By B. Opic

This gives a dialogue of Hardy-type inequalities. They play an immense function in numerous branches of research corresponding to approximation idea, differential equations, concept of functionality areas and so on. The one-dimensional case is handled nearly thoroughly. a variety of techniques are defined and a few extensions are given (eg the case of estaimates concerning greater order derivatives, or the dependence at the type of funcions for which the inequality should still hold). The N-dimensional case is handled through the one-dimensional case in addition to by utilizing applicable exact methods.

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28). Assume that CL < m . 28) for (b - a) < CL ess sup 1 f(x) = 1 , we obtain = CL a

10. The case the case 1 < p,q < °° llr/q' dtl 0 dv*l1-p/ 1/r dx l dx J1 < . < p < W 1 [Jfdt ) . , 0 < q < 1 Up to now we have dealt only with . Recently, G. 7 alto for the case mentioned above see Section 9. - 3. 15 Let us start with an auxiliary assertion. The Minkowski A modification of the Minkowski integral inequality. 1. 1) ll U a c K(x,y) dyJ Kr(x,Y) dx] l/r c a holds for every non-negative measurable function r ? , G. H. HARDY, J. E. LITTLE- WOOD, G. POLYA [1] (Theorem 202) or N. DUNFORD, J.

6. Comments. 18) is finite. 14. 2 - - especially as concerns Lemma a modification of the former proofs given by B. MUCKENHOUPT [1] (the case p = q ) and J. S. BRADLEY [1], V. C. MAZ'JA [1]. 24). 18) (J. CL < p1/q(p')I/p' BL S. BRADLEY [17, V. S. KOKILASHVILI [1] and - by another method - P. 19) CL < q1/q(q')1/P'B (V. C. MAZ'JA [1]). 2. 3) defined as k(q,p) g(l + P = inf g(s) s>1 provided < p < q < - 1 Consequently, the constant . CL up to now best estimate of ; leads to the k(q,p) this estimate is due to B.