October  2020, 40(10): 5755-5764. doi: 10.3934/dcds.2020244

A simple proof of the Adams type inequalities in $ {\mathbb R}^{2m} $

Department of Mathematics, FPT University, Ha Noi, Vietnam

Received  July 2019 Revised  April 2020 Published  June 2020

We provide the simple proof of the Adams type inequalities in whole space $ {\mathbb R}^{2m} $. The main tools are the Fourier rearrangement technique introduced by Lenzmann and Sok [16], a Hardy–Rellich type inequality for radial functions, and the sharp Moser–Trudinger type inequalities in $ {\mathbb R}^2 $.

Citation: Van Hoang Nguyen. A simple proof of the Adams type inequalities in $ {\mathbb R}^{2m} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5755-5764. doi: 10.3934/dcds.2020244
References:
[1]

S. Adachi and K. Tanaka, Trudinger type inequalities in $\bf R^N$ and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057.  doi: 10.1090/S0002-9939-99-05180-1.  Google Scholar

[2]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math., 128 (1988), 385-398.  doi: 10.2307/1971445.  Google Scholar

[3]

Ad imurthi and O. Druet, Blow–up analysis in dimension $2$ and a sharp form of Trudinger–Moser inequality, Comm. Partial Differential Equations, 29 (2004), 295-322.  doi: 10.1081/PDE-120028854.  Google Scholar

[4]

Ad imurthi and K. Sandeep, A singular Moser–Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar

[5]

Ad imurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger–Moser inequality in $\mathbb{R}^N$ and its applications, Int. Math. Res. Not. IMRN, 2010 (2010), 2394-2426.  doi: 10.1093/imrn/rnp194.  Google Scholar

[6]

J. Bertrand and K. Sandeep, Adams inequality on pinched Hadamard manifolds, preprint, arXiv: 1809.00879v3. Google Scholar

[7]

E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in $L_p(\Omega)$, Math. Z., 227 (1998), 511-523.  doi: 10.1007/PL00004389.  Google Scholar

[8]

A. DelaTorre and G. Mancini, Improved Adams–type inequalities and their extremals in dimension 2m, preprint, arXiv: 1711.00892v2. Google Scholar

[9]

L. Fontana and C. Morpurgo, Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on $\mathbb{R}^n$, Nonlinear Anal., 167 (2018), 85-122.  doi: 10.1016/j.na.2017.10.012.  Google Scholar

[10]

N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Mathematical Surveys and Monographs, 187. American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/187.  Google Scholar

[11]

S. IbrahimN. Masmoudi and K. Nakanishi, Trudinger–Moser inequality on the whole plane with the exact growth condition, J. Eur. Math. Soc. (JEMS), 17 (2015), 819-835.  doi: 10.4171/JEMS/519.  Google Scholar

[12]

D. Karmakar and K. Sandeep, Adams inequality on the hyperbolic space, J. Funct. Anal., 270 (2016), 1792-1817.  doi: 10.1016/j.jfa.2015.11.019.  Google Scholar

[13]

N. Lam and G. Lu, Sharp singular Adams inequalities in high order Sobolev spaces, Methods Appl. Anal., 19 (2012), 243-266.  doi: 10.4310/MAA.2012.v19.n3.a2.  Google Scholar

[14]

N. Lam and G. Lu, Sharp Adams type inequalities in Sobolev spaces $W^{m,\frac{n}{m}} (\mathbb{R}^n)$ for arbitrary integer $m$, J. Differential Equations, 253 (2012), 1143-1171.  doi: 10.1016/j.jde.2012.04.025.  Google Scholar

[15]

N. Lam and G. Lu, A new approach to sharp Moser–Trudinger and Adams type inequalities: A rearrangement–free argument, J. Differential Equations, 255 (2013), 298-325.  doi: 10.1016/j.jde.2013.04.005.  Google Scholar

[16]

