Some imbedding theorems for weighted Sobolev spaces of banach-valued functions
by
R.A. Bandaliev
Baku State University, Baku, 370069, Azerbaijan, Rasim Mukhtar str. 10

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) = æ è ó õ Rn  ||f(x)||Bp\omega(x)dx ö ø 1/p   ,     1 <= p <= \infty

(for p = \infty the usual modification is understood). We put Lp(Rn;B) for \omega = 1. We denote the norm by ||·||Lp(Rn;B) (respectively, ||·||p, B ).
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)+ n å j=1  ||Djljf||Lp, \omegaj(Rn;B).

Let a=(a₁, ..., a_n), a_i > 0, i=1, ..., n and let a function \rho(x) be a positive solution of the equation \sum_i=1ⁿ x_i²\rho^-2a_i=1.
Let K_\alpha be a given function on Rⁿ with the following properties:
a)   K_\alpha(x) = \rho(x)^{\alpha- |a|}, if 0 < \alpha < |a|;
b)   if \alpha = 0, then K₀(t^a x) = t^-|a|K₀(x), and
\int_S K₀(x)\sum_i=1ⁿ a_i x_i² d\sigma(x)=0,       \int₀¹ \omega_K0(t)[ dt/t] < \infty,
where \omega_K0(t)=sup{|K₀(x)- K₀(y)|; x, y in S,   |x-y| <= t} and S is unit sphere on Rⁿ.
Consider the operator T_\alpha f(x) = \int_RⁿK_\alpha(x-y) f(y)dy.

Theorem 1 Let 1 < p <= q < \infty, 1 < p < [(|a|)/(\alpha)], \alpha = |a|([ 1/p]- [ 1/q]). Let also f in L_{p, \omega}(Rⁿ;B), and let a weight pair (\omega(\rho(x)), \omega₁(\rho(x))) satisfies the following conditions :
\omega(t) and \omega₁(t) are positive functions on (0, \infty), and there is a constant C > 0 such that
(sup_{t < \tau <= 8t}\omega₁(\tau))^p/q <= Cinf_{t < \tau <= 8t}\omega(\tau);
sup_{t > 0}(\int_t^\infty\omega(\tau)\tau^{-1- |a|q/p'}d\tau)^p/q(\int₀^t\omega₁(\tau)^1-p'\tau^{|a|- 1}d\tau)^p-1 < \infty
sup_{t > 0}(\int₀^t\omega₁((\tau)\tau^{|a|- 1}d\tau)^p/q(\int_t^\infty\omega^1-p'(\tau)\tau^{-1- |a|p'/q} d\tau)^p-1 < \infty;
Then

æ è ó õ Rn  ||T\alpha f(x)||Bq \omega1(\rho(x))dx ö ø 1/q   <= C æ è ó õ Rn  ||f(x)||Bp \omega(\rho(x))dx ö ø 1/p

(1)

the inequality (1) is valid also for every almost x in Rⁿ, where C independent of f.
If, in addition, B is \zeta-convex banach lattice and the kernel K₀ satisfies condition b), then the inequality (1) is valid also for 1 < p = q < \infty.

The weight w is said to belong to A_p(Rⁿ), 1 < p < \infty if

sup B in Rn  æ è |B|-1 ó õ B  w(x)dx ö ø æ è |B|-1 ó õ B  w(x)1- p'dx ö ø p-1   < \infty.

Theorem 2 Let 1 < p <= q < \infty, \alpha = |a|([ 1/p]- [ 1/q]), the radial function j in A_1+[ q/(p')](Rⁿ), and let v(t) and w(t) be positive monotone on (0, \infty) functions, \omega = vj, \omega₁ = u j.
Let \omega and \omega₁ satisfying the condition 1) or 2):
1)   v and u are increasing functions on (0, \infty), and
sup_{t > 0} (\int_t^\infty \omega(\tau)\tau^{-1- |a|p'/q}d\tau)^p/q(\int₀^t/2 \omega₁(\tau)^{1- p'} \tau^{|a|- 1}d\tau)^{p- 1} < \infty;
2)   v and u are decreasing functions on (0, \infty), and
sup_{t > 0} (\int₀^t/2\omega₁(\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 the inequality (1) holds.
If, in addition, B is \zeta-convex banach lattice and the kernel K₀ satisfies condition b), then the inequality (1) is valid also for 1 < p = q < \infty.

Remark. We note that theorem 1 in the scalar case on the homogeneous groups was obtained independently by V.S.Guliev [1], [2] and V.M.Kokilashvili, A.Meskhi [3], A.Meskhi [4]. Theorem 2 in the scalar case for j \equiv 1 was obtained by V.S.Guliev [2] and generalized case by V.M.Kokilashvili, A.Meskhi [3], A.Meskhi [4].

Theorem 3 Let B be a Banach space, k=(k₁, ..., k_n), l=(l₁, ..., l_n) > 0, k=(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). Also, let the weight pairs (\omega_j, \omega), j=0, 1, ..., n, satisfies conditions of theorem 1 or theorem 2 . Then for k < 1 the continuous imbedding

Dk Wp, \omega0, \omega1, ..., \omeganl1, ..., ln(Rn;B)\hookrightarrow Lq, \omega(Rn;B).

(2)

is valid. If, in addition, B is \zeta-convex banach lattice, then the inequality (2) is valid also for 1 < p = q < \infty, k=1.

1. V.S.Guliev, Imbedding theorems for weighted Sobolev spaces of B-valued functions.
Dokl. Ros. Akad. Nauk, 1994, v.338, 4, p.264-268.

2. V.S.Guliev, Integral operators on function spaces on the homogeneous groups and on domains in Rⁿ.
Doctor's dissertation, Moscow, Mat. Inst. Steklov, 1994, p.1-329. (Russian)

3. V.Kokilashvili., A.Meskhi, Two-weighted inequalities for singular integrals defined on homogeneous groups.
Proc.of A Razmadze Mathematical Institute, 1997, v.112, p.57-90.

4. A.Meskhi, Two-weighted inequalities for potentials defined on homogeneous groups.
Proc.of A Razmadze Mathematical Institute, 1997, v.112, p.90- 111.

Date received: June 11, 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-05.