Some imbedding theorems for weighted Sobolev spaces of banach-valued functions.
by
R.A. Bandaliev
Institute of Mathematics & Mechanics of Acad. Sci. Azerbaijan, Baku

Let Rⁿ-n-dimensional Euclidian spaces of point x=(x₁, ... , x_n), k, l in N₀, N₀ = N \cup {0}. Let B-be a Banach space, \omega a positive measurable function defined on Rⁿ. Denote by L_{p, \omega}(Rⁿ;B) the space of strongly measurable functions on Rⁿ with values in B with finite norm ||f||_{Lp, \omega(Rⁿ;B)}^p = \int_Rⁿ||f(x)||_B^p\omega(x)dx, 1 <= p < \infty.

We say that a Banach space B is \zeta-convex if there exists symmetric function \zeta(\xi, \eta) on B×B, that is convex with respect to each of the variables and satisfies the conditions \zeta(0, 0) > 0, \zeta(\xi, \eta) <= ||\xi+\eta||B for ||\xi||B = ||\eta||B=1 (see [1]).
We define the anisotropic Sobolev space Wp, \omega0, \omega1, ..., \omeganl1, ..., ln(Rn;B), l=(l1, ..., ln) >= 0, li >= 0, i = 1, ... , n the integers, as the set of B-valued functions f(x), x in G, that have generalized derivatives Dljjf, with values in B and finite norm
||f;Wp, \omega0, \omega1, ..., \omeganl1, ..., ln(Rn;B)|| = ||f||Lp, \omega0(Rn;B)+\sumj=1n||Djljf||Lp, \omegaj(Rn;B).

Theorem. Let B be a Banach space, k=(k₁, ..., k_n), l=(l₁, ..., l_n) > 0, æ = (k, 1/l) <= 1, (k+1/p-1/q, 1/l)=1, 1 < p <= q < \infty, a=(a₁, ..., a_n), a_i=1/l_i, i=1, ..., n, and let weight functions \omega, \omega₀, \omega₁, \omega_n depend only on \rho(x), \omega(t) = v(t) j(t), \omega_j(t) = v_j(t) j(t), the radial function j in A_1+[ q/(p')](Rⁿ), the weight pairs (\omega_j, \omega), j=0, 1, ..., n, satisfies condition a) or b):
a) v(t) and v_j(t), j=0, 1, ..., n be positive incresing function on (0, \infty), and
sup_{t > 0}( \int_t^\infty \omega_j(\tau) \tau^{-1-[(|a|q)/(p')]} d\tau)^{p/ q} (\int₀^t/2 \omega(\tau)^1-p'\tau^{|a| -1}d\tau)^p-1 < \infty,
b) v(t) and v_j(t), j=0, 1, ..., n be positive decreasing function on (0, \infty), and
sup_{t > 0} (\int₀^t/2 \omega_j(\tau) \tau^{|a| -1}d\tau)^{p/ q} ( \int_t^\infty\omega(\tau)^{1- p'} \tau^{-1-[(|a|p')/q]} d\tau) ^p-1 < \infty.
Then for æ < 1 the continuous imbedding
D^k W_{p, \omega0, \omega1, ..., \omegan}^{l₁, ..., l_n}(Rⁿ;B)\hookrightarrow L_{q, \omega}(Rⁿ;B) is valid. If, in addition, B is \zeta-convex banach lattice, then the imbedding is valid also for 1 < p = q < \infty, æ = 1.

1. V.S.Guliev, Imbedding theorems for weighted Sobolev spaces of B-valued functions. // Dokl. Ros. Akad. Nauk, 1994, v.338, No 4, p.264-268.

Date received: August 12, 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 # caia-07.