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Abstract. It is shown that for Lp, 0 < p < 1, the Peetre K-functional between Lp and W; is identically zero. Useful measures that are equivalent to the moduli of smoothness are found. The equivalence results that are given are valid for 0

Ul'yanov-type inequalities are extended to include many measures of smoothness. Many of the results are valid for L p , p > 0. Observations and extensions for needed Nikolskii-type inequalities are given.