E. Lenzmann and J. Sok, A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, Int. Math. Res. Not., 2020, arXiv: 1805.06294v1. doi: 10.1093/imrn/rnz274.  Google Scholar

[17]

Y. Li and B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in $\mathbb{R}^n$, Indiana Univ. Math. J., 57 (2008), 451-480.  doi: 10.1512/iumj.2008.57.3137.  Google Scholar

[18]

E. H. Lieb and M. Loss, Analysis, 2nd edn, Graduate Studies in Mathematics, Vol. 14, Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[19]

G. LuH. Tang and M. Zhu, Best constants for Adams' inequalities with the exact growth condition in $\mathbb{R}^n$, Adv. Nonlinear Stud., 15 (2015), 763-788.  doi: 10.1515/ans-2015-0402.  Google Scholar

[20]

G. Lu and Y. Yang, Adams' inequalities for bi–Laplacian and extremal functions in dimension four, Adv. Math., 220 (2009), 1135-1170.  doi: 10.1016/j.aim.2008.10.011.  Google Scholar

[21]

G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068.  doi: 10.1142/S0219199710004111.  Google Scholar

[22]

G. ManciniK. Sandeep and C. Tintarev, Trudinger–Moser inequality in the hyperbolic space ${\mathbb H}^N$, Adv. Nonlinear Anal., 2 (2013), 309-324.  doi: 10.1515/anona-2013-0001.  Google Scholar

[23]

N. Masmoudi and F. Sani, Trudinger–Moser inequalities with the exact growth condition in $\mathbb{R}^N$ and applications, Comm. Partial Differential Equations, 40 (2015), 1408-1440.  doi: 10.1080/03605302.2015.1026775.  Google Scholar

[24]

N. Masmoudi and F. Sani, Adams' inequality with the exact growth condition in $\mathbb{R}^4$, Comm. Pure Appl. Math., 67 (2014), 1307-1335.  doi: 10.1002/cpa.21473.  Google Scholar

[25]

N. Masmoudi and F. Sani, Higher order Adams' inequality with the exact growth condition, Commun. Contemp. Math., 20 (2018), 1750072, 33 pp. doi: 10.1142/S0219199717500729.  Google Scholar

[26]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[27]

Q. A. Ngo and V. H. Nguyen, Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space, Rev. Mat. Iberoam., in press, arXiv: 1606.07094v2. Google Scholar

[28]

B. Opic and A. Kufner, Hardy–Type Inequalities, Pitman Research Notes in Mathematics Series, vol.219, Longman Scientific and Technical, Harlow, 1990.  Google Scholar

[29]

S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk. SSSR, 165 (1965), 36-39.   Google Scholar

[30]

B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in $\mathbb{R}^2$, J. Funct. Anal., 219 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[31]

B. Ruf and F. Sani, Sharp Adams-type inequalities in $\mathbb{R}^n$, Trans. Amer. Math. Soc., 365 (2013), 645-670.  doi: 10.1090/S0002-9947-2012-05561-9.  Google Scholar

[32]

C. Tarsi, Adams' inequality and limiting Sobolev embeddings into Zygmund spaces, Potential Anal., 37 (2012), 353-385.  doi: 10.1007/s11118-011-9259-4.  Google Scholar

[33]

A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy–Rellich inequalities and related improvements, Adv. Math., 209 (2007), 407-459.  doi: 10.1016/j.aim.2006.05.011.  Google Scholar

[34]

C. Tintarev, Trudinger–Moser inequality with remainder terms, J. Funct. Anal., 266 (2014), 55-66.  doi: 10.1016/j.jfa.2013.09.009.  Google Scholar

[35]

N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

[36]

D. Yafaev, Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144.  doi: 10.1006/jfan.1999.3462.  Google Scholar

[37]

Y. Yang, A sharp form of Moser–Trudinger inequality in high dimension, J. Funct. Anal., 239 (2006), 100-126.  doi: 10.1016/j.jfa.2006.06.002.  Google Scholar

