FRACTIONAL ADAMS-MOSER-TRUDINGER TYPE INEQUALITY WITH SINGULAR TERM IN LORENTZ SPACE AND $L^P$ SPACE

Yan Wu; Guanglan Wang

doi:10.11948/20230094

2024 Volume 14 Issue 1

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Yan Wu, Guanglan Wang. FRACTIONAL ADAMS-MOSER-TRUDINGER TYPE INEQUALITY WITH SINGULAR TERM IN LORENTZ SPACE AND $L^P$ SPACE[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 133-145. doi: 10.11948/20230094

Citation:

Yan Wu, Guanglan Wang. FRACTIONAL ADAMS-MOSER-TRUDINGER TYPE INEQUALITY WITH SINGULAR TERM IN LORENTZ SPACE AND $L^P$ SPACE[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 133-145. doi: 10.11948/20230094

FRACTIONAL ADAMS-MOSER-TRUDINGER TYPE INEQUALITY WITH SINGULAR TERM IN LORENTZ SPACE AND $L^P$ SPACE

Yan Wu^1,,
Guanglan Wang^1, ,

1.
School of Mathematics and Statistics, Linyi University, West side of the north section of Gongye Dadao, 276005, China

Author Bio: Email: wuyan@lyu.edu.cn(Y. Wu)
Corresponding author: Email: wangguanglan@lyu.edu.cn(G. Wang)
Fund Project: The authors were supported by National Natural Science Foundation of China (12271232) and National Science Foundation of Shandong (ZR2022MA18)

Abstract

Abstract

For fractional derivatives $ (-\Delta)^{\frac{s}{2}} $, we establish Adams-Moser- Trudinger type inequalities with singular term $ \frac{1}{|x|^\alpha} $ under Lorentz norm and $ L^p $ norm on bounded open domains, and get the sharpness of all inequalities. Furthermore, we obtain the sharpness with a more general method.
- Adams-Moser-Trudinger type inequality /
- fractional derivative /
- Lorentz space
MSC: 35B33, 35J60

