September  2021, 20(9): 3259-3297. doi: 10.3934/cpaa.2021105

Admissible function spaces for weighted Sobolev inequalities

1. 

Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India

2. 

The Institute of Mathematical Sciences (HBNI), Chennai 600113, India

* Corresponding author

Received  December 2020 Revised  May 2021 Published  September 2021 Early access  June 2021

Let
$ k,N\in \mathbb{N} $
with
$ 1\le k\le N $
and let
$ \Omega = \Omega_1 \times \Omega_2 $
be an open set in
$ \mathbb{R}^k \times \mathbb{R}^{N-k} $
. For
$ p\in (1,\infty) $
and
$ q \in (0,\infty), $
we consider the following weighted Sobolev type inequality:
$\begin{align} \int_{\Omega} |g_1(y)||g_2(z)| |u(y,z)|^q \, {\rm d}y {\rm d}z \leq C \left( \int_{\Omega} | \nabla u(y,z) |^p \, {\rm d}y {\rm d}z \right)^{\frac{q}{p}}, \quad \forall \, u \in \mathcal{C}^1_c(\Omega), \\(0.1)\end{align}$
for some
$ C>0 $
. Depending on the values of
$ N,k,p,q $
we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for
$ (g_1, g_2) $
so that (0.1) holds. Furthermore, we give a sufficient condition on
$ g_1,g_2 $
so that the best constant in (0.1) is attained in the Beppo-Levi space
$ \mathcal{D}^{1,p}_0(\Omega) $
-the completion of
$ \mathcal{C}^1_c(\Omega) $
with respect to
$\|\nabla u\|_{L p(\Omega)}$
.
Citation: T. V. Anoop, Nirjan Biswas, Ujjal Das. Admissible function spaces for weighted Sobolev inequalities. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3259-3297. doi: 10.3934/cpaa.2021105
References:
[1]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. Math. (2), 128 (1988), 385-398.  doi: 10.2307/1971445.  Google Scholar

[2]

Adimurthi, N. Chaudhuri and M. Ramaswamy., An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc., 130 (2002), 489-505.  doi: 10.1090/S0002-9939-01-06132-9.  Google Scholar

[3]

W. Allegretto., Principal eigenvalues for indefinite-weight elliptic problems in ${{{\mathbb R}}}^n$, Proc. Amer. Math. Soc., 116 (1992), 701-706.  doi: 10.2307/2159436.  Google Scholar

[4]

W. Allegretto and Y. X. Huang, Eigenvalues of the indefinite-weight $p$-Laplacian in weighted spaces, Funkcial. Ekvac., 38 (1995), 233-242.   Google Scholar

[5]

T. V. Anoop., A note on generalized Hardy-Sobolev inequalities, Int. J. Anal., pages Art. ID 784398, 9, 2013. doi: 10.1155/2013/784398.  Google Scholar

[6]

T. V. Anoop and U. Das, The compactness and the concentration compactness via $p$-capacity, Annali di Matematica Pura ed Applicata (1923 -), 2021. doi: 10.1007/s10231-021-01098-2.  Google Scholar

[7]

T. V. AnoopU. Das and A. Sarkar, On the generalized Hardy-Rellich inequalities, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 897-919.  doi: 10.1017/prm.2018.128.  Google Scholar

[8]

T. V. AnoopP. Drábek and S. Sasi, Weighted quasilinear eigenvalue problems in exterior domains, Calc. Var. Partial Differ. Equ., 53 (2015), 961-975.  doi: 10.1007/s00526-014-0773-2.  Google Scholar

[9]

T. V. AnoopM. Lucia and M. Ramaswamy, Eigenvalue problems with weights in Lorentz spaces, Calc. Var. Partial Differ. Equ., 36 (2009), 355-376.  doi: 10.1007/s00526-009-0232-7.  Google Scholar

[10]

M. Badiale and E. Serra, Critical nonlinear elliptic equations with singularities and cylindrical symmetry, Rev. Mat. Iberoam., 20 (2004), 33-66.  doi: 10.4171/RMI/379.  Google Scholar

