Bernstein's Inequality for the Weyl Derivatives of Trigonometric Polynomials in the Space $L_0$

A logarithmic asymptotics for the behavior with respect to $n$ of the exact constant in Bernstein's inequality for the Weyl derivative of positive noninteger order of trigonometric polynomials of order $n$ in the space $L_0$ is obtained. It turns out that the order in $n$ of the behavior of this constant for positive noninteger orders of the derivatives has exponential growth in contrast to the power growth in the well-studied case of classical derivatives of positive integer order.