Some properties of spaces with multiwieghted derivatives.
by
Kalybay Aigerim
Institute of Mathematics, Kazakhstan, Almaty
Coauthors: Oinarov Ryskul (Institute of Mathematics, Kazakhstan, Almaty)

Let n be the natural number and [`(\alpha)]=(\alpha₀, \alpha₁, ..., \alpha_n) be a set of real numbers.

For the function f:(0, 1) --> R we define the differential operation:

D[`(\alpha)]0f(t) \equiv t\alpha0f(t)  ,   D[`(\alpha)]if(t) = t\alphai\fracddtD[`(\alpha)]i-1f(t),   i=1, 2, ..., n  ,

where each derivative is in generalized sense.

The operation D_[`(\alpha)]ⁱf(t), 0 <= i <= n is called \alpha - multiweighted i order derivative of f.

We denote by W_{p, [`(\alpha)]</sub><sup>n</sup> \equiv W<sub>p, [`(\alpha)]}ⁿ(0, 1), 1 < p < \infty the space of the functions f:(0, 1) --> R, which have \alpha - multiweighted n order derivatives on (0, 1) and for which the norm:

||f||Wp, [`(\alpha)]n = ||D[`(\alpha)]nf||p+ n-1 å \iota = 0  |f(i)(1)|

is finite.

The space W_{p, [`(\alpha)]}ⁿ was investigated in [1, 2], and if \alpha_n=\gamma, \alpha_i=0, i=0, 1, ..., n-1 it will become the well known space W_{p, \gamma}ⁿ (see [3]).

For the functions f in W_{p, [`(\alpha)]}ⁿ we study the question on existence of the finite limit:

lim t --> 0+  n-1 å j=i  Ki+1, j(t, t0)D[`(\alpha)]jf(t) = Bit0f(0)  ,

where 0 <= i <= n-1, t₀ > 0, K_{i+1, j}(t, t₀)=\int_t^t₀t_i+1^-\alpha_i+1\int_t^t_i+1t_i+2^-\alpha_i+2...\int_t^t_j-1t_j^-\alpha_jdt_jdt_j-1...dt_i+1 for i < j, K_{i+1, j}(t, t₀) \equiv 1 for i=j.

We prove that there exists B_i^t₀f(0), i=0, 1, ..., n-1 for each f in W_{p, [`(\alpha)]</sub><sup>n</sup> if and only if when max<sub>0 <= i <= n-1</sub>\gamma<sub>i</sub> < 1-\frac1p, where \gamma<sub>i</sub>=\alpha<sub>n</sub>+\sum<sub>k=i+1</sub><sup>n-1</sup>(\alpha<sub>k</sub>-1),   i=0, 1, ..., n-2, \gamma<sub>n-1</sub>=\alpha<sub>n</sub>, and more over the norm ||f||<sub>Wp, [`(\alpha)]ⁿ} is equivalent to the functional:

||D[`(\alpha)]nf||p+ k å i=0  |Bit0f(0)|+ l å i=0  |f(i)(1)|  ,     k+l >= n  ,     0 <= k, l <= n-1  .

It's known [3], that generally speaking there not exists the finite limit lim_{t --> 0+}f⁽ⁱ⁾(t), 0 <= i <= n-1 for f from Wⁿ_{p, \gamma} when \gamma > n-\frac1p. We show that for \gamma > n-\frac1p there exists the set [`(\alpha)]=(\alpha<sub>0</sub>, \alpha<sub>1</sub>, ..., \alpha<sub>n</sub>) such that W<sup>n</sup><sub>p, \gamma</sub>\hookrightarrowW<sup>n</sup><sub>p, [`(\alpha)]</sub> and in addition there exists B_i^t₀f(0), i=0, 1, ..., n-1 for each f in W_{p, [`(\alpha)]</sub><sup>n</sup> and W<sup>n</sup><sub>p, \gamma</sub>={f:f in W<sup>n</sup><sub>p, [`(\alpha)]},   lim_{t --> 0+}t^\gamma-n+if⁽ⁱ⁾(t)=0,   i=0, 1, ..., n-1}.

References.

[1]. Baideldinov B.L., Oinarov R. Inner properties of one weighted class. - Proceedings of International Conference "Functional spaces, theory of approximations, nonlinear analysis", Moscow, 1995.

[2]. Baideldinov B.L. Multiweighted spaces and its application to differential equations. - Proceedings of International Conference in functional spaces, Moscow, 1998, p. 29 - 33.

[3]. Kudryavtscev L.D. On equivalent norms in weighted spaces. - Proceedings of Mathematical Institute of Academy of Sciences, USSR, v.170, 1984, p. 161 - 190.

Date received: July 6, 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 # cahv-60.