[38]

Q. YangD. Su and Y. Kong, Sharp Moser–Trudinger inequalities on Riemannian manifolds with negative curvature, Ann. Mat. Pura Appl., 195 (2016), 459-471.  doi: 10.1007/s10231-015-0472-4.  Google Scholar

[39]

V. I. Yudovič, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk. SSSR, 138 (1961), 805-808.   Google Scholar

show all references

References:
[1]

S. Adachi and K. Tanaka, Trudinger type inequalities in $\bf R^N$ and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057.  doi: 10.1090/S0002-9939-99-05180-1.  Google Scholar

[2]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math., 128 (1988), 385-398.  doi: 10.2307/1971445.  Google Scholar

[3]

Ad imurthi and O. Druet, Blow–up analysis in dimension $2$ and a sharp form of Trudinger–Moser inequality, Comm. Partial Differential Equations, 29 (2004), 295-322.  doi: 10.1081/PDE-120028854.  Google Scholar

[4]

Ad imurthi and K. Sandeep, A singular Moser–Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar

[5]

Ad imurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger–Moser inequality in $\mathbb{R}^N$ and its applications, Int. Math. Res. Not. IMRN, 2010 (2010), 2394-2426.  doi: 10.1093/imrn/rnp194.  Google Scholar

[6]

J. Bertrand and K. Sandeep, Adams inequality on pinched Hadamard manifolds, preprint, arXiv: 1809.00879v3. Google Scholar

[7]

E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in $L_p(\Omega)$, Math. Z., 227 (1998), 511-523.  doi: 10.1007/PL00004389.  Google Scholar

[8]

A. DelaTorre and G. Mancini, Improved Adams–type inequalities and their extremals in dimension 2m, preprint, arXiv: 1711.00892v2. Google Scholar

[9]

L. Fontana and C. Morpurgo, Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on $\mathbb{R}^n$, Nonlinear Anal., 167 (2018), 85-122.  doi: 10.1016/j.na.2017.10.012.  Google Scholar

[10]

N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Mathematical Surveys and Monographs, 187. American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/187.  Google Scholar

[11]

S. IbrahimN. Masmoudi and K. Nakanishi, Trudinger–Moser inequality on the whole plane with the exact growth condition, J. Eur. Math. Soc. (JEMS), 17 (2015), 819-835.  doi: 10.4171/JEMS/519.  Google Scholar

[12]

D. Karmakar and K. Sandeep, Adams inequality on the hyperbolic space, J. Funct. Anal., 270 (2016), 1792-1817.  doi: 10.1016/j.jfa.2015.11.019.  Google Scholar

[13]

N. Lam and G. Lu, Sharp singular Adams inequalities in high order Sobolev spaces, Methods Appl. Anal., 19 (2012), 243-266.  doi: 10.4310/MAA.2012.v19.n3.a2.  Google Scholar

[14]

N. Lam and G. Lu, Sharp Adams type inequalities in Sobolev spaces $W^{m,\frac{n}{m}} (\mathbb{R}^n)$ for arbitrary integer $m$, J. Differential Equations, 253 (2012), 1143-1171.  doi: 10.1016/j.jde.2012.04.025.  Google Scholar

[15]

N. Lam and G. Lu, A new approach to sharp Moser–Trudinger and Adams type inequalities: A rearrangement–free argument, J. Differential Equations, 255 (2013), 298-325.  doi: 10.1016/j.jde.2013.04.005.  Google Scholar

[16]

E. Lenzmann and J. Sok, A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, Int. Math. Res. Not., 2020, arXiv: 1805.06294v1. doi: 10.1093/imrn/rnz274.  Google Scholar

[17]

Y. Li and B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in $\mathbb{R}^n$, Indiana Univ. Math. J., 57 (2008), 451-480.  doi: 10.1512/iumj.2008.57.3137.  Google Scholar

[18]

