The imbedding theorems on the Lizorkin-Triebel-Morrey type space
by
V.S. Guliev
Baku State University, Baku, 370069, Azerbaijan, Rasim Mukhtar str. 10
Coauthors: A.M. Najafov

The spaces F_{p, \theta}^l(Rⁿ), l in (0, \infty)ⁿ, 1 <= p, \theta <= \infty called Lizorkin-Triebel spaces, were introduced in [1] in terms of property of Fourier integral decompositions of functions; by its reserived different descriptions this spaces, studing as they imbedding and others properties. In [2, 3] pursuance of systematic research this spaces for 0 <= p, \theta <= \infty. In [4, 5] introduced new variants descriptions norm for space F_{p, \theta}^l(G), studing equivalent norms on these spaces, and imbeddings of them in the case of an open set G subset Rⁿ satisfies the flexible horn condition.

In the work is constucted Lizorkin-Triebel-Morrey type space F_{p, \theta, a, æ, \tau}^l(G) in the case the set G satisfies the flexible horn condition and studing as with point of view theory imbeddings some properties function from construction space.

Let æ , l in (0, \infty)ⁿ, \frac1l _* =\frac1n\sum_i=1ⁿ\frac1l_i, \lambda = \fracl _* l in (0, \infty)ⁿ, G subset Rⁿ-satisfy condition flexible \lambda-horn introduced by O.V.Besov, p in (1, \infty)ⁿ, \theta in (1, \infty), a in [0, 1]ⁿ, t in (0, \infty), \tau in [1, \infty], [\upsilon]₁=min{1, \upsilon}, m_i-natural, k_i- nonnegative integer number , m_i > l_i > k_i >= 0, i=1, 2, ... , n, \upsilon₀-fixed positive number.

The Lizorkin-Triebel-Morrey type space F_{p, \theta, a, æ , \tau}^l(G) is called space function f in L^loc(G) with norm

|| f|| Fp, \theta, a, æ , \taul(G)=||f|| p, a, æ , \tau;G+ n å i=1  || ì í î 1 ó õ 0  [t(ki-li)\lambdai\deltaimi-ki(t\lambda)Dikif(·)] \theta\fracdtt ü ý þ \frac1\theta   || p, a, æ, \tau,

where

|| f|| p, a, æ , \tau= sup x in G  ì í î \vartheta0 ó õ 0  [[\upsilon]1-\sumj=1n\fracæ jajpj|| f|| p, G\upsilon(x)]\tau\fracd\upsilon\upsilon ü ý þ \frac1\tau   ,

\delta_i^m_i(t^\lambda)f(x)=\int_-1¹| \Delta_i^m_i(t^\lambda_iu, G_t^\lambda)f(x)| du.
At \tau = \infty, with space F_{p, \theta, a, æ , \infty}^l(G)=F_{p, \theta, a, æ} ^l(G), is called Lizorkin-Triebel-Morrey space .

Let 1 < p <= q <= \infty, l¹ in (0, \infty)ⁿ, 1 < \theta < \theta₁ < \infty, 1 <= \tau₁ < \tau₂ <= \infty, \alpha = (\alpha₁, \alpha₂, ... , \alpha_n), \alpha_j >= 0-nonnegative integer number, j=1, 2, ... , n, f in F_{p, \theta, a, æ , \tau1}^l(G), \epsilon = 1-\sum_j=1ⁿ[ \alpha_j+( 1-æ _ja_j) ( \frac1p_j-\frac1q_j) ] \frac1l_j > 0, \epsilon₀=1-\sum_j=1ⁿ[ \alpha_j+(1-æ _ja_j)\frac1p_j] \frac1l_j, then D^\alpha:F_{p, \theta, a, æ , \tau1}^l(G) --> L_{q, b, æ , \tau2}(G);

|| D\alphaf|| q, G <= C T(\epsilon-1)l * || f|| p, a, æ , \tau1;G+

+C T\epsilonl * n å i=1  || { [t(ki-li)\lambdai\deltami-ki(t\lambda)Dikif] \theta\fracdtt} \frac1\theta|| p, a, æ , \tau1,

|| D\alphaf|| q, b, æ , \tau2;G <= C ||f|| Fp, \theta, a, æ , \tau1l(G), (p <= q < \infty);

but, if \epsilonl _* > l _* ¹, then

D\alpha:Fp, \theta, a, æ , \tau1l(G) --> Fq, \theta1, b, æ , \tau2l1(G),    D\alpha:Fp, \theta, a, æ , \tau1l(G) --> Bq, \theta1, b, æ , \tau2l1(G);

|| D\alphaf|| Fq, \theta1l1(G) <= C T(\epsilon-1)l * -l * 1|| f|| p, a, æ , \tau1;G+

+C T\epsilonl * -l * 1 n å i=1  ||{ [ t(ki-li)\lambdai\deltami-ki(t\lambda)Dikif] \theta\fracdtt} \frac1\theta|| p, a, æ , \tau1, (p <= q < \infty);

|| D\alphaf|| Fq, \theta1, b, æ , \tau2l1(G) <= C || f|| Fp, \theta, a, æ , \tau1l(G), (p <= q < \infty);

|| D\alphaf|| Bq, \theta1l1(G) <= C T(\epsilon-1)l * -l * 1|| f|| p, a, æ , \tau1;G+

+C T\epsilonl * -l * 1 n å i=1  ||{ [ t(ki-li)\lambdai\deltami-ki(t\lambda)Dikif] \theta\fracdtt} \frac1\theta|| p, a, æ , \tau1;

|| D\alphaf|| Bq, \theta1, b, æ , \tau2l1(G) <= C || f|| Fp, \theta, a, æ , \tau1l(G), (p <= q < \infty);

In particular, si \epsilon₀ > 0, then generalized derivatve D^\alphaf condition on G and

sup x in G  | D\alphaf| <= C T(\epsilon0-1)l * || f|| p, a, æ , \tau1;G+

+C T\epsilon0l * n å i=1  || {[ t(ki-li)\lambdai\deltami-ki(t\lambda)Dikif] \theta\fracdtt} \frac1\theta|| p, a, æ , \tau1

where T-arbitary number from (0, min(1, T₀)], b=(b₁, b₂, ... , b_n), b_j number satisfy :
0 <= b_j <= 1,     if     \epsilon₀ > 0,
0 <= b_j < 1,     if     \epsilon₀=0,
0 <= b_j <= 1+\fracl _* \epsilon₀(1-a_j)\sum_j=1ⁿ(\lambda_j-æ _ja_j)=a_j+\fracl _* \epsilon(1-a_j)\sum_j=1ⁿ(\lambda_j-æ_ja_j),     if     \epsilon₀ < 0. Are proved, that for f in F_{p, \theta, a, æ, \tau}^l(G) generalized derivative D^\alphaf satisfy on G condition Holder in metric L_q.

1. Lizorkin P.I. The operators trusfum of a differentiation in theory imbedding.//Material Sov.-Cech. symposium by theory of function. Novosibirsk, 1971-Novosibirsk, AN SSSR, 1972, p. 135-139.

2.H.Triebel. Interpolation theory. Function spaces. Differential operators. Berlin 1978.

3. H.Triebel. Theory of function spaces. Basel-Boston-Stuttgart 1983.

4. Besov O.V. Integral representations of functions and imbedding theorems for the domain with condition of the flexible horn. Trudy Mat. Inst. Steklov, 1984, v.170, p.12-30.

5. Besov O.V. The spaces of Sobolev-Liouville and Lizorkin-Triebel on the domain. Trudy Mat. Inst. Steklov, 1990, v.192, p.20-34.

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-07.