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Exponential return times in a zero-entropy process
Improving sharp Sobolev type inequalities by optimal remainder gradient norms
1. | Dipartimento di Matematica "U.Dini", Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy |
2. | Dipartimento di Matematica, Seconda Università di Napoli, Viale Lincoln 5, 81100 Caserta, Italy |
References:
[1] |
Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its applications,, Proc. Amer. Math. Soc., 130 (2002), 489.
doi: 10.1090/S0002-9939-01-06132-9. |
[2] |
Adimurthi and M. J. Esteban, An improved Hardy-Sobolev inequality in $W^{1,p}$ and its applications to Schrödinger operators,, NoDEA Nonlinear Differential Equation Appl., 12 (2005), 243.
doi: 10.1007/S00030-005-0009-4. |
[3] |
A. Alvino, Sulla disuguaglianza di Sobolev in spazi di Lorentz,, Boll. Un. Mat. Ital., 5 (1977), 148.
|
[4] |
A. Alvino, P.-L.Lions and G. Trombetti, On optimization problems with prescribed rearrangements,, Nonlinear Anal., 13 (1989), 185.
doi: 10.1016/0362-546X(89)90043-6. |
[5] |
A. Alvino, R. Volpicelli and B. Volzone, On Hardy inequalities with a remainder term,, Ric. Mat., 59 (2010), 265.
doi: 10.1007/s11587-010-0086-5. |
[6] |
T. Aubin, Problèmes isopérimetriques et espaces de Sobolev,, J. Diff. Geom., 11 (1976), 573.
|
[7] |
G. Barbatis, S. Filippas and A. Tertikas, Series expansion for $L^p$ Hardy inequalities,, Indiana Univ. Math. J., 52 (2003), 171.
doi: 0.1512/iumj.2003.52.2207. |
[8] |
G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants,, Trans. Amer. Math. Soc., 356 (2004), 2169.
doi: 10.1090/S0002-9947-03-03389-0. |
[9] |
R. Benguria, R. Frank and M. Loss, The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space,, Math. Res. Lett., 15 (2008), 613.
|
[10] |
C. Bennett and R. Sharpley, "Interpolation of Operators,", Academic Press, (1988).
|
[11] |
H.Brezis and E.Lieb, Sobolev inequalities with remainder terms,, J. Funct. Anal., 62 (1985), 73.
doi: 10.1016/0022-1236(85)90020-5. |
[12] |
H. Brezis and M. Marcus, Hardy's inequalities revisited,, Ann. Sc. Norm. Super. Pisa, 25 (1997), 217.
|
[13] |
H. Brezis, M. Marcus and I. Shafrir, Extremal functions for Hardy's inequality with weight,, J. Funct. Anal., 171 (2000), 177.
doi: 10.1006/jfan.1999.3504. |
[14] |
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.
doi: 10.1002/cpa.3160360405. |
[15] |
H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities,, Comm. Part. Diff. Eq., 5 (1980), 773.
doi: 10.1080/03605308008820154. |
[16] |
J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions,, J. Reine Angew. Math., 384 (1988), 419.
doi: 10.1515/crll.1988.384.153. |
[17] |
A. Cianchi and A. Ferone, Best remainder norms in Sobolev-Hardy inequalities,, Indiana Univ. Math. J., 58 (2009), 1051.
doi: 10.1512/iumj.2009.58.3561. |
[18] |
G. Crasta, I. Fragalà and F. Gazzola, Some estimates for the torsional rigidity of composite rods,, Math. Nachr., 280 (2007), 242.
doi: 0.1002/mana.200410478. |
[19] |
M. Cwikel and E. Pustylnik, Sobolev type embeddings in the limiting case,, J. Fourier Anal. Appl., 4 (1998), 433.
doi: 10.1007/BF02498218. |
[20] |
A. Detalla, T. Horiuchi and H. Ando, Missing terms in Hardy-Sobolev inequalities and its applications,, Far East J. Math. Sci., 14 (2004), 333.
