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An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation
A general approach to weighted $L^{p}$ Rellich type inequalities related to Greiner operator
Department of Mathematics, Faculty of Humanities and Social Sciences, Istanbul Ticaret University, Beyoglu, 34445, Istanbul, Turkey |
$L^{p}$ |
$a$ |
$b$ |
$\vartheta $ |
$Δ _{k}$ |
${\Delta _k}\left( {a|{\Delta _k}\vartheta {|^{p - 2}}{\Delta _k}\vartheta {\rm{ }}} \right) \ge b{\vartheta ^{p - 1}}$ |
$\mathbb{R}^{2n+1}, $ |
$a$ |
$b$ |
$L^{p}$ |
$p>1$ |
$Δ _{k} = \sum\nolimits_{j = 1}^n {} \left(X_{j}^{2}+Y_{j}^{2}\right) $ |
${X_j} = \frac{\partial }{{\partial {x_j}}} + 2k{y_j}|z{{\rm{|}}^{2k - 2}}\frac{\partial }{{\partial l}}, \;\;\;\;{Y_j} = \frac{\partial }{{\partial {y_j}}} - 2k{x_j}|z{|^{2k - 2}}\frac{\partial }{{\partial l}}, \;\;\;\;j = 1, ..., n, $ |
$\left( z, l\right) = \left( x, y, l\right) ∈\mathbb{R}^{2n+1} = \mathbb{R}^{n}×\mathbb{R}^{n}×\mathbb{R}, $ |
$|z{\rm{|}} = \sqrt {\sum\nolimits_{j = 1}^n {} \left( {x_j^2 + y_j^2} \right)} $ |
$k≥ 1$ |
$L^{p}$ |
$L^{p}$ |
$Δ _{k}$ |
$Ω $ |
$\mathbb{R}^{2n+1}$ |
References:
[1] |
Adimurthi, M. Grossi and S. Santra,
Optimal Hardy-Rellich inequalities, maximum principle
and related eigenvalue problem, J. Funct. Anal., 240 (2006), 36-83.
doi: 10.1016/j.jfa.2006.07.011. |
[2] |
Adimurthi and S. Santra,
Generalized Hardy-Rellich inequalities in critical dimension and its
applications, Commun. Contemp. Math., 11 (2009), 367-394.
doi: 10.1142/S0219199709003405. |
[3] |
S. Ahmetolan and I. Kombe,
A sharp uncertainty principle and Hardy-Poincaré inequalities
on sub-Riemannian manifolds, Math. Inequal. Appl., 15 (2012), 457-467.
doi: 10.7153/mia-15-40. |
[4] |
S. Ahmetolan and I. Kombe,
Hardy and Rellich type inequalities with two weight functions, Math. Inequal. Appl., 19 (2016), 937-948.
doi: 10.7153/mia-19-68. |
[5] |
W. Allegretto and Y. X. Huang,
A Picone's identity for the p−Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.
doi: 10.1016/S0362-546X(97)00530-0. |
[6] |
G. Barbatis,
Improved Rellich inequalities for the polyharmonic operator, Indiana Univ. Math. J., 55 (2006), 1401-1422.
doi: 10.1512/iumj.2006.55.2752. |
[7] |
R. Beals, B. Gaveau and P. Greiner,
On a geometric formula for the fundamental solution of
sub-elliptic Laplacians, Math. Nachr., 181 (1996), 81-163.
doi: 10.1002/mana.3211810105. |
[8] |
R. Beals, B. Gaveau and P. Greiner,
Uniform hypoelliptic Green's functions, J. Math. Pures Appl., 77 (1998), 209-248.
doi: 10.1016/S0021-7824(98)80069-X. |
[9] |
D. M. Bennett,
An extension of Rellich's inequality, Proc. Amer. Math. Soc., 106 (1989), 987-993.
doi: 10.2307/2047283. |
[10] |
P. Caldiroli and R. Musina,
Rellich inequalities with weights, Calc. Var. Partial Differential Equations, 45 (2012), 147-164.
doi: 10.1007/s00526-011-0454-3. |
[11] |
E. B. Davies and A. M. Hinz,
Explicit constants for Rellich inequalities in Lp(Ω), Math. Z., 227 (1998), 511-523.
doi: 10.1007/PL00004389. |
[12] |
A. Detalla, T. Horiuchi and H. Ando,
Sharp remainder terms of the Rellich inequality and its
application, Bull. Malays. Math. Sci. Soc., 35 (2012), 519-528.
|
[13] |
G. B. Folland,
A fundamental solution for a sub-elliptic operator, Bull. Amer. Math. Soc., 79 (1973), 373-376.
