By B. Opic
This offers a dialogue of Hardy-type inequalities. They play an immense function in a variety of branches of research equivalent to approximation conception, differential equations, concept of functionality areas and so on. The one-dimensional case is handled virtually thoroughly. a number of methods are defined and a few extensions are given (eg the case of estaimates regarding better order derivatives, or the dependence at the category of funcions for which the inequality may still hold). The N-dimensional case is handled through the one-dimensional case in addition to by utilizing applicable particular methods.
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13) z zq/p'+l + Qw Since . 12). (ii) function Let us fix A > K . 16) J dt < w(t) a Moreover, x z0(x) - z1(x) = f(x) - P! 13). 5). 4. Remark. 5) is solvable. 4). 1 be satisfied. 5. Lemma. 18) (l + q 11/p, k(q,p) _ (1 + p Proof. 17) will be proved by contradiction. 3. 19). (ii) BL = w If ally. 17) holds trivi- , w E W(a,b) BL < - Then, due to the definition of . BL and the we have , b 0 < J w(y) dy < - for t E (a,b) t Consequently, the function b ,/q tf (r f(t) = sBL,IJ w(y) dy, -p- vl-p,(y) dy t is continuous on (a,b) J a for every s E (1,-) b dyJP,/q .
14 is proved completely. 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.
Denote . x E (a,b) , since the assumption leads to a contradiction with E (a,b) , S(x0) = 0 v (F W(a,b) . 17) then > 0 0< Define 50 . IM . and if we denote Mn = {x E (a,E); IMnI x E (a,b) v-1(x) > S(E) - n l , Therefore, there exists a subset I <- . 18) J a 1 n1 v(x) dx CL f(x) v(x) dx = CL I CL IMnI [S(E) - I . 12) yields . 20) Since vn E W(a,b) x E (a,b) for every vn(x) ? e. 5) holds for the pair v ) with the same constant CL . 19). 11) is proved. p = The case , 1 q = - , . 21) BL = BL(a,b,w,v,°°,1) = ,v-1II sup (x,b).