|
[21] |
J. Dolbeault, M. J. Esteban, M. Loss and L. Vega, An analytical proof of Hardy-like inequalities related to the Dirac operator,, J. Funct. Anal., 216 (2004), 1.
doi: 10.1016/j.jfa.2003.09.010. |
[22] |
D. E. Edmunds, R. A. Kerman and L. Pick, Optimal Sobolev imbeddings involving rearrangement invariant quasi-norms,, J. Funct. Anal., 170 (2000), 307.
doi: 10.1006/jfan.1999.3508. |
[23] |
S. Filippas, V. G. Maz'ya and A. Tertikas, On a question of Brezis and Marcus,, Calc. Var. Partial Differential Equations, 25 (2006), 491.
doi: 10.1007/s00526-005-0353-6. |
[24] |
S. Filippas, V. G. Maz'ya and A. Tertikas, Critical Hardy-Sobolev inequalities,, J. Math. Pures Appl., 87 (2007), 37.
doi: 10.1016/j.matpur.2006.10.007. |
[25] |
F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, Trans. Amer. Math. Soc., 356 (2004), 2149.
doi: 10.1090/S0002-9947-03-03395-6. |
[26] |
N. Ghoussoub and A. Moradifam, On the best possible remaining term in the Hardy inequality,, Proc. Natl. Acad. Sci. USA, 105 (2008), 13746.
doi: 10.1073/pnas.0803703105. |
[27] |
E. Giarrusso and D. Nunziante, Symmetrization in a class of first-order Hamilton-Jacobi equations,, Nonlinear Anal., 8 (1984), 289.
doi: 10.1016/0362-546X(84)90031-2. |
[28] |
A. Gogatishvili, B. Opic and L. Pick, Weighted inequalities for Hardy-type operators involving suprema,, Collect. Math., 57 (2006), 227.
|
[29] |
A. Gogatishvili and L. Pick, Discretization and antidiscretization of rearrangement-invariant norms,, Publ. Mat., 47 (2003), 311.
|
[30] |
K. Hansson, Imbedding theorems of Sobolev type in potential theory,, Math. Scand., 45 (1979), 77.
|
[31] |
M. Hoffman-Ostenhof, T. Hoffman-Ostenhof and A. Laptev, A geometrical version of Hardy's inequality,, J. Funct. Anal., 189 (2002), 539.
|
[32] |
B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Math. 1150, (1150).
|
[33] |
V. M. Maz'ya and T. O. Shaposhnikova, "Sobolev Spaces: with Applications to Elliptic Partial Differential Equations,", Springer-Verlag, (2011).
doi: 10.1007/978-3-642-15564-2. |
[34] |
J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.
|
[35] |
R. O'Neil, Convolution operators in $L(p,q)$ spaces,, Duke Math. J., 30 (1963), 129.
doi: 10.1215/S0012-7094-63-03015-1. |
[36] |
J. Peetre, Espaces d' interpolation et théorème de Soboleff,, Ann. Inst. Fourier, 16 (1966), 279.
doi: 10.5802/aif.232. |
[37] |
S. I. Pohozaev, On the imbedding Sobolev theorem for $p=n$,, Doklady Conference, (1965), 158. Google Scholar |
[38] |
G. Sinnamon and V. D. Stepanov, The weighted Hardy inequality: new proofs and the case $p=1$,, J. London Math. Soc., 54 (1996), 89.
|
[39] |
G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353.
doi: 10.1007/BF02418013. |
[40] |
J. Tidblom, A Hardy inequality in the half-space,, J. Funct. Anal., 221 (2005), 482.
doi: 10.1016/j.jfa.2004.09.014. |
[41] |
N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.
|
[42] |
V. I. Yudovic, Some estimates connected with integral operators and with solutions of elliptic equations,, Dokl. Akad. Nauk SSSR, 138 (1961), 805.
