The norm estimation problem for Fourier operators acting from
Lwp
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Lυq
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$$\mathbb{T}$$
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p
≦
q
< ∞ was investigated. These results has been generalized to the two-dimensional case and applied to obtain generalizations of the Bernstein inequality for trigonometric polynomials of one and two variables. Also, the rates of convergence of Cesaro and Abel-Poisson means of functions
f
∈
Lwp
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p
=
q
and
υ
≡
w
. The generalized Bernstein inequality applied to estimate the order of best trigonometric approximation of the derivative of functions
f
∈
Lwp
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\end{document}) in the space
Lυq
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\begin{document}
$$\mathbb{T}$$
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