Limits embedding theorems on the anisotropic Sobolev-Bessel space.
by
A.Kh. Narimanov
Azerbaijan Architecture and Building University, Baku , Azerbaijan
Coauthors: V.S. Guliev

In this work is proved boundedness the anisotropic Fourier-Bessel singular integral operators acts boundedly in the space L_p^\gamma(R₊ⁿ). As well proved limits embedding theorems on the Sobolev-Bessel space W_{p, \gamma}^{l₁, ... , l_n}(Rⁿ₊).

Let Rⁿ be the n-dimensional Euclidean space of points x=(x₁, ... , x_n), |x| = (\sum_i=1ⁿ x_i²)^[ 1/2], R₊ⁿ={x in Rⁿ; x_n > 0}, \gamma > 0 and let given vector a=(a₁, ... , a_n), a_i >= 1 (i=1, ... , n), |a| = \sum_i=1ⁿ a_i, |x|_a=max_{1 <= i <= n}|x_i|^[ 1/(a_i)].

By L_p^\gamma(R₊ⁿ) denote a space of measurable functions f, with the finite norm

|| f|| Lp\gamma (R+n) = æ è ó õ R+n  | f(x)|p xn\gamma dx ö ø [ 1/p]   ,     1 <= p < \infty.

T^y f(x) of operator generalized shift introduced by B.M.Levitan

Ty f(x)=C\gamma ó õ \pi 0  f æ è x'-y', Ö   xn2+yn2-2xnyncos\alpha   ö ø sin\gamma-1\alphad\alpha,

where x=(x', x_n), y=(y', y_n), C_\gamma = [(\Gamma([(\gamma+1)/2]))/(\Gamma([(\gamma)/2])\Gamma([ 1/2]))].

Consider the space W_{p, \gamma}^l(Rⁿ₊) with norm

|| f|| Wp, \gammal (R+n)=||f|| Lp\gamma(R+n)+ n-1 å i=1  ||Dliif||Lp\gamma(R+n)+||Bnlnf||Lp\gamma(R+n),

where B_n=[(\partial²)/(\partialx_n²)]+[(\gamma)/(x_n)][(\partial)/(\partialx_n)] differential operator of Bessel.

Theorem. Let the function f in W_{p, \gamma}^l(Rⁿ₊), 1 < p < \infty, l=(l₁, ... , l_n), \nu = (\nu₁, ... , \nu_n), l_i > 0, \nu_i >= 0, i=1, ... , n the integer numbers, such that |\nu:l|=1.

Then for \gamma =/= 1, 3, ... , 2l_n-1 the continius embedding

DBn\nuWp, \gammal(R+n) subset \succ Lp\gamma(R+n),

is valid, where D_Bn^\nu = D₁^\nu₁ ... D_n-1^\nu_n-1B_n^\nu_n.

Moreover

|| DBn\nuf|| Lp\gamma(R+n) <= C || f||Wp, \gammal(R+n),

where C independent on f.

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-06.