By B. Opic
This gives a dialogue of Hardy-type inequalities. They play an incredible function in a variety of branches of study comparable to approximation conception, differential equations, conception of functionality areas and so forth. 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 regarding better order derivatives, or the dependence at the classification of funcions for which the inequality should still hold). The N-dimensional case is handled through the one-dimensional case in addition to through the use of applicable unique methods.
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MAZ'JA . 9. The case This case has been treated in literature only in . 15, which was proved by V. G. 7 (i)). e. q < p, , and the corresponding necessary was also investigated by V. G. 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.
I) The foregoing result is due to P. 17). 18). 5 (v) were ex- tended by P. 11). 15) cover also the result of G. A. 1. 8. 14. e. 17), appeared probably for the first time in G. TOMASELLI  (1969). 19) K = 1 inf sup f a 4) and ql/q AL = CL (cf. 6). ] Now, we shall show that our considerations are not purely formal. However, we start with an auxiliary lemma. 3. Lemma. w E W(a,b) . 7) Proof. e. 7) is not true, and denote I = (a,b) , S(x) = ess sup w(t) a
4) and ql/q AL = CL (cf. 6). ] Now, we shall show that our considerations are not purely formal. However, we start with an auxiliary lemma. 3. Lemma. w E W(a,b) . 7) Proof. e. 7) is not true, and denote I = (a,b) , S(x) = ess sup w(t) a