Atlas: The properties of functions on the boundary surfaces by A.D. Dzhabrailov

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3rd International ISAAC Congress
August 20-25, 2001
Freie Universitaet Berlin
Berlin, Germany

Organizers
ISAAC Board

The properties of functions on the boundary surfaces
by
A.D. Dzhabrailov
Azerbaijan Architecture and Building University, Baku 370069, Azerbaijan, Rasim Mukthar str. 10

Consider the spaces W_p^{á r
ñ}(G;s), B_{p, q}^{á r
ñ}(G;s) of functions f=f(x) of points x=(x₁;¼;x_s) Î E_n of several groups of variables x_k=(x_{k, 1};¼;x_{k, n_k}) Î E_{n_k} (k=1, 2, ¼, s), defined in domain G Ì E_n=E_n₁×¼×E_{n_s} (n=n₁+n₂+¼+n_s; 1 £ s £ n), satisfying the condition ßemihorn".

These spaces introduced as the completion the set of sufficiently smooth of finite functions in E_n with respect to the norm

|| f|| _{W_p^{á r
ñ}( G;s)}=
å
i Î Q
|| f|| _{L_p^{á rⁱ
ñ}( G;s)}, || f|| _{B_{p, q}^{á r
ñ}(G;s)}=
å
i Î Q
|| f||_{L_{p, q}^{á rⁱ
ñ}(G;s)},
(1)
where sum is taken of all possible vectors i=(i₁, ¼, i_n) Î Q with coordinates i_k Î {0, 1, ¼, n_k} for all k=1, 2, ¼, s.

The vector rⁱ=(r₁^i₁;¼;r_s^i_s) (i Î Q) related vector r=(r₁;¼;r_s) (r_k=(r_{k, 1};¼;r_{k, n_k}); k=1, 2, ¼, s) so that r_k^i_k=(0;¼;r_{k, i_k};0;¼;0) (k=1, 2, ¼, s).

Let m=m₁+m₂+¼+m_a < n₁+n₂+¼+n_s, (1 £ a £ s £ n) where 1 £ m_k £ n_k (k=1, 2, ¼, a). The class P¹ is defined the m-dimensional surface G_m and be introduced the consept of L_p-traces of functions f=f(x) and their derivatives on surface G_m as limit of contraction the corresponding functions on surface G_m+h Ì G for |h|® 0 in L_p(G_m+h) space. The existens of traces of functions f=f(x) on the surface G_m is denote by f|_{G_m} Î L_p(G_m).

Are founded conditions, at which are proved the exists the traces of functions on surface G_m and has proved validity the integral inequalities type of imbedding theorems

|| Dⁿf|_{G_m}||_{W_q^{< r >}(G_m;a)} £ C||f||_{W_p^{< r >}(G;s)}, || Dⁿf|_{G_m}||_{B_{q, q^*}^{< r >}(G_m;a)} £ C||f||_{B_{p, q}^{< r >}(G;s)}.

Note that in the case s=1 the results of the report corespond to the investigations of traces of functions from the known S.L.Sobolev-L.N.Slobodetskii and S.M.Nikol'skii-O.V.Besov spaces, and in the case s=n to these investigations of boundary properties of functions from corresponding functional spaces with dominated mixed derivative cited in papers by S.M.Nikol'skii, A.D.Dzhabrailov, P.I.Lizorkin and others.

1. F.G.Maksudov, A.D.Dzhabrailov. Method integral representation in theory of spaces. Baku, Ëlm" 2000, V. 1, 200 pp.

Date received: June 12, 2001