Hardy-type Inequalities with several weights
by
B.L. Baideldinov
Institute of Mathematics, Pushkin str. 125, Almaty 480100, Kazakhstan

Let w and r be measurable functions on (0, 1) which are such that r(·)w^-1(·)=r(·) [ 1/w(·)] is integrable on (t, 1) for all t in (0, 1). We shall consider the semi-normed space W_pⁿ(w, r) of all functions y, for which

||wy(n)+ ry(n-1)||p < \infty,   ||f||p = æ è 1 ó õ 0  |f(t)|pdt ö ø 1/p   ,   1 < p < \infty.

W_pⁿ(w, r) is a Banach space with respect to norm

||y||Wpn(w, r)=||wy(n)+ry(n-1)||p+ n-1 å i=0  |y(i)(1)|.

Let \rho(·) be a nonnegative measurable function on (0, 1), which is such that ||\rho||_q < \infty, 1 < q < \infty. Let L_{q, \rho} denote the weighted Lebesque space with norm ||f||_{q, \rho}=||f \rho||_q.

Moreover, define p'=[ p/(p-1)],

A1 = sup 0 < x < 1  æ è 1 ó õ x  |w-p'(t)|exp{-p' 1 ó õ t  r(\xi)w-1(\xi)d\xi}dt ö ø 1/p   ×

× æ è x ó õ 0  \rhoq(t) æ è x ó õ t  (\tau-t)n-2exp{ 1 ó õ \tau  r(\xi)w-1(\xi)d\xi}d\tau ö ø q   dt ö ø 1/q   ,

A2= sup 0 < x < 1  æ è 1 ó õ x  |w-p'(t)| æ è t ó õ x  (t-\tau)n-2exp{ t ó õ \tau  r(\xi)w-1(\xi)d\xi}d\tau ö ø p'   dt ö ø 1/p'   æ è x ó õ 0  \rhoq(t)dt ö ø 1/q   .

The main result of this talk reads as follows:

Theorem. Let 1 < q <= p < \infty, n > 1 be a natural number. W_pⁿ(w, r) is continuously embedded in L_{q, \rho} if and only if A=max(A₁, A₂) < \infty. The norm of imbedding is equivalent to A.

When r(·) \equiv 0, y⁽ⁱ⁾(1)=0, i=0, 1, ... , n-1 we've obtained the estimation of Riemann-Liouville operator which was investigated in [1]. Note, that the proof of this Theorem refers to the estimation of the integral operator not satisfying the conditions from [2], [3]. The estimates of the operators of corresponding type were investigated in [4] in more general case.

Moreover in our talk we intend to introduce some other generalization of Hardy type inequalities with several weights.

References

[1].V.D.Stepanov, Math. USSR-Izv.36(1991).

[2].Steven Bloom and Ron Kerman, Proc.Amer.Math.Soc.113(1991), 135-141.

[3].R.Oinarov, Dokl.Akad.Nauk SSR, 319(1991), 1076-1078.

[4].B.L.Baideldinov and R.Oinarov, Dokl.Akad.Nauk Kazakhstan, 1996, N6, 19-22.

Date received: August 8, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-53.