[11]

M. Badiale and G. Tarantello, A Sobolev-{H}ardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163 (2002), 259-293.  doi: 10.1007/s002050200201.  Google Scholar

[12]

C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. (Rozprawy Mat.), 175: 67, 1980.  Google Scholar

[13] G. Bertin, Dynamics of Galaxies, Cambridge University Press, Cambridge, 2000.   Google Scholar
[14]

J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull., 21 (1978), 405-408.  doi: 10.4153/CMB-1978-071-7.  Google Scholar

[15]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[16]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.   Google Scholar

[17]

H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Commun. Partial Differ. Equ., 5 (1980), 773-789.  doi: 10.1080/03605308008820154.  Google Scholar

[18]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.   Google Scholar

[19]

R. E. Castillo and H. Rafeiro, An introductory course in Lebesgue spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, [Cham], 2016. doi: 10.1007/978-3-319-30034-4.  Google Scholar

[20]

S. Chanillo and R. L. Wheeden, $L^p$-estimates for fractional integrals and Sobolev inequalities with applications to Schrödinger operators, Commun. Partial Differ. Equ., 10 (1985), 1077-1116.  doi: 10.1080/03605308508820401.  Google Scholar

[21]

L. Ciotti, Dynamical models in astrophysics, Lecture Notes, Scuola Normale Superiore, Pisa, 2001. Google Scholar

[22]

D. E. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-07731-3.  Google Scholar

[23]

D. E. Edmunds and H. Triebel, Sharp Sobolev embeddings and related Hardy inequalities: the critical case, Math. Nachr., 207 (1999), 79-92.  doi: 10.1002/mana.1999.3212070105.  Google Scholar

[24]

C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 129-206.  doi: 10.1090/S0273-0979-1983-15154-6.  Google Scholar

[25]

C. Fefferman and D. H. Phong, Lower bounds for Schrödinger equations, In Conference on Partial Differential Equations (Saint Jean de Monts, 1982), pages Conf. No. 7, 7. Soc. Math. France, Paris, 1982.  Google Scholar

[26]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233.  doi: 10.1006/jfan.2001.3900.  Google Scholar

[27]

G. B. Folland, Real Analysis, Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, second edition, 1999.  Google Scholar

[28]

N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.  doi: 10.1007/s00208-010-0510-x.  Google Scholar

[29]

N. Ghoussoub and A. Moradifam, Functional inequalities: new perspectives and new applications, volume 187 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/187.  Google Scholar

[30]

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.  doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar

[31]

K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand., 45 (1979), 77-102.  doi: 10.7146/math.scand.a-11827.  Google Scholar

[32] G. H. HardyJ. E. Littlewood and G. Pólya, Inequalities, Cambridge, at the University Press, 1952.   Google Scholar
[33]

L. Hörmander and J. L. Lions, Sur la complétion par rapport à une intégrale de Dirichlet, Math. Scand., 4 (1956), 259-270.  doi: 10.7146/math.scand.a-10474.  Google Scholar

[34]

R. A. Hunt, On $L(p, q)$ spaces, Enseign. Math. (2), 12: 249–276, 1966.  Google Scholar

[35]

B. KawohlM. Lucia and S. Prashanth, Simplicity of the principal eigenvalue for indefinite quasilinear problems, Adv. Differ. Equ., 12 (2007), 407-434.   Google Scholar

[36]

R. Kerman and E. Sawyer, Weighted norm inequalities for potentials with applications to Schrödinger operators, Fourier transforms, and Carleson measures, Bull. Amer. Math. Soc. (N.S.), 12 (1985), 112-116.  doi: 10.1090/S0273-0979-1985-15306-6.  Google Scholar

[37]

A. Kufner, L. Maligranda and L. E. Persson, The Hardy inequality, Vydavatelský Servis, Plzeň, 2007.  Google Scholar

[38]

