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Journals / Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi / 2021 / Cilt: 21 - Sayı: 3

On Continuity of Functions Involving the Translation Operator on the Space $A_{p1,;q1}^{p2,;q2}left(G,;wright)$

$A_{p1,;q1}^{p2,;q2}left(G,;wright)$ Uzayı Üzerinde Öteleme Operatörünü İçeren FonksiyonlarınSürekliliği Üzerine

İbrahim DEĞİRMEN
Nilay DEĞİRMEN
Pages
549–558
Publication Date
2022-10-12
DOI
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⚡ TL;DR(Too Long; Didn't Read)

Also, by usingthese two results, it has been obtained that for an $h;in$ $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ the mapping $srightarrow L_sh$ fromG to $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ and the mapping $srightarrowleft|left|L_shright|right|$ from G to $mathbb{R}^+cup;left{0right}$ are continuous.

📌 3 Key Takeaways

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Objective & Hypothesis:Let G be a unimodular locally compact group and $w;in;B_p$ where $p=;minleft{p1,;p2right}$.
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Key Findings & Data:In thispaper, it has been proved that for every weight w that satisfies the condition $W;in;triangle_2left(Gright)$ andbelongs to the space $L^1left(Gright)$ or not, there can be a dense set everywhere in the space $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ with respect to the norm $left|left|.right|right|$ and for any elementh in these sets the function $srightarrow L_sh$ from the groupG to the space $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ is continuous.
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Conclusion & Impact:Also, by usingthese two results, it has been obtained that for an $h;in$ $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ the mapping $srightarrow L_sh$ fromG to $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ and the mapping $srightarrowleft|left|L_shright|right|$ from G to $mathbb{R}^+cup;left{0right}$ are continuous.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

Let G be a unimodular locally compact group and $w;in;B_p$ where $p=;minleft{p1,;p2right}$. In thispaper, it has been proved that for every weight w that satisfies the condition $W;in;triangle_2left(Gright)$ andbelongs to the space $L^1left(Gright)$ or not, there can be a dense set everywhere in the space $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ with respect to the norm $left|left|.right|right|$ and for any elementh in these sets the function $srightarrow L_sh$ from the groupG to the space $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ is continuous. Here, the set of simplefunctions inG and a subset of this set with support in a set of finite measure is used. Also, by usingthese two results, it has been obtained that for an $h;in$ $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ the mapping $srightarrow L_sh$ fromG to $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ and the mapping $srightarrowleft|left|L_shright|right|$ from G to $mathbb{R}^+cup;left{0right}$ are continuous.

Özet

G ünimodüler yerel kompakt grup ve $p=;minleft{p1,;p2right}$ olmak üzere $w;in;B_p$ olsun. Bu makalede, ilk olarak $W;in;triangle_2left(Gright)$ şartını sağlayan ve $L^1left(Gright)$ uzayına ait olan ya da olmayan her w ağırlığı için $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ uzayında $left|left|.right|right|$ normuna göre her yerde yoğun olan bir küme bulunabildiği ve bu kümelerdeki herhangi bir h elemanı için G grubundan $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ uzayına tanımlı $srightarrow L_sh$ fonksiyonunun sürekli olduğu ispatlanmıştır.Burada, Güzerinde tanımlı basit fonksiyonların kümesi ve bu kümenin sonlu ölçümlü bir kümede desteğe sahip bir alt kümesi kullanılır. Ayrıca bu iki sonuç yardımıyla her $h;in$ $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ için Ggrubundan $A_{p1,;q1}^{p2,;q2}left(G,;wright)$ uzayına $srightarrow L_sh$ tanımlı ve Ggrubundan $mathbb{R}^+cup;left{0right}$ kümesine tanımlı $srightarrowleft|left|L_shright|right|$ fonksiyonlarının sürekli olduğu elde edilmiştir.

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Cite This Article

DEĞİRMEN, İ., DEĞİRMEN, N. (2022). On Continuity of Functions Involving the Translation Operator on the Space $A_{p1,;q1}^{p2,;q2}left(G,;wright)$. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, Cilt: 21 - Sayı: 3, 549–558. https://ajindex.com/articles/429655

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References (11)

  1. Arino, M. A. and Muckenhoupt, B., 1990. Maximal functions classical Lorentz spaces and Hardy' s inequality with weights for nonincreasing functions. Transactions of the American Mathematical Society, 320(2), 727-735.
  2. Avcı, H. and Gürkanlı, A. T., 2007. Multipliers and tensor products of L( p,q) Lorentz spaces. Acta Mathematica Scientia, 27(B)(1), 107-116.
  3. Bonsall, F. F. and Duncan, J., 1973, Complete Normed Algebras, 80, Springer Verlag, Berlin, 230-237.
  4. Carro, M. J., Raposo, J. A. and Soria, J., 2007, Recent Developments in the Theory of Lorentz Spaces and Weighted Inequalities, 187, no. 877, Managing editor Robert Guralnick, Memoirs of the American Mathematical Society. Değirmen, N. ve Değirmen, İ., 2021. A_{p1,q1}^{p2,q2}(G,w) uzayı ve bazı topolojik özellikleri üzerine. Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 11(2), 1468-1480.
  5. Folland, G. B., 1995, A Course in Abstract Harmonic Analysis, CRS Press,
  6. Boca Raton, Florida, 36-47.
  7. Halmos, P. R., 1974,
  8. Measure Theory, Second edition, Springer Verlag, New York, 73-183.
  9. Hunt, R. 1966. On L( p,q) spaces. L’Enseignement Mathématique, 12, 249-276.
  10. Li, H. and Sun, Q., 2012. Multipliers and tensor products of the weighted Lorentz spaces $A_G^{p,;q}left(wright)$. Georgian Mathematical Journal, 19, 721-740.
  11. Yap, L. Y. H., 1969. Some remarks on convolution operators and L( p,q) spaces. Duke Mathematical Journal, 36, 647-658.