Weighted effects some integral operators and its applications
by
V.S. Guliev
Baku State University, Baku, 370069, Azerbaijan, Rasim Mukthar str. 10

Two-weighted L_p -estimates with monotone weights are obtained in [1] for integral operators of potential type and for singular integral operators (see [1], Theorems 1.8, 1.9 ). With the help of these theorems we obtain weighted imbedding theorems for the Sobolev spaces W_{p, \omega0, ... , \omegan}^{l₁, ... , l_n}(Rⁿ). Weight ëffects" are also discovered for integral operators arising on the basis of the integral representation of Il'in and Besov for domains in Rⁿ satisfying the l-horn condition (see Theorem 2.2), as well as for a parabolic potential and a parabolic singular integral (see Theorems 3.1 and  3.3). With the help of these theorems we obtain weighted imbedding theorems for the spaces W_{p, \omega0, ... , \omegan}^{l₁, ... , l_n}(G) defined on a domain G with nonsmooth boundary (see Theorem 2.3). In particular, this theorem leads to weighted imbedding theorems for the Sobolev spaces W_{p, \omega0, ... , \omegan}^{l₁, ... , l_n}(G) for weights of exponential growth.

Let Rⁿ-n-dimensional Euclidian spaces of point x=(x₁, ... , x_n), R₊₊ⁿ = {x: x in Rⁿ, x_i > 0, i = 1, ... , n}, \rho(x) = \sum_{i = 1}ⁿ |x_i|^1/a_i, a_i > 0, i = 1, ... , n, |a| = \sum_{i = 1}ⁿ a_i. Let \omega a positive measurable function defined on R₊₊ⁿ. Denote by L_{p, \omega}(R₊₊ⁿ) the space of measurable functions on R₊₊ⁿ with finite norm ||f||_{Lp, \omega(R++ⁿ)}^p = \int_R++ⁿ|f(x)|^p\omega(x)dx, 1 <= p < \infty.

We define the anisotropic Sobolev space Wp, \omega0, \omega1, ... , \omeganl1, ... , ln(R++n), l=(l1, ... , ln), li >= 0, i = 1, ... , n the integers, as the set of functions f(x), x in R++n, that have generalized derivatives Dljjf, and finite norm
|| f ||Wp, \omega0, \omega1, ... , \omeganl1, ... , ln(R++n) = ||f||Lp, \omega0(R++n)+ \sumj=1n ||Dljf||Lp, \omegaj(R++n).

Theorem 1 Let 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')], and the weight pairs (\omega_j, \omega), j=0, 1, ... , n, satisfies condition :

v(t) and v_j(t), j=0, 1, ... , n be positive increasing functions on (0, \infty), and
existsC > 0:    for all\tau in (0, \infty),   \omega(\tau)^p/q <= C\omega_j(\tau/2),

Then for æ <= 1 the continuous imbedding is valid
D^k W_{p, \omega0, \omega1, ... , \omegan}^{l₁, ... , l_n}(R₊₊ⁿ)\hookrightarrow L_{q, \omega}(R₊₊ⁿ)

[1]. Guliev V.S. Two-weighted inequalities for integral operators in L_p-spaces, and their appl. Tr. Steklov Ins. of the Ros. Akad. Nauk, 1993, v.204, p.113-136.

Date received: June 18, 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-28.