|
[43] |
J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, J. Funct. Anal., 173 (2000), 103.
doi: 10.1006/jfan.1999.3556. |
show all references
References:
[1] |
Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its applications,, Proc. Amer. Math. Soc., 130 (2002), 489.
doi: 10.1090/S0002-9939-01-06132-9. |
[2] |
Adimurthi and M. J. Esteban, An improved Hardy-Sobolev inequality in $W^{1,p}$ and its applications to Schrödinger operators,, NoDEA Nonlinear Differential Equation Appl., 12 (2005), 243.
doi: 10.1007/S00030-005-0009-4. |
[3] |
A. Alvino, Sulla disuguaglianza di Sobolev in spazi di Lorentz,, Boll. Un. Mat. Ital., 5 (1977), 148.
|
[4] |
A. Alvino, P.-L.Lions and G. Trombetti, On optimization problems with prescribed rearrangements,, Nonlinear Anal., 13 (1989), 185.
doi: 10.1016/0362-546X(89)90043-6. |
[5] |
A. Alvino, R. Volpicelli and B. Volzone, On Hardy inequalities with a remainder term,, Ric. Mat., 59 (2010), 265.
doi: 10.1007/s11587-010-0086-5. |
[6] |
T. Aubin, Problèmes isopérimetriques et espaces de Sobolev,, J. Diff. Geom., 11 (1976), 573.
|
[7] |
G. Barbatis, S. Filippas and A. Tertikas, Series expansion for $L^p$ Hardy inequalities,, Indiana Univ. Math. J., 52 (2003), 171.
doi: 0.1512/iumj.2003.52.2207. |
[8] |
G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants,, Trans. Amer. Math. Soc., 356 (2004), 2169.
doi: 10.1090/S0002-9947-03-03389-0. |
[9] |
R. Benguria, R. Frank and M. Loss, The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space,, Math. Res. Lett., 15 (2008), 613.
|
[10] |
C. Bennett and R. Sharpley, "Interpolation of Operators,", Academic Press, (1988).
|
[11] |
H.Brezis and E.Lieb, Sobolev inequalities with remainder terms,, J. Funct. Anal., 62 (1985), 73.
doi: 10.1016/0022-1236(85)90020-5. |
[12] |
H. Brezis and M. Marcus, Hardy's inequalities revisited,, Ann. Sc. Norm. Super. Pisa, 25 (1997), 217.
|
[13] |
H. Brezis, M. Marcus and I. Shafrir, Extremal functions for Hardy's inequality with weight,, J. Funct. Anal., 171 (2000), 177.
doi: 10.1006/jfan.1999.3504. |
[14] |
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.
doi: 10.1002/cpa.3160360405. |
[15] |
H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities,, Comm. Part. Diff. Eq., 5 (1980), 773.
doi: 10.1080/03605308008820154. |
[16] |
J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions,, J. Reine Angew. Math., 384 (1988), 419.
doi: 10.1515/crll.1988.384.153. |
[17] |
A. Cianchi and A. Ferone, Best remainder norms in Sobolev-Hardy inequalities,, Indiana Univ. Math. J., 58 (2009), 1051.
doi: 10.1512/iumj.2009.58.3561. |
[18] |
G. Crasta, I. Fragalà and F. Gazzola, Some estimates for the torsional rigidity of composite rods,, Math. Nachr., 280 (2007), 242.
doi: 0.1002/mana.200410478. |
[19] |
M. Cwikel and E. Pustylnik, Sobolev type embeddings in the limiting case,, J. Fourier Anal. Appl., 4 (1998), 433.
doi: 10.1007/BF02498218. |
[20] |
A. Detalla, T. Horiuchi and H. Ando, Missing terms in Hardy-Sobolev inequalities and its applications,, Far East J. Math. Sci., 14 (2004), 333.
