Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient

Sudheer Khan; Wang Shu; Monica Abhidha

doi:doi:10.11648/j.sjams.20200805.11

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Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient

Sudheer Khan, Wang Shu, Monica Abhidha

Published in Science Journal of Applied Mathematics and Statistics (Volume 8, Issue 5)

Received: 1 July 2020 Accepted: 16 July 2020 Published: 21 September 2020

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Abstract

Our aim in this study is to give the Gagliardo-Nirenberg Inequality as a consequence of pointwise estimates for the function in terms of the Riesz potential of the gradient. Our aim here is to discuss boundedness of Reisz potential in term of maximal functions and to give the proof for Gagliardo-Nirenberg Inequality in term of Reisz potential. We will extend our result to discuss weak type estimate for Gagliaro-Nirenberg Sobolev inequality. Further, in this paper we are interested to extract Sobolev type inequality in terms of Riesz potentials for α is equal to one and to extend our work for weak type estimates when p is equal to one.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 8, Issue 5)
DOI	10.11648/j.sjams.20200805.11
Page(s)	53-58
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Gagliardo-Nirenberg Inequality, Hardy Littlewood Maximal Function, Riesz Potential, Sobolev Type Inequality

References

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[2]	Armin Schikorra, Daniel Spectory: An L1-type estimate for Riesz potentials, Revista Matematica Iberoamericana 33 (2017), no. 1, 291-304.
[3]	Petteri Harjulehto, Ritva Hurri-Syrjänen, Pointwise estimates to the modified Riesz potential, manuscripta math. 156, 521-543 (2018).
[4]	Hajłasz, P., and Z. Liu: Sobolev spaces, Lebesgue points and maximal functions.-J. Fixed Point Theory Appl. 13: 1, 2013, 259-269.
[5]	Gilbarg D., and N. S. Trudinger: Elliptic Partial Differential Equations of Second Order.-Springer-Verlag, Berlin, 1983.
[6]	Tanaka, H.: A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function.-Bull. Austral. Math. Soc. 65: 2, 2002, 253-258.
[7]	P. Haj lasz, A new characterization of the Sobolev space, Studia Math. 159 (2003), no. 2, 263-275.
[8]	Maz’ya, V.: Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011.
[9]	P. Koskela and E. Saksman, Pointwise characterizations of Hardy-Sobolev functions, Math. Res. Lett. 15 (2008), no. 4, 727-744.
[10]	J. M. Aldaz and J. P_erez L_azaro, Functions of bounded variation, the derivative of the one-dimensional maximal function, and applications to inequalities, Trans. Amer. Math. Soc. 359, no. 5 (2007), 2443-2461.
[11]	N. Kalton, S. Mayboroda, and M. Mitrea, Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin spaces and applications to problems in partial differential equations, pp. 121-177 in Interpolation theory and applications (Miami, FL, 2006), edited by L. De Carli and M. Milman, Contemp. Math. 445, Amer. Math. Soc., Providence, RI, 2007.
[12]	Y. Mizuta, Potential theory in Euclidean spaces, Gakk_otosho, Tokyo, 1996.
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[14]	Carlos p_erez, Tiago picon, Olli saari, regularity of maximal functions on hardy-sobolev spaces, Mathematics Subject Classification. 2010, 42B25, 42B30, 46E35.
[15]	P. Hajlasz and J. Onninen, on boundedness of maximal functions in sobolev spaces, Annales Academie Scientiarum Fennicae Mathematica, Volume 29, 2004, 167-176.
[16]	Ondřej Kurka, on the variation of the hardy-Littlewood maximal function, Annales Academiæ Scientiarum Fennicæ Mathematica, Volumen 40, 2015, 109-133.
[17]	G. Leoni, A First Course in Sobolev Spaces, AMS, 2009.
[18]	Inan Cinar, The Gagliardo-Nirenberg-Sobolev Inequality for Non-Isotropic Riesz Potentials, Journal of Mathematics and System Science 5 (2015) 83-85.

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Sudheer Khan, Wang Shu, Monica Abhidha. (2020). Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient. Science Journal of Applied Mathematics and Statistics, 8(5), 53-58. https://doi.org/10.11648/j.sjams.20200805.11

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ACS Style

Sudheer Khan; Wang Shu; Monica Abhidha. Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient. Sci. J. Appl. Math. Stat. 2020, 8(5), 53-58. doi: 10.11648/j.sjams.20200805.11

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AMA Style

Sudheer Khan, Wang Shu, Monica Abhidha. Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient. Sci J Appl Math Stat. 2020;8(5):53-58. doi: 10.11648/j.sjams.20200805.11

