By B. Opic

This offers a dialogue of Hardy-type inequalities. They play a major position in a number of branches of study reminiscent of approximation thought, differential equations, conception of functionality areas and so on. 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 greater order derivatives, or the dependence at the type of funcions for which the inequality may still hold). The N-dimensional case is handled through the one-dimensional case in addition to through the use of applicable unique techniques.

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3). 7. Remark. 14). 7 (i) is solvable (with a finite constant ,, I CL ) f BL < m Let us prove it. 7 (i) is solvable with CL < m . e. 9) 1 holds. 14) - CL ). 13). 16), too. 15. The first concerns the 'if' part. 8. Lemma. ber < q < p < w 1 AL = AL(a,b,w,v,q,p) holds for every v, w E W(a,b) . 19) is finite. 21) Proof. 10, it suffices . 12). 21). 9. Lemma. Let 30 1 < q < p < - and v, w t W(a,b). 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 .

G(xk) = (1 - 2-k)IBI Set . , , there exists an increasing sequence such that . 8) implies S(x1) + n < w(x) 0 Since this inequality holds for every 48 k E N , we have w(x) = - for all x E C ICI . >0 However, this contradicts the assumption w E W(a,b) , since . 4. Lemma. Let v, w C W(a,b) p = Let . 6) and Zet BL CL . 5). 9) BL = CL . Proof. 10) BL < ( 5 . 2) for . 10) is proved. 14) . 15) Then (cf. 10) for S(x) > 0 for some for every x0 E (a,b) implies that for p = 1 ). Denote . x E (a,b) , since the assumption leads to a contradiction with E (a,b) , S(x0) = 0 v (F W(a,b) .

The assumption BL implies that the integral t vl-p' (y) dy a is finite for every t E (a,b) . 4) h(t) = II v 1-p Consequently, the function 1/(p's) ' (y) dyI a where s is a fixed number from 0 < h(t) < Let f E M+(a,b) . (1,00) for every 00 satisfies t E (a,b) . 10. 3. Remark. 14. 7 (i) has a solution. 4. Lemma. 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 .