|
[21] |
J. Dolbeault, M. J. Esteban, M. Loss and L. Vega, An analytical proof of Hardy-like inequalities related to the Dirac operator,, J. Funct. Anal., 216 (2004), 1.
doi: 10.1016/j.jfa.2003.09.010. |
[22] |
D. E. Edmunds, R. A. Kerman and L. Pick, Optimal Sobolev imbeddings involving rearrangement invariant quasi-norms,, J. Funct. Anal., 170 (2000), 307.
doi: 10.1006/jfan.1999.3508. |
[23] |
S. Filippas, V. G. Maz'ya and A. Tertikas, On a question of Brezis and Marcus,, Calc. Var. Partial Differential Equations, 25 (2006), 491.
doi: 10.1007/s00526-005-0353-6. |
[24] |
S. Filippas, V. G. Maz'ya and A. Tertikas, Critical Hardy-Sobolev inequalities,, J. Math. Pures Appl., 87 (2007), 37.
doi: 10.1016/j.matpur.2006.10.007. |
[25] |
F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, Trans. Amer. Math. Soc., 356 (2004), 2149.
doi: 10.1090/S0002-9947-03-03395-6. |
[26] |
N. Ghoussoub and A. Moradifam, On the best possible remaining term in the Hardy inequality,, Proc. Natl. Acad. Sci. USA, 105 (2008), 13746.
doi: 10.1073/pnas.0803703105. |
[27] |
E. Giarrusso and D. Nunziante, Symmetrization in a class of first-order Hamilton-Jacobi equations,, Nonlinear Anal., 8 (1984), 289.
doi: 10.1016/0362-546X(84)90031-2. |
[28] |
A. Gogatishvili, B. Opic and L. Pick, Weighted inequalities for Hardy-type operators involving suprema,, Collect. Math., 57 (2006), 227.
|
[29] |
A. Gogatishvili and L. Pick, Discretization and antidiscretization of rearrangement-invariant norms,, Publ. Mat., 47 (2003), 311.
|
[30] |
K. Hansson, Imbedding theorems of Sobolev type in potential theory,, Math. Scand., 45 (1979), 77.
|
[31] |
M. Hoffman-Ostenhof, T. Hoffman-Ostenhof and A. Laptev, A geometrical version of Hardy's inequality,, J. Funct. Anal., 189 (2002), 539.
|
[32] |
B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Math. 1150, (1150).
|
[33] |
V. M. Maz'ya and T. O. Shaposhnikova, "Sobolev Spaces: with Applications to Elliptic Partial Differential Equations,", Springer-Verlag, (2011).
doi: 10.1007/978-3-642-15564-2. |
[34] |
J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.
|
[35] |
R. O'Neil, Convolution operators in $L(p,q)$ spaces,, Duke Math. J., 30 (1963), 129.
doi: 10.1215/S0012-7094-63-03015-1. |
[36] |
J. Peetre, Espaces d' interpolation et théorème de Soboleff,, Ann. Inst. Fourier, 16 (1966), 279.
doi: 10.5802/aif.232. |
[37] |
S. I. Pohozaev, On the imbedding Sobolev theorem for $p=n$,, Doklady Conference, (1965), 158. Google Scholar |
[38] |
G. Sinnamon and V. D. Stepanov, The weighted Hardy inequality: new proofs and the case $p=1$,, J. London Math. Soc., 54 (1996), 89.
|
[39] |
G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353.
doi: 10.1007/BF02418013. |
[40] |
J. Tidblom, A Hardy inequality in the half-space,, J. Funct. Anal., 221 (2005), 482.
doi: 10.1016/j.jfa.2004.09.014. |
[41] |
N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.
|
[42] |
V. I. Yudovic, Some estimates connected with integral operators and with solutions of elliptic equations,, Dokl. Akad. Nauk SSSR, 138 (1961), 805.
|
[43] |
J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, J. Funct. Anal., 173 (2000), 103.
doi: 10.1006/jfan.1999.3556. |
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