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@article{10.11648/j.sjams.20200805.11,
  author = {Sudheer Khan and Wang Shu and Monica Abhidha},
  title = {Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {8},
  number = {5},
  pages = {53-58},
  doi = {10.11648/j.sjams.20200805.11},
  url = {https://doi.org/10.11648/j.sjams.20200805.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20200805.11},
  abstract = {Our aim in this study is to give the Gagliardo-Nirenberg Inequality as a consequence of pointwise estimates for the function in terms of the Riesz potential of the gradient. Our aim here is to discuss boundedness of Reisz potential in term of maximal functions and to give the proof for Gagliardo-Nirenberg Inequality in term of Reisz potential. We will extend our result to discuss weak type estimate for Gagliaro-Nirenberg Sobolev inequality. Further, in this paper we are interested to extract Sobolev type inequality in terms of Riesz potentials for α is equal to one and to extend our work for weak type estimates when p is equal to one.},
 year = {2020}
}

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TY  - JOUR
T1  - Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient
AU  - Sudheer Khan
AU  - Wang Shu
AU  - Monica Abhidha
Y1  - 2020/09/21
PY  - 2020
N1  - https://doi.org/10.11648/j.sjams.20200805.11
DO  - 10.11648/j.sjams.20200805.11
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 53
EP  - 58
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20200805.11
AB  - Our aim in this study is to give the Gagliardo-Nirenberg Inequality as a consequence of pointwise estimates for the function in terms of the Riesz potential of the gradient. Our aim here is to discuss boundedness of Reisz potential in term of maximal functions and to give the proof for Gagliardo-Nirenberg Inequality in term of Reisz potential. We will extend our result to discuss weak type estimate for Gagliaro-Nirenberg Sobolev inequality. Further, in this paper we are interested to extract Sobolev type inequality in terms of Riesz potentials for α is equal to one and to extend our work for weak type estimates when p is equal to one.
VL  - 8
IS  - 5
ER  -

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Author Information

Sudheer Khan

College of Applied Sciences, Beijing University of Technology, Beijing, China
Wang Shu

College of Applied Sciences, Beijing University of Technology, Beijing, China
Monica Abhidha

College of Applied Sciences, Beijing University of Technology, Beijing, China

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APA Style

Sudheer Khan, Wang Shu, Monica Abhidha. (2020). Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient. Science Journal of Applied Mathematics and Statistics, 8(5), 53-58. https://doi.org/10.11648/j.sjams.20200805.11

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ACS Style

Sudheer Khan; Wang Shu; Monica Abhidha. Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient. Sci. J. Appl. Math. Stat. 2020, 8(5), 53-58. doi: 10.11648/j.sjams.20200805.11

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AMA Style

Sudheer Khan, Wang Shu, Monica Abhidha. Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient. Sci J Appl Math Stat. 2020;8(5):53-58. doi: 10.11648/j.sjams.20200805.11

Copy | Download

@article{10.11648/j.sjams.20200805.11,
  author = {Sudheer Khan and Wang Shu and Monica Abhidha},
  title = {Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {8},
  number = {5},
  pages = {53-58},
  doi = {10.11648/j.sjams.20200805.11},
  url = {https://doi.org/10.11648/j.sjams.20200805.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20200805.11},
  abstract = {Our aim in this study is to give the Gagliardo-Nirenberg Inequality as a consequence of pointwise estimates for the function in terms of the Riesz potential of the gradient. Our aim here is to discuss boundedness of Reisz potential in term of maximal functions and to give the proof for Gagliardo-Nirenberg Inequality in term of Reisz potential. We will extend our result to discuss weak type estimate for Gagliaro-Nirenberg Sobolev inequality. Further, in this paper we are interested to extract Sobolev type inequality in terms of Riesz potentials for α is equal to one and to extend our work for weak type estimates when p is equal to one.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient
AU  - Sudheer Khan
AU  - Wang Shu
AU  - Monica Abhidha
Y1  - 2020/09/21
PY  - 2020
N1  - https://doi.org/10.11648/j.sjams.20200805.11
DO  - 10.11648/j.sjams.20200805.11
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 53
EP  - 58
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20200805.11
AB  - Our aim in this study is to give the Gagliardo-Nirenberg Inequality as a consequence of pointwise estimates for the function in terms of the Riesz potential of the gradient. Our aim here is to discuss boundedness of Reisz potential in term of maximal functions and to give the proof for Gagliardo-Nirenberg Inequality in term of Reisz potential. We will extend our result to discuss weak type estimate for Gagliaro-Nirenberg Sobolev inequality. Further, in this paper we are interested to extract Sobolev type inequality in terms of Riesz potentials for α is equal to one and to extend our work for weak type estimates when p is equal to one.
VL  - 8
IS  - 5
ER  -

Copy | Download