N. Lam, G. Lu and L. Zhang, Factorizations and Hardy's type identities and inequalities on upper half spaces, Calc. Var. Partial Differ. Equ., 58 (2019), . doi: 10.1007/s00526-019-1633-x.  Google Scholar

[39]

N. Lam, G. Lu and L. Zhang, Geometric Hardy's inequalities with general distance functions., J. Funct. Anal., 279(8): 108673, 35, 2020. doi: 10.1016/j.jfa.2020.108673.  Google Scholar

[40]

J. Lehrbäck and A. V. Vähäkangas, In between the inequalities of Sobolev and Hardy, J. Funct. Anal., 271 (2016), 330-364.  doi: 10.1016/j.jfa.2016.04.028.  Google Scholar

[41]

Y. Li, On the positive solutions of the Matukuma equation, Duke Math. J., 70 (1993), 575-589.  doi: 10.1215/S0012-7094-93-07012-3.  Google Scholar

[42]

Y. Li and W. M. Ni., On conformal scalar curvature equations in Rn, Duke Math. J., 57 (1988), 895-924.  doi: 10.1215/S0012-7094-88-05740-7.  Google Scholar

[43]

R. L. Long and F. S. Nie, Weighted Sobolev inequality and eigenvalue estimates of Schrödinger operators, In Harmonic analysis (Tianjin, 1988), volume 1494 of Lecture Notes in Math., pages 131–141. Springer, Berlin, 1991. doi: 10.1007/BFb0087765.  Google Scholar

[44]

G. G. Lorentz, Some new functional spaces, Ann. Math. (2), 51 (1950), 37-55.  doi: 10.2307/1969496.  Google Scholar

[45]

F. Mamedov and Y. Shukurov, A Sawyer-type sufficient condition for the weighted Poincaré inequality, Positivity, 22 (2018), 687-699.  doi: 10.1007/s11117-017-0537-2.  Google Scholar

[46]

A. Manes and A. M. Micheletti., Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), 7 (1973), 285-301.   Google Scholar

[47]

V. G. Maz'ja, Sobolev Spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-09922-3.  Google Scholar

[48]

V. Maz'ya, Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces, Contemp. Math., 338 (2003), 307-340.  doi: 10.1090/conm/338/06078.  Google Scholar

[49]

B. Muckenhoupt, Hardy's inequality with weights, Studia Math., 44 (1972), 31-38.  doi: 10.4064/sm-44-1-31-38.  Google Scholar

[50]

E. S. Noussair and C. A. Swanson, Solutions of Matukuma's equation with finite total mass, Indiana Univ. Math. J., 38 (1989), 557-561.  doi: 10.1512/iumj.1989.38.38026.  Google Scholar

[51]

R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142.   Google Scholar

[52]

C. Pérez., Two weighted norm inequalities for Riesz potentials and uniform $L^p$-weighted Sobolev inequalities, Indiana Univ. Math. J., 39(1): 31–44, 1990. doi: 10.1512/iumj.1990.39.39004.  Google Scholar

[53] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27. Princeton University Press, Princeton, N. J., 1951.   Google Scholar
[54]

K. Sandeep, On a noncompact minimization problem of Hardy-Sobolev type, Adv. Nonlinear Stud., 2 (2002), 81-91.  doi: 10.1515/ans-2002-0106.  Google Scholar

[55]

E. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc., 308 (1988), 533-545.  doi: 10.2307/2001090.  Google Scholar

[56]

E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math., 114 (1992), 813-874.  doi: 10.2307/2374799.  Google Scholar

[57]

G. J. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl., 160 (1991), 434-445.  doi: 10.1016/0022-247X(91)90316-R.  Google Scholar

[58]

A. Szulkin and M. Willem, Eigenvalue problems with indefinite weight, Studia Math., 135 (1999), 191-201.   Google Scholar

[59]

H. Tanaka, Two-weight norm inequalities for product fractional integral operators, Bull. Sci. Math., 166: 102940, 18, 2021. doi: 10.1016/j.bulsci.2020.102940.  Google Scholar

[60]

