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For functions defined on the entire real axis or a semiaxis, we obtain Kolmogorov-type inequalities that estimate the Lp-norms (1 ≤ p < ∞) of fractional derivatives in terms of the Lp-norms of functions (or the Lp-norms of their truncated derivatives) and their Lp-moduli of continuity and establish their sharpness for p = 1: Applications of the obtained inequalities are given.
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