lambda(n)} (where lambda(n) is a(n) or b(n) respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients lambda(n) and the sum function phi(x) under the condition that {lambda(n)} is an element of MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of phi(x) in L-p norm.