H. Triebel, Higher Analysis, Hochschulbücher für Mathematik. [University Books for Mathematics]. Johann Ambrosius Barth Verlag GmbH, Leipzig, 1992.  Google Scholar

[61]

N. Visciglia, A note about the generalized Hardy-Sobolev inequality with potential in $L^{p, d}({{\mathbb R}}^n)$, Calc. Var. Partial Differential Equations, 24(2): 167–184, 2005. doi: 10.1007/s00526-004-0319-0.  Google Scholar

[62]

E. Yanagida and S. Yotsutani, Global structure of positive solutions to equations of Matukuma type, Arch. Rational Mech. Anal., 134(3): 199–226, 1996. doi: 10.1007/BF00379534.  Google Scholar

[63]

L. S. Yu, Nonlinear $p$-Laplacian problems on unbounded domains, Proc. Amer. Math. Soc., 115(4): 1037–1045, 1992. doi: 10.2307/2159352.  Google Scholar

show all references

References:
[1]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. Math. (2), 128 (1988), 385-398.  doi: 10.2307/1971445.  Google Scholar

[2]

Adimurthi, N. Chaudhuri and M. Ramaswamy., An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc., 130 (2002), 489-505.  doi: 10.1090/S0002-9939-01-06132-9.  Google Scholar

[3]

W. Allegretto., Principal eigenvalues for indefinite-weight elliptic problems in ${{{\mathbb R}}}^n$, Proc. Amer. Math. Soc., 116 (1992), 701-706.  doi: 10.2307/2159436.  Google Scholar

[4]

W. Allegretto and Y. X. Huang, Eigenvalues of the indefinite-weight $p$-Laplacian in weighted spaces, Funkcial. Ekvac., 38 (1995), 233-242.   Google Scholar

[5]

T. V. Anoop., A note on generalized Hardy-Sobolev inequalities, Int. J. Anal., pages Art. ID 784398, 9, 2013. doi: 10.1155/2013/784398.  Google Scholar

[6]

T. V. Anoop and U. Das, The compactness and the concentration compactness via $p$-capacity, Annali di Matematica Pura ed Applicata (1923 -), 2021. doi: 10.1007/s10231-021-01098-2.  Google Scholar

[7]

T. V. AnoopU. Das and A. Sarkar, On the generalized Hardy-Rellich inequalities, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 897-919.  doi: 10.1017/prm.2018.128.  Google Scholar

[8]

T. V. AnoopP. Drábek and S. Sasi, Weighted quasilinear eigenvalue problems in exterior domains, Calc. Var. Partial Differ. Equ., 53 (2015), 961-975.  doi: 10.1007/s00526-014-0773-2.  Google Scholar

[9]

T. V. AnoopM. Lucia and M. Ramaswamy, Eigenvalue problems with weights in Lorentz spaces, Calc. Var. Partial Differ. Equ., 36 (2009), 355-376.  doi: 10.1007/s00526-009-0232-7.  Google Scholar

[10]

M. Badiale and E. Serra, Critical nonlinear elliptic equations with singularities and cylindrical symmetry, Rev. Mat. Iberoam., 20 (2004), 33-66.  doi: 10.4171/RMI/379.  Google Scholar

[11]

M. Badiale and G. Tarantello, A Sobolev-{H}ardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163 (2002), 259-293.  doi: 10.1007/s002050200201.  Google Scholar

[12]

C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. (Rozprawy Mat.), 175: 67, 1980.  Google Scholar

[13] G. Bertin, Dynamics of Galaxies, Cambridge University Press, Cambridge, 2000.   Google Scholar
[14]

J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull., 21 (1978), 405-408.  doi: 10.4153/CMB-1978-071-7.  Google Scholar

[15]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[16]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.   Google Scholar

[17]

H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Commun. Partial Differ. Equ., 5 (1980), 773-789.  doi: 10.1080/03605308008820154.  Google Scholar

[18]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.   Google Scholar

[19]

R. E. Castillo and H. Rafeiro, An introductory course in Lebesgue spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, [Cham], 2016. doi: 10.1007/978-3-319-30034-4.  Google Scholar

