The Bernstein-Szego Inequality for Fractional Derivatives of Trigonometric Polynomials

On the set A, of trigonometric polynomials of degree n >= 1 with complex coefficients, we consider the Szego operator D-theta(alpha) defined by the relation D-theta(alpha) f(n) (t) = cos theta D-alpha f(n) (t) sin theta D-alpha f(n) (t) for alpha, theta is an element of R N, where alpha >= 0. Here, D-alpha f(n) P f and D(alpha)f(n) are the Weyl fractional derivatives of (real) order a of the polynomial f of its conjugate /Th. In particular, we that, if a > n In 2n, then, for any 0 E N, the sharp inequality Mcos 0 D' f sin 0 D prove p f L, holds on the set 3;-Th in the spaces Lp for all p >= 0. For classical derivatives (of integer order a > 1), this inequality was obtained by Szego in the uniform norm (p = infinity) in 1928 and by Zygmund for 1 <= p <= infinity in 1931-1935. For fractional derivatives of (real) order alpha >= 1 and 1 <= p <= infinity, the inequality was proved by Kozko in 1998.