E. H. Lieb and M. Loss, Analysis, 2nd edn, Graduate Studies in Mathematics, Vol. 14, Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[19]

G. LuH. Tang and M. Zhu, Best constants for Adams' inequalities with the exact growth condition in $\mathbb{R}^n$, Adv. Nonlinear Stud., 15 (2015), 763-788.  doi: 10.1515/ans-2015-0402.  Google Scholar

[20]

G. Lu and Y. Yang, Adams' inequalities for bi–Laplacian and extremal functions in dimension four, Adv. Math., 220 (2009), 1135-1170.  doi: 10.1016/j.aim.2008.10.011.  Google Scholar

[21]

G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068.  doi: 10.1142/S0219199710004111.  Google Scholar

[22]

G. ManciniK. Sandeep and C. Tintarev, Trudinger–Moser inequality in the hyperbolic space ${\mathbb H}^N$, Adv. Nonlinear Anal., 2 (2013), 309-324.  doi: 10.1515/anona-2013-0001.  Google Scholar

[23]

N. Masmoudi and F. Sani, Trudinger–Moser inequalities with the exact growth condition in $\mathbb{R}^N$ and applications, Comm. Partial Differential Equations, 40 (2015), 1408-1440.  doi: 10.1080/03605302.2015.1026775.  Google Scholar

[24]

N. Masmoudi and F. Sani, Adams' inequality with the exact growth condition in $\mathbb{R}^4$, Comm. Pure Appl. Math., 67 (2014), 1307-1335.  doi: 10.1002/cpa.21473.  Google Scholar

[25]

N. Masmoudi and F. Sani, Higher order Adams' inequality with the exact growth condition, Commun. Contemp. Math., 20 (2018), 1750072, 33 pp. doi: 10.1142/S0219199717500729.  Google Scholar

[26]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[27]

Q. A. Ngo and V. H. Nguyen, Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space, Rev. Mat. Iberoam., in press, arXiv: 1606.07094v2. Google Scholar

[28]

B. Opic and A. Kufner, Hardy–Type Inequalities, Pitman Research Notes in Mathematics Series, vol.219, Longman Scientific and Technical, Harlow, 1990.  Google Scholar

[29]

S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk. SSSR, 165 (1965), 36-39.   Google Scholar

[30]

B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in $\mathbb{R}^2$, J. Funct. Anal., 219 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[31]

B. Ruf and F. Sani, Sharp Adams-type inequalities in $\mathbb{R}^n$, Trans. Amer. Math. Soc., 365 (2013), 645-670.  doi: 10.1090/S0002-9947-2012-05561-9.  Google Scholar

[32]

C. Tarsi, Adams' inequality and limiting Sobolev embeddings into Zygmund spaces, Potential Anal., 37 (2012), 353-385.  doi: 10.1007/s11118-011-9259-4.  Google Scholar

[33]

A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy–Rellich inequalities and related improvements, Adv. Math., 209 (2007), 407-459.  doi: 10.1016/j.aim.2006.05.011.  Google Scholar

[34]

C. Tintarev, Trudinger–Moser inequality with remainder terms, J. Funct. Anal., 266 (2014), 55-66.  doi: 10.1016/j.jfa.2013.09.009.  Google Scholar

[35]

N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

[36]

D. Yafaev, Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144.  doi: 10.1006/jfan.1999.3462.  Google Scholar

[37]

Y. Yang, A sharp form of Moser–Trudinger inequality in high dimension, J. Funct. Anal., 239 (2006), 100-126.  doi: 10.1016/j.jfa.2006.06.002.  Google Scholar

[38]

Q. YangD. Su and Y. Kong, Sharp Moser–Trudinger inequalities on Riemannian manifolds with negative curvature, Ann. Mat. Pura Appl., 195 (2016), 459-471.  doi: 10.1007/s10231-015-0472-4.  Google Scholar

[39]

V. I. Yudovič, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk. SSSR, 138 (1961), 805-808.   Google Scholar

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