On an Integral of Marcinkiewicz
Daniel Waterman · Transactions of the American Mathematical Society · 1959
Let/(0) be a function of period 2w and class LT, p>l.By F( 6) is meant a primitive of/(f), i.e., F(6) = f f(x)dx + C. J o Marcinkiewicz [3] defined the function Pr(6) = j J \[F(B + t) + F(0 -0 -2F(d)Y/r+l)dtl ( r * rF(d + t) + F(8 -t)-2F(9)1 r\ w and demonstrated the inequalities(1) IW|P ^ Ap\\f\l, P = 2 11/11, = ^J!m,||p, k^2.Since addition of a constant to f(6) does not alter p.p, the second inequality clearly requires f f(6)d6 = 0.