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

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

Van Hoang Nguyen

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 $.

Keywords: Adams inequality, Adams inequality with exact growth, Fourier rearrangement, Hardy–Rellich inequality.
Mathematics Subject Classification: Primary:46E35, 26D10;Secondary:35A23.
Citation: Van Hoang Nguyen. A simple proof of the Adams type inequalities in $ {\mathbb R}^{2m} $. Discrete & Continuous Dynamical Systems, 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. Ibrahim, N. 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. Lu, H. 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. Mancini, K. 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. Yang, D. 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. Ibrahim, N. 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. Lu, H. 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. Mancini, K. 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. Yang, D. 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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