Download PDF by B. Opic: Hardy-type inequalities

By B. Opic

ISBN-10: 0470215844

ISBN-13: 9780470215845

ISBN-10: 0582051983

ISBN-13: 9780582051980

This gives a dialogue of Hardy-type inequalities. They play an immense function in quite a few branches of study equivalent to approximation concept, differential equations, thought of functionality areas and so forth. The one-dimensional case is handled nearly thoroughly. a number of techniques are defined and a few extensions are given (eg the case of estaimates regarding better 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 acceptable particular ways.

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Example text

Ity 1 < p,q < - v, w E W(a,b) . 7) Proof. e. 9) b111/4 [J(Hf)(x) w(x) dx] b( < CLIJ fp(x) v(x) dx) a a holds for every f E M+(a,b) b . E E (a,b) Let b j(HLf)q(x) w(x) dx Z J( a be fixed. 10) If w(x) dx] f(x) dx] -< CLIJ fP(x) v(x) dx] a f E M+(a,b) 1/p (b ll II E for every ( a Let us take . 12) vn(x) = v(x) + n(1 + x2/(p'-l)) Obviously vn E W(a,b) too, and for , x x E (a,b) b dt J a we have vn-p'(t) dt = n J n E N define For . 15) for {Ju(x)j q q w(x) dx] n c- N ll 1/ p b ( < CLIIIu'(x)Ip vn(x) dx] a for every and a u E ACL(a,b) .

9. The case This case has been treated in literature only in . 15, which was proved by V. G. 7 (i)). e. q < p, [1], 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.

11) holds with the (finite) constant u e ACL(a,b) . 24) 1 where Proof. 25) I J x for every x c (a,b) numbers such that . e. 27) f fp(x) v(x) dx > 0 n . 29) r/q fn( x l f w(t) dtI an vl-p1-p ,(t) dtir/q' v ( An J an l '(x) dx = j an 31 . 11) in view of . 27). 24). 10. Remark. 15. The procedure used here is a small modification of the original proof of V. G. MAZ'JA [1]. In the conclusion of this Section we give a proof of an auxiliary assertion, which will be useful in the sequel. 11. Lemma. Let 1 < q < p < m and v, w E W(a,b) .

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Hardy-type inequalities by B. Opic


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