The imbedding theorems on the Besov-Djabrailov-Morrey type space and its applications
by
A.M. Najafov
Baku State University, Baku, 370069, Azerbaijan, Rasim Mukhtar str. 10

In the work is constructed Besov-Djabrailov-Morrey type space B_{p, q, a, æ, t}^l(Q, G), received the new inteqral representation and studying as differential so difference- differential properties function from this spaces. By means of proven theorems imbedding studying solution (smothness solution) for one class quasielliptic equation.

Let open set G Ì Rⁿ satisfy condition flexible l-horn introduced by O.V.Besov [1]. e_n={1, 2, ¼, n}, e_n⁰=e_nÈ{0}, e Ì Q Ì e_n, where Q-for all fixed sets. Let u, h Î (0, ¥)ⁿ, i, j, i¢ Î e_n\Q, h_i=h_j, u_i=u_j, [u_j ]₁=min{1, u_j }, j=1, ¼, n.

For every x Î Rⁿ we assume

Iuæ (x)={y:| yj-xj| < ujæ j, j=1, 2, ¼, n},       Guæ (x)=GÇIuæ (x).

Let m=(m₁, ¼, m_n), m_j-natural number, k=(k₁, ¼, k_n), k_j-integer nonnegative number, m^eÈ{i}=(m₁^eÈ{i}, ¼, m_n^eÈ{i}); m_j^eÈ{i}=m_j at j Î eÈ{i}, m_j^eÈ{i}=0 at j Î e_n\(eÈ{i}), m₀ = l₀ = k₀=0, h₀, u₀-fixed positive vector. The space B_{p, q, a, æ , t}^l(Q, G) with p Î [1, ¥)ⁿ, q Î [1, ¥], t Î [1, ¥], a Î [0, 1]ⁿ and l, l, æ Î (0, ¥)ⁿ is defined to be the Banach space of locally integrable functions of f on G with norm (m_j > l_j-k_j > 0, j=1, 2, ¼, n):

|| f|| Bp, q, a, æ , tl(Q, G) =

= å e Ì Q  å i Î en0\Q  ì í î h0 ó õ 0  æ ç è || DmeÈ{i}(h, Gh)DkeÈ{i}f|| p, a, æ , t Õ j Î eÈ{i}  hjlj-kj ö ÷ ø q   Õ j Î eÈ{i}  dhj hj ü ý þ [1/(q)]   ,

where

|| f||p, a, æ, t= sup x Î G  ì í î J0 ó õ 0  é ë n Õ j=1  [uj]1-[(æ j aj)/(pj)]|| f||p, Guæ(x) ù û t   Õ j Î QÈ{i}  duj uj ü ý þ [1/(t)]

Note, that the space B_{p, q, 0, æ , ¥}^l(Q, G) º B_{p, q}^l(Q, G) was introduced and studying by A.D.Djabrailov in [2]. In the case Q=Æ the space B_{p, q, a, æ , t}^l(Q, G) was introduced by V.S.Guliev and studying in [3], but in the case Q º e_n and | Q| = n-1 (| Q| - quantity elements of set Q) the space B_{p, q, a, æ , t}^l(Q, G) º S_{p, q, a, æ , t}^lB(G) was introduced and studying in [4].

The theorems imbedding type are proved (q < q₁,  t₁ < t₂)

1) D^a:B_{p, q, a, æ 1, t1}^l(Q, G)® L_{q, b, æ , t2}(G)

2) D^a:B_{p, q, a, æ 1, t1}^l(Q, G)® B_{q, q1, b, æ , t2}^l1(Q, G)

3) Proved, that generalized derivative D^af on G satisfies condition Holder in metrics L_q for f from construction space.

Let r=(r₁, ¼, r_n), r_j > 0-integer, a = (a₁, ¼, a_n), b = (b₁, ¼, b_n) and r₀=0. Consider the following equation

å \Sb aj, bj £ rj j Î e Ì Q \endSb  å \Sb (a, [1/r])en\Q £ 1 (b, [1/r])en\Q £ 1 i, i¢ Î en0\Q\endSb  DaeÈ{i}( aaeÈ{i} beÈ{i¢}(x)DbeÈ{i¢}u) =

= å \Sb aj £ rj j Î e Ì Q \endSb  å \Sb (a, [1/r])en\Q £ 1 i Î en0\Q \endSb  DaeÈ{i}faeÈ{i}.

Propose, that a_{a^eÈ{i} b^eÈ{i¢}}(x)-bounded, measurable function on G Ì Rⁿ, a_(ab)^eÈ{i} º a_(ba)^eÈ{i} and

å \Sb aj, bj = rj j Î e Ì Q \endSb  å \Sb (a, [1/r])en\Q = (b, [1/r])en\Q=1 i, i¢ Î en0\Q\endSb  (-1)| aeÈ{i}|aaeÈ{i}beÈ{i¢}(x)xaeÈ{i}xbeÈ{i¢} ³

³ c1 å \Sb aj=rj j Î e Ì Q \endSb  å \Sb (a, [1/r])en\Q=1 i Î en0\Q \endSb  | xaeÈ{i}| 2,     c1=const > 0,

Let bounded domain G Ì Rⁿ satisfies condition flexible l-horn. Let also f_a^eÈ{i} Î L₂(G), for a_j < r_j, (a, [1/r])_en\Q < 1 and f_a^eÈ{i} Î L_p(G) for a_j=r_j (a, [1/r])_en\Q=1 for some p > 2. Then equation (1) we have :

1) generalized solution u Î W₂^r(Q, G);

2) that solution continuous in G and satisfies condition Holder in any subdomain, compact imbedding in G.

1. O.V.Besov, V.P.Il'yin and S.M. Nicol'skiy, Integral representations of functions and imbedding theorems, "Nauka", Moscow, 1996.

2. A.D.Djabrailov, On one Integral representation of smooth function and some classe of functional spaces. Dokl. Akad. nauk SSSR, v.166, No 6, 1966.

3. V.S.Guliev, A.M.Najafov, On some imbedding theorems of Besov-Morrey and Lizorkin-Triebel-Morrey type spaces. In:"Proc. Voronej winter school on Theory of Functions and related problems. Abstracts. Voronej. 1999. p.71.

4. Najafov A.M. The imbedding theorems for the space of Besov-Morrey type. Proc. of inst. of mathematics and mechanics. Baku, 2000, XII vol, p.97-104.

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-06.