By B. Opic
This offers a dialogue of Hardy-type inequalities. They play a huge position in a number of branches of study akin to approximation thought, differential equations, idea of functionality areas and so on. The one-dimensional case is handled virtually thoroughly. numerous 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 should still hold). The N-dimensional case is handled through the one-dimensional case in addition to by utilizing acceptable distinct methods.
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Additional resources for Hardy-type inequalities
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. OPIC and is published here for the first time. 12). 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) .
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  (Theorem 202) or N. DUNFORD, J. T. SCHWARTZ  (Chap. VI, 21 Section 11). 2) b dxJ dYJ a a 4' E M+(a,b) 1/r b j Y(y) < If 'P(x) dxJ dy y . 13. 2. Lemma. ber < p < q < m 1 BL = BL(a,b,w,v,q,p) holds for every and v, w E W(a,b) .
G. H. HARDY, J. E. LITTLE- WOOD, G. POLYA  (Theorem 202) or N. DUNFORD, J. T. SCHWARTZ  (Chap. VI, 21 Section 11). 2) b dxJ dYJ a a 4' E M+(a,b) 1/r b j Y(y) < If 'P(x) dxJ dy y . 13. 2. Lemma. ber < p < q < m 1 BL = BL(a,b,w,v,q,p) holds for every and v, w E W(a,b) . 18) is finite. 3) with k(q,p) Proof. 24). 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) .
Hardy-type inequalities by B. Opic