RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Ufimsk. Mat. Zh.: Year: Volume: Issue: Page: Find

 Ufimsk. Mat. Zh., 2011, Volume 3, Issue 3, Pages 140–151 (Mi ufa109)

About the unimprobality of the limiting embedding theorem for different metrics in the Lorentz spaces with Hermite's weight

E. S. Smailova, A. I. Takuadinab

a Institute of Applied Mathematics National Academy of Sciences of Kazakhstan, Karaganda, Kazakhstan
b Karaganda State Medical University, Karaganda, Kazakhstan

Abstract: In this article we obtained inequality of different metrics in the Lorentz spaces with Hermit's weight for multiple algebraic polynomials. On this basis we established a sufficient condition of embedding of different metrics in the Lorenz spaces with Hermite's weight. Its unimprobality is shown in terms of the “extreme function”.
Let $f\in L_{p,\theta}(\mathbb R_n;\rho_n)$, $1\leq p<+\infty$, $1\leq\theta\leq+\infty$. The sequense $t\{l_k\}_{k=0}^{+\infty}\subset\mathbb N$ is such that $l_0=1$ and $l_{k+1}\cdot l_k^{-1}>a_0>1$, $\forall k\in\mathbb Z^+$. $f(\bar x)=\sum_{k=0}^{+\infty}\Delta_{l_k,…,l_k}(f;\bar x)$ is some presentation of the functions in the metric $L_{p,\theta}(\mathbb R_n;\rho_n)$, where $\Delta_{l_0,…,l_0}(f;\bar x)=T_{1,…,1},\Delta_{l_k,…,l_k}(f;\bar x)=T_{l_k,…,l_k}(\bar x)-T_{l_{k-1},…,l_{k-1}}(\bar x)$, $\forall k\in\mathbb N$. Here
$$T_{l_k,…,l_k}(\bar x)=\sum_{m_1=0}^{l_k-1}…\sum_{m_n=0}^{l_k-1}a_{m_1,…,m_n}\prod^n_{i=1}x^{m_i}_i$$
are algebraic polynomials for all $k\in\mathbb Z^+$.
$1^0$. If the series
$$A(f)_{p\theta}=\sum_{k=0}^{+\infty}l_k^{\tau(\frac n{2p}-\frac n{2q})}\|\Delta_{l_k,…,l_k}(f)\|_{L_{p,\theta}(\mathbb R_n;\rho_n)}^\tau$$
converge under some $q$ and $\tau$: $p<q<+\infty$, $0<\tau<+\infty$, then $f\in L_{q,\tau}(\mathbb R_n;\rho_n)$ and we have the inequality
$$\|f\|_{L_{q,\tau}(\mathbb R_n;\rho_n)}\leq C_{pq\theta\tau n}\times(A(f)_{p\theta})^\frac1\tau.$$

$2^0$. The condition $1^0$ is unimprovable in the sense that there exists a function $f_0\in L_{p,\theta}(\mathbb R_n;\rho_n)$ and $A(f_0)_{p\theta}$ diverges for it and $f_0\notin L_{q,\tau}(\mathbb R_n;\rho_n)$. At the same time, the function $f_0\in L_{q-\varepsilon,\tau}(\mathbb R_n;\rho_n)$ for all $\varepsilon>0$: $p<(q-\varepsilon)<q$.

Keywords: Lorentz's space, Hermitte's weight, nonincreasing rearrangement, inequality of different metrics, theorem in embedding, non improving.

Full text: PDF file (438 kB)
References: PDF file   HTML file

Bibliographic databases:
UDC: 517.51

Citation: E. S. Smailov, A. I. Takuadina, “About the unimprobality of the limiting embedding theorem for different metrics in the Lorentz spaces with Hermite's weight”, Ufimsk. Mat. Zh., 3:3 (2011), 140–151

Citation in format AMSBIB
\Bibitem{SmaTak11} \by E.~S.~Smailov, A.~I.~Takuadina \paper About the unimprobality of the limiting embedding theorem for different metrics in the Lorentz spaces with Hermite's weight \jour Ufimsk. Mat. Zh. \yr 2011 \vol 3 \issue 3 \pages 140--151 \mathnet{http://mi.mathnet.ru/ufa109} \zmath{https://zbmath.org/?q=an:1249.46023}