A simple proof of the Adams type inequalities in <inline-formula><tex-math id="M1">\begin{document} {\mathbb R}^{2m} \end{document}</tex-math></inline-formula>

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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 and 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.
[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.
[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.
[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.
[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.
[6]	J. Bertrand and K. Sandeep, Adams inequality on pinched Hadamard manifolds, preprint, arXiv: 1809.00879v3.
[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.
[8]	A. DelaTorre and G. Mancini, Improved Adams–type inequalities and their extremals in dimension 2m, preprint, arXiv: 1711.00892v2.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068. doi: 10.1142/S0219199710004111.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	B. Opic and A. Kufner, Hardy–Type Inequalities, Pitman Research Notes in Mathematics Series, vol.219, Longman Scientific and Technical, Harlow, 1990.
[29]	S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk. SSSR, 165 (1965), 36-39.
[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.
[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.
[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.
[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.
[34]	C. Tintarev, Trudinger–Moser inequality with remainder terms, J. Funct. Anal., 266 (2014), 55-66. doi: 10.1016/j.jfa.2013.09.009.
[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.
[36]	D. Yafaev, Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144. doi: 10.1006/jfan.1999.3462.
[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.
[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.
[39]	V. I. Yudovič, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk. SSSR, 138 (1961), 805-808.

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.
[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.
[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.
[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.
[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.
[6]	J. Bertrand and K. Sandeep, Adams inequality on pinched Hadamard manifolds, preprint, arXiv: 1809.00879v3.
[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.
[8]	A. DelaTorre and G. Mancini, Improved Adams–type inequalities and their extremals in dimension 2m, preprint, arXiv: 1711.00892v2.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068. doi: 10.1142/S0219199710004111.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	B. Opic and A. Kufner, Hardy–Type Inequalities, Pitman Research Notes in Mathematics Series, vol.219, Longman Scientific and Technical, Harlow, 1990.
[29]	S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk. SSSR, 165 (1965), 36-39.
[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.
[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.
[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.
[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.
[34]	C. Tintarev, Trudinger–Moser inequality with remainder terms, J. Funct. Anal., 266 (2014), 55-66. doi: 10.1016/j.jfa.2013.09.009.
[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.
[36]	D. Yafaev, Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144. doi: 10.1006/jfan.1999.3462.
[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.
[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.
[39]	V. I. Yudovič, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk. SSSR, 138 (1961), 805-808.

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