ON ZYGMUNDS LEMMA AND LACUNARY FOURIER-SERIES

Concerning the lacunary Fourier series SIGMA-c(k) exp(in(k) x) of f is-an-element-of L2 with lacunae satisfying the condition B2, Zygmund has proved that lambda(-1) \E\ parallel-to f parallel-to 2(2) less-than-or-equal-to parallel-to f chi(E) parallel-to 2(2) less-than-or-equal-to lambda \E parallel-to f parallel-to 2(2) if n1 is large depending on given lambda > 1 real and E subset-of [- pi, pi] of positive measure; chi(E) being the characteristic function of E. We apply this to show that we also have parallel-to f chi(E) parallel-to 4 less-than-or-equal-to C parallel-to f chi(E) parallel-to 2 for such f. This, together with the convexity argument of Littlewood, then leads us to a generalization of above Zygmund's lemma. The generalized version then helps us to estimate the tails of SIGMA \f(n(k))\beta, 0 < beta less-than-or-equal-to 1, in terms of local mean modulus of continuity; which in turn gives Szasz type local theorems for the absolute convergence of lacunary Fourier series.