doi: 10.1090/S0002-9904-1973-13171-4. |
[14] |
V. A. Galaktionov and I. V. Kamotski,
On nonexistence of Baras-Goldstein type for higherorder parabolic equations with singular potentials, Trans. Am. Math. Soc., 362 (2010), 4117-4136.
doi: 10.1090/S0002-9947-10-04855-5. |
[15] |
N. Garofalo and E. Lanconelli,
Frequency functions on the Heisenberg group, the uncertainty
principle and unique continuation, Ann Inst Fourier (Grenoble), 40 (1990), 313-356.
|
[16] |
F. Gazzola, H. C. Grunau and E. Mitidieri,
Hardy inequalities with optimal constants and
remainder terms, Trans. Amer. Math. Soc., 356 (2004), 2149-2168.
doi: 10.1090/S0002-9947-03-03395-6. |
[17] |
P. C. Greiner,
A fundamental solution for a nonelliptic partial differential operator, Canad. J. Math., 31 (1979), 1107-1120.
doi: 10.4153/CJM-1979-101-3. |
[18] |
I. Kombe,
On the nonexistence of positive solutions to nonlinear degenerate parabolic equations with singular coefficients, Appl. Anal., 85 (2006), 467-478.
doi: 10.1080/00036810500404967. |
[19] |
B. Lian,
Some sharp Rellich type inequalities on nilpotent groups and application, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 59-74.
doi: 10.1016/S0252-9602(12)60194-5. |
[20] |
P. Lindqvist,
On the equation $\text{div}(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u = 0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.
doi: 10.2307/2048375. |
[21] |
G. Metafune, M. Sobajima and S. C. Motohiro,
Weighted Calderón-Zygmund and Rellich
inequalities in Lp, Math. Ann., 361 (2015), 313-366.
doi: 10.1007/s00208-014-1075-x. |
[22] |
A. Moradifam,
Optimal weighted Hardy-Rellich inequalities on $H^{2}\cap H_{0}^{1}$, J. Lond. Math. Soc., 85 (2012), 22-40.
doi: 10.1112/jlms/jdr045. |
[23] |
R. Musina,
Weighted Sobolev spaces of radially symmetric functions, Annali di Matematica, 193 (2014), 1629-1659.
doi: 10.1007/s10231-013-0348-4. |
[24] |
P. Niu, Y. Ou and J. Han,
Several Hardy type inequalities with weights related to generalized
Greiner operator, Canad. Math. Bull., 53 (2010), 153-162.
doi: 10.4153/CMB-2010-029-9. |
[25] |
P. Niu, H. Zhang and Y. Wang,
Hardy type and Rellich type inequalities on the Heisenberg
group, Proc. Amer. Math. Soc., 129 (2001), 3623-3630.
doi: 10.1090/S0002-9939-01-06011-7. |
[26] |
F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. Ⅲ, Erven P. Noordhoff N.V.,
Groningen; North-Holland Publishing Co., Amsterdam, 1956, pp. 243–250. |
[27] |
A. Tertikas and N. Zographopoulos,
Best constants in the Hardy-Rellich inequalities and
related improvements, Adv. in Math., 209 (2007), 407-459.
doi: 10.1016/j.aim.2006.05.011. |
[28] |
A. Yener, General weighted Hardy type inequalities related to Greiner operators, to appear in Rocky Mountain J. Math., https://projecteuclid.org/euclid.rmjm/1528164034. |
show all references
References:
[1] |
Adimurthi, M. Grossi and S. Santra,
Optimal Hardy-Rellich inequalities, maximum principle
and related eigenvalue problem, J. Funct. Anal., 240 (2006), 36-83.
doi: 10.1016/j.jfa.2006.07.011. |
[2] |
Adimurthi and S. Santra,
Generalized Hardy-Rellich inequalities in critical dimension and its
applications, Commun. Contemp. Math., 11 (2009), 367-394.
doi: 10.1142/S0219199709003405. |
[3] |
S. Ahmetolan and I. Kombe,
A sharp uncertainty principle and Hardy-Poincaré inequalities
on sub-Riemannian manifolds, Math. Inequal. Appl., 15 (2012), 457-467.
doi: 10.7153/mia-15-40. |
[4] |
S. Ahmetolan and I. Kombe,
Hardy and Rellich type inequalities with two weight functions, Math. Inequal. Appl., 19 (2016), 937-948.
doi: 10.7153/mia-19-68. |
[5] |
W. Allegretto and Y. X. Huang,
A Picone's identity for the p−Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.
doi: 10.1016/S0362-546X(97)00530-0. |
[6] |
G. Barbatis,
Improved Rellich inequalities for the polyharmonic operator, Indiana Univ. Math. J., 55 (2006), 1401-1422.
doi: 10.1512/iumj.2006.55.2752. |
[7] |
R. Beals, B. Gaveau and P. Greiner,
On a geometric formula for the fundamental solution of
sub-elliptic Laplacians, Math. Nachr., 181 (1996), 81-163.
