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A simple proof of the Adams type inequalities in $ {\mathbb R}^{2m} $
Department of Mathematics, FPT University, Ha Noi, Vietnam |
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 [
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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