[20]

S. Chanillo and R. L. Wheeden, $L^p$-estimates for fractional integrals and Sobolev inequalities with applications to Schrödinger operators, Commun. Partial Differ. Equ., 10 (1985), 1077-1116.  doi: 10.1080/03605308508820401.  Google Scholar

[21]

L. Ciotti, Dynamical models in astrophysics, Lecture Notes, Scuola Normale Superiore, Pisa, 2001. Google Scholar

[22]

D. E. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-07731-3.  Google Scholar

[23]

D. E. Edmunds and H. Triebel, Sharp Sobolev embeddings and related Hardy inequalities: the critical case, Math. Nachr., 207 (1999), 79-92.  doi: 10.1002/mana.1999.3212070105.  Google Scholar

[24]

C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 129-206.  doi: 10.1090/S0273-0979-1983-15154-6.  Google Scholar

[25]

C. Fefferman and D. H. Phong, Lower bounds for Schrödinger equations, In Conference on Partial Differential Equations (Saint Jean de Monts, 1982), pages Conf. No. 7, 7. Soc. Math. France, Paris, 1982.  Google Scholar

[26]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233.  doi: 10.1006/jfan.2001.3900.  Google Scholar

[27]

G. B. Folland, Real Analysis, Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, second edition, 1999.  Google Scholar

[28]

N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.  doi: 10.1007/s00208-010-0510-x.  Google Scholar

[29]

N. Ghoussoub and A. Moradifam, Functional inequalities: new perspectives and new applications, volume 187 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/187.  Google Scholar

[30]

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.  doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar

[31]

K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand., 45 (1979), 77-102.  doi: 10.7146/math.scand.a-11827.  Google Scholar

[32] G. H. HardyJ. E. Littlewood and G. Pólya, Inequalities, Cambridge, at the University Press, 1952.   Google Scholar
[33]

L. Hörmander and J. L. Lions, Sur la complétion par rapport à une intégrale de Dirichlet, Math. Scand., 4 (1956), 259-270.  doi: 10.7146/math.scand.a-10474.  Google Scholar

[34]

R. A. Hunt, On $L(p, q)$ spaces, Enseign. Math. (2), 12: 249–276, 1966.  Google Scholar

[35]

B. KawohlM. Lucia and S. Prashanth, Simplicity of the principal eigenvalue for indefinite quasilinear problems, Adv. Differ. Equ., 12 (2007), 407-434.   Google Scholar

[36]

R. Kerman and E. Sawyer, Weighted norm inequalities for potentials with applications to Schrödinger operators, Fourier transforms, and Carleson measures, Bull. Amer. Math. Soc. (N.S.), 12 (1985), 112-116.  doi: 10.1090/S0273-0979-1985-15306-6.  Google Scholar

[37]

A. Kufner, L. Maligranda and L. E. Persson, The Hardy inequality, Vydavatelský Servis, Plzeň, 2007.  Google Scholar

[38]

N. Lam, G. Lu and L. Zhang, Factorizations and Hardy's type identities and inequalities on upper half spaces, Calc. Var. Partial Differ. Equ., 58 (2019), . doi: 10.1007/s00526-019-1633-x.  Google Scholar

[39]

N. Lam, G. Lu and L. Zhang, Geometric Hardy's inequalities with general distance functions., J. Funct. Anal., 279(8): 108673, 35, 2020. doi: 10.1016/j.jfa.2020.108673.  Google Scholar

[40]

J. Lehrbäck and A. V. Vähäkangas, In between the inequalities of Sobolev and Hardy, J. Funct. Anal., 271 (2016), 330-364.  doi: 10.1016/j.jfa.2016.04.028.  Google Scholar

[41]

Y. Li, On the positive solutions of the Matukuma equation, Duke Math. J., 70 (1993), 575-589.  doi: 10.1215/S0012-7094-93-07012-3.  Google Scholar

[42]