doi: 10.1002/mana.3211810105. |
[8] |
R. Beals, B. Gaveau and P. Greiner,
Uniform hypoelliptic Green's functions, J. Math. Pures Appl., 77 (1998), 209-248.
doi: 10.1016/S0021-7824(98)80069-X. |
[9] |
D. M. Bennett,
An extension of Rellich's inequality, Proc. Amer. Math. Soc., 106 (1989), 987-993.
doi: 10.2307/2047283. |
[10] |
P. Caldiroli and R. Musina,
Rellich inequalities with weights, Calc. Var. Partial Differential Equations, 45 (2012), 147-164.
doi: 10.1007/s00526-011-0454-3. |
[11] |
E. B. Davies and A. M. Hinz,
Explicit constants for Rellich inequalities in Lp(Ω), Math. Z., 227 (1998), 511-523.
doi: 10.1007/PL00004389. |
[12] |
A. Detalla, T. Horiuchi and H. Ando,
Sharp remainder terms of the Rellich inequality and its
application, Bull. Malays. Math. Sci. Soc., 35 (2012), 519-528.
|
[13] |
G. B. Folland,
A fundamental solution for a sub-elliptic operator, Bull. Amer. Math. Soc., 79 (1973), 373-376.
doi: 10.1090/S0002-9904-1973-13171-4. |
[14] |
V. A. Galaktionov and I. V. Kamotski,
On nonexistence of Baras-Goldstein type for higherorder parabolic equations with singular potentials, Trans. Am. Math. Soc., 362 (2010), 4117-4136.
doi: 10.1090/S0002-9947-10-04855-5. |
[15] |
N. Garofalo and E. Lanconelli,
Frequency functions on the Heisenberg group, the uncertainty
principle and unique continuation, Ann Inst Fourier (Grenoble), 40 (1990), 313-356.
|
[16] |
F. Gazzola, H. C. Grunau and E. Mitidieri,
Hardy inequalities with optimal constants and
remainder terms, Trans. Amer. Math. Soc., 356 (2004), 2149-2168.
doi: 10.1090/S0002-9947-03-03395-6. |
[17] |
P. C. Greiner,
A fundamental solution for a nonelliptic partial differential operator, Canad. J. Math., 31 (1979), 1107-1120.
doi: 10.4153/CJM-1979-101-3. |
[18] |
I. Kombe,
On the nonexistence of positive solutions to nonlinear degenerate parabolic equations with singular coefficients, Appl. Anal., 85 (2006), 467-478.
doi: 10.1080/00036810500404967. |
[19] |
B. Lian,
Some sharp Rellich type inequalities on nilpotent groups and application, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 59-74.
doi: 10.1016/S0252-9602(12)60194-5. |
[20] |
P. Lindqvist,
On the equation $\text{div}(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u = 0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.
doi: 10.2307/2048375. |
[21] |
G. Metafune, M. Sobajima and S. C. Motohiro,
Weighted Calderón-Zygmund and Rellich
inequalities in Lp, Math. Ann., 361 (2015), 313-366.
doi: 10.1007/s00208-014-1075-x. |
[22] |
A. Moradifam,
Optimal weighted Hardy-Rellich inequalities on $H^{2}\cap H_{0}^{1}$, J. Lond. Math. Soc., 85 (2012), 22-40.
doi: 10.1112/jlms/jdr045. |
[23] |
R. Musina,
Weighted Sobolev spaces of radially symmetric functions, Annali di Matematica, 193 (2014), 1629-1659.
doi: 10.1007/s10231-013-0348-4. |
[24] |
P. Niu, Y. Ou and J. Han,
Several Hardy type inequalities with weights related to generalized
Greiner operator, Canad. Math. Bull., 53 (2010), 153-162.
doi: 10.4153/CMB-2010-029-9. |
[25] |
P. Niu, H. Zhang and Y. Wang,
Hardy type and Rellich type inequalities on the Heisenberg
group, Proc. Amer. Math. Soc., 129 (2001), 3623-3630.
doi: 10.1090/S0002-9939-01-06011-7. |
[26] |
F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. Ⅲ, Erven P. Noordhoff N.V.,
Groningen; North-Holland Publishing Co., Amsterdam, 1956, pp. 243–250. |
[27] |
A. Tertikas and N. Zographopoulos,
Best constants in the Hardy-Rellich inequalities and
related improvements, Adv. in Math., 209 (2007), 407-459.
doi: 10.1016/j.aim.2006.05.011. |
[28] |
A. Yener, General weighted Hardy type inequalities related to Greiner operators, to appear in Rocky Mountain J. Math., https://projecteuclid.org/euclid.rmjm/1528164034. |
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