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References

[1]	D. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math., 1988, 128, 383–398. Google Scholar
[2]	Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, Nolinear Differential Equations Application, 2007, 13, 585–603. Google Scholar
[3]	Adimurthi and K. Tintarev, On a version of Moser-Trudinger inequality with Mobius shift invariance, Calc. Var. Partial Differential Equations, 2010, 39, 203–212. doi: 10.1007/s00526-010-0307-5 CrossRef Google Scholar
[4]	A. Alberico, Moser type inequalities for higher-order derivatives in Lorentz spaces, Potential Anal., 2008, 28, 389–400. doi: 10.1007/s11118-008-9085-5 CrossRef Google Scholar
[5]	A. Alvino, V. Ferone and G. Trombetti, Moser-type inequalities in Lorentz spaces, Potential Anal., 1996, 5, 273–299. doi: 10.1007/BF00282364 CrossRef Google Scholar
[6]	R. Arora, J. Giacomoni, T. Mukherjee and K. Sreenadh, Adams-Moser-Trudinger inequality in the Cartesian product of Sobolev spaces and its applications, Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A Matematicas, 2020, 111, 114–118. Google Scholar
[7]	A. Cianchi, V. Musil and L. Pick, On the existence of extremals for Moser-type inequalities in Gauss space, International mathematics research notices: IMRN, 2022, 2, 1494–1453. Google Scholar
[8]	S. Cingolani and T. Weth, Trudinger-Moser-type inequality with logarithmic convolution potentials, The Journal of the London Mathematical Society, 2022, 105, 1897–1935. doi: 10.1112/jlms.12549 CrossRef Google Scholar
[9]	M. Han, Bifurcation Theory of Limit Cycles, Science Press, Beijing, 2013. Google Scholar
[10]	S. Helgason, Geometric Analysis on Symmetric Spaces, Second edition, Mathematical Surveys and Monographs, 39, American Mathematical Society, Providence, RI, 2008. Google Scholar
[11]	J. Huang, P. Li, Y. Liu and S. Shi, Fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations, Nonlinear Analysis, 2020, 195, 1–45. Google Scholar
[12]	A. Hyder, Moser functions and fractional Moser-Trudinger type inequalities, Nonlinear Anal., 2016, 146, 185–210. doi: 10.1016/j.na.2016.08.024 CrossRef Google Scholar
[13]	S. M. Imran, S. Asghar and M. Mushtaq, Mixed Convection Flow Over an Unsteady Stretching Surface in a Porous Medium with Heat Source, Mathematical Problems in Engineering, 2012. DOI: 10.1155/2012/485418. CrossRef Google Scholar
[14]	N. Lam and G. Lu, Sharp singular Adams inequalities in high order Sobolev spaces, Methods Appl. Anal., 2012, 19, 243–266. doi: 10.4310/MAA.2012.v19.n3.a2 CrossRef Google Scholar
[15]	C. Li and J. Llibre, Uniqueness of limit cycles for Lienard differential equations of degree four, J. Diff. Eqs., 2012, 252, 3142–3162. doi: 10.1016/j.jde.2011.11.002 CrossRef Google Scholar
[16]	E. H. Lieb and M. Loss, Analysis, Second edition. Graduate Studies in Mathematics. 14. American Mathematical Society, Providence, RI, 2001. ISBN: 0-8218-2783-9. Google Scholar
[17]	G. Lu and H. Tang, Sharp singular Trudinger-Moser inequalities in Lorentz-Sobolev spaces, Adv. Nonlinear Stud., 2016, 16, 581–601. doi: 10.1515/ans-2015-5046 CrossRef Google Scholar
[18]	G. Lu, H. Tang and M. Zhu, Best constants for Adams' inequalities with the exact growth condition in $\mathbb{R}^n$, Adv. Nonlinear Stud., 2015, 15, 763–788. doi: 10.1515/ans-2015-0402 CrossRef $\mathbb{R}^n$" target="_blank">Google Scholar
[19]	G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 2010, 12, 1055–1068. doi: 10.1142/S0219199710004111 CrossRef Google Scholar
[20]	L. Martinazzi, Fractional Admas-Moser-Trudinger type inequalities, Nonlinear Anal., 2015, 127, 263–278. doi: 10.1016/j.na.2015.06.034 CrossRef Google Scholar
[21]	J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 1971, 20, 1077–1092. doi: 10.1512/iumj.1971.20.20101 CrossRef Google Scholar
[22]	R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 1963, 30, 129–142. $L(p, q)$ spaces" target="_blank">Google Scholar
[23]	S. I. Pohozaev, The Sobolev embedding in the case $pl=n$, Proceeding of the Technical Scientific Conference on Advances of Scientific Research 1964–1965. Mathematics Section, Moskov. Energet. Inst., 1965, 158–170. Google Scholar
[24]	P. A. Ruiz and F. Baudoin, Gagliardo-Nirenberg, Trudinger-Moser and Morrey inequalities on Dirichlet spaces, Journal of Mathematical Analysis and Applications, 2021, 2, 202–227. Google Scholar
[25]	S. Shi, Some integrability estimates for solutions of the fractional p-Laplace equation, J. Nonlinear Sci. Appl., 2017, 10, 5585–5592. doi: 10.22436/jnsa.010.10.38 CrossRef Google Scholar
[26]	S. Shi, Some notes on supersolutions of fractional p-Laplace equation, J. Math. Anal. Appl., 2018, 463, 1052–1074. doi: 10.1016/j.jmaa.2018.03.064 CrossRef Google Scholar
[27]	S. Shi and Z. Fu, Compactness of the commutators of fractional hardy operator with rough kernel, Analysis in Theory and Applications, Anal. Theory Appl., 2021, 37, 347–361. doi: 10.4208/ata.2021.lu80.05 CrossRef Google Scholar
[28]	S. Shi, L. Zhang and G. Wang, Fractional non-linear regularity, Potential and Balayage, The Journal of Geometric Analysis, 2022, 32, Art. Number 221. Google Scholar
[29]	N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech., 1967, 17, 473–484. Google Scholar
[30]	G. Wang and D. Ye, A Hardy-Moser-Trudinger inequality, Adv. Math., 2012, 230, 294–320. doi: 10.1016/j.aim.2011.12.001 CrossRef Google Scholar
[31]	J. Xiao and Z. Zhai, Fractional Sobolev, Moser-Trudinger, Moser-Sobolev inequality under lorentz norms, J. Math. Sci., 2010, 166, 357–376. doi: 10.1007/s10958-010-9872-6 CrossRef Google Scholar
[32]	V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 1961, 2, 746–749. Google Scholar