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






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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
Received: 13.07.2011

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}


Linking options:
  • http://mi.mathnet.ru/eng/ufa109
  • http://mi.mathnet.ru/eng/ufa/v3/i3/p140

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Уфимский математический журнал
    Number of views:
    This page:202
    Full text:57
    References:40
    First page:2

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020