Chebyshev polynomial iterations and approximate solutions of linear operator equations

Using several types of reversed Hölder inequalities (which hold for functions defined on (0,infty) satisfying one or two conditions of quasi-monotonicity) the authors employ results from their previous paper [Acta Sci. Math. (Szeged) {bf 59} (1994), no.~1-2, 221--239; [msn] MR1285442 (95f:26021) [/msn]] to obtain reversed Hardy inequalities for other ranges of parameters. More precisely, they prove that under suitable assumptions on the parameters A, B, p, q the inequalities |xsp A(Hsb if)(x)|sb{Lsp q}le C|xsp B f(x)|sb{Lsp p}quad(i=1,2) (where 0<p<qleinfty, (Hsb1f)(x)=(1/x)intsb0sp xf, (Hsb2f)(x)=(1/x)intsb xsp infty f, f satisfies some quasi-monotonicity conditions and the constant C does not depend on f) are valid. Moreover, for some ranges of the parameters i, A, B, p, q, the best possible constants C are found. Unfortunately, the method does not enable one to find the best possible constants in all cases; thus the question of finding them in some cases remains open. par In the last section of the paper the authors give several applications. They derive some embeddings between homogeneous Besov spaces and Lorentz spaces, including estimates of the embedding constants. par There are similar results in the literature [see V. D. Stepanov, Trans. Amer. Math. Soc. {bf 338} (1993), no.~1, 173--186; [msn] MR1097171 (93j:26012) [/msn]; C. J. Neugebauer, Publ. Mat. {bf 35} (1991), no.~2, 429--447; [msn] MR1201565 (93m:26042) [/msn]], which involve general weighted functions w, v instead of power type functions xsp A, xsp B; however, the best possible constants C in such generality are not known.