Y. Li and W. M. Ni., On conformal scalar curvature equations in Rn, Duke Math. J., 57 (1988), 895-924.  doi: 10.1215/S0012-7094-88-05740-7.  Google Scholar

[43]

R. L. Long and F. S. Nie, Weighted Sobolev inequality and eigenvalue estimates of Schrödinger operators, In Harmonic analysis (Tianjin, 1988), volume 1494 of Lecture Notes in Math., pages 131–141. Springer, Berlin, 1991. doi: 10.1007/BFb0087765.  Google Scholar

[44]

G. G. Lorentz, Some new functional spaces, Ann. Math. (2), 51 (1950), 37-55.  doi: 10.2307/1969496.  Google Scholar

[45]

F. Mamedov and Y. Shukurov, A Sawyer-type sufficient condition for the weighted Poincaré inequality, Positivity, 22 (2018), 687-699.  doi: 10.1007/s11117-017-0537-2.  Google Scholar

[46]

A. Manes and A. M. Micheletti., Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), 7 (1973), 285-301.   Google Scholar

[47]

V. G. Maz'ja, Sobolev Spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-09922-3.  Google Scholar

[48]

V. Maz'ya, Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces, Contemp. Math., 338 (2003), 307-340.  doi: 10.1090/conm/338/06078.  Google Scholar

[49]

B. Muckenhoupt, Hardy's inequality with weights, Studia Math., 44 (1972), 31-38.  doi: 10.4064/sm-44-1-31-38.  Google Scholar

[50]

E. S. Noussair and C. A. Swanson, Solutions of Matukuma's equation with finite total mass, Indiana Univ. Math. J., 38 (1989), 557-561.  doi: 10.1512/iumj.1989.38.38026.  Google Scholar

[51]

R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142.   Google Scholar

[52]

C. Pérez., Two weighted norm inequalities for Riesz potentials and uniform $L^p$-weighted Sobolev inequalities, Indiana Univ. Math. J., 39(1): 31–44, 1990. doi: 10.1512/iumj.1990.39.39004.  Google Scholar

[53] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27. Princeton University Press, Princeton, N. J., 1951.   Google Scholar
[54]

K. Sandeep, On a noncompact minimization problem of Hardy-Sobolev type, Adv. Nonlinear Stud., 2 (2002), 81-91.  doi: 10.1515/ans-2002-0106.  Google Scholar

[55]

E. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc., 308 (1988), 533-545.  doi: 10.2307/2001090.  Google Scholar

[56]

E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math., 114 (1992), 813-874.  doi: 10.2307/2374799.  Google Scholar

[57]

G. J. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl., 160 (1991), 434-445.  doi: 10.1016/0022-247X(91)90316-R.  Google Scholar

[58]

A. Szulkin and M. Willem, Eigenvalue problems with indefinite weight, Studia Math., 135 (1999), 191-201.   Google Scholar

[59]

H. Tanaka, Two-weight norm inequalities for product fractional integral operators, Bull. Sci. Math., 166: 102940, 18, 2021. doi: 10.1016/j.bulsci.2020.102940.  Google Scholar

[60]

H. Triebel, Higher Analysis, Hochschulbücher für Mathematik. [University Books for Mathematics]. Johann Ambrosius Barth Verlag GmbH, Leipzig, 1992.  Google Scholar

[61]

N. Visciglia, A note about the generalized Hardy-Sobolev inequality with potential in $L^{p, d}({{\mathbb R}}^n)$, Calc. Var. Partial Differential Equations, 24(2): 167–184, 2005. doi: 10.1007/s00526-004-0319-0.  Google Scholar

[62]

E. Yanagida and S. Yotsutani, Global structure of positive solutions to equations of Matukuma type, Arch. Rational Mech. Anal., 134(3): 199–226, 1996. doi: 10.1007/BF00379534.  Google Scholar

[63]

L. S. Yu, Nonlinear $p$-Laplacian problems on unbounded domains, Proc. Amer. Math. Soc., 115(4): 1037–1045, 1992. doi: 10.2307/2159352.  Google Scholar

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