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Journal articles on the topic 'Hardy-Sobolev Inequality'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 June 2024

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1

Ruzhansky, Michael, Durvudkhan Suragan, and Nurgissa Yessirkegenov. "Euler semigroup, Hardy–Sobolev and Gagliardo–Nirenberg type inequalities on homogeneous groups." Semigroup Forum 101, no. 1 (June 10, 2020): 162–91. http://dx.doi.org/10.1007/s00233-020-10110-9.

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Abstract In this paper we describe the Euler semigroup $$\{e^{-t\mathbb {E}^{*}\mathbb {E}}\}_{t>0}$$ { e - t E ∗ E } t > 0 on homogeneous Lie groups, which allows us to obtain various types of the Hardy–Sobolev and Gagliardo–Nirenberg type inequalities for the Euler operator $$\mathbb {E}$$ E . Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and $$|\cdot |$$ | · | -radial weighted Hardy–Sobolev type inequality are established.

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2

Liu, Min, Fengli Jiang, and Zhenyu Guo. "Fractional Hardy–Sobolev Inequalities with Magnetic Fields." Advances in Mathematical Physics 2019 (December 12, 2019): 1–5. http://dx.doi.org/10.1155/2019/6595961.

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A fractional Hardy–Sobolev inequality with a magnetic field is studied in the present paper. Under appropriate conditions, the achievement of the best constant of the fractional magnetic Hardy–Sobolev inequality is established.

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3

Dou, Jingbo. "Weighted Hardy–Littlewood–Sobolev inequalities on the upper half space." Communications in Contemporary Mathematics 18, no. 05 (July 18, 2016): 1550067. http://dx.doi.org/10.1142/s0219199715500674.

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In this paper, we establish a weighted Hardy–Littlewood–Sobolev (HLS) inequality on the upper half space using a weighted Hardy type inequality on the upper half space with boundary term, and discuss the existence of extremal functions based on symmetrization argument. As an application, we can show a weighted Sobolev–Hardy trace inequality with [Formula: see text]-biharmonic operator.

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4

Dou, Jingbo, and Meijun Zhu. "Reversed Hardy–Littewood–Sobolev Inequality." International Mathematics Research Notices 2015, no. 19 (December 4, 2014): 9696–726. http://dx.doi.org/10.1093/imrn/rnu241.

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5

Ruzhansky, Michael, Bolys Sabitbek, and Durvudkhan Suragan. "Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups." Bulletin of Mathematical Sciences 10, no. 03 (July 4, 2020): 2050016. http://dx.doi.org/10.1142/s1664360720500162.

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In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: [Formula: see text] which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.

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6

Yang, Ying, and Zhi-ying Deng. "Existence of Solutions for Quasilinear Coupled Systemswith Multiple Critical Nonlinearities." Journal of Mathematics and Informatics 23 (2022): 23–39. http://dx.doi.org/10.22457/jmi.v23a03212.

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This paper is dedicated to investigating a quasilinear elliptic systemwith p- Laplacian in R N , which involves critical Hardy-Littlewood-Sobolev nonlinearities and critical Sobolev nonlinearities. Based upon the Hardy-Littlewood-Sobolev inequality and variational methods, we obtain the attainability of the corresponding best constants and the existence of nontrivial solutions.

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7

ADIMURTHI. "HARDY-SOBOLEV INEQUALITY IN H1(Ω) AND ITS APPLICATIONS." Communications in Contemporary Mathematics 04, no. 03 (August 2002): 409–34. http://dx.doi.org/10.1142/s0219199702000713.

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In this article, we have determined the remainder term for Hardy–Sobolev inequality in H1(Ω) for Ω a bounded smooth domain and studied the existence, non existence and blow up of first eigen value and eigen function for the corresponding Hardy–Sobolev operator with Neumann boundary condition.

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8

Secchi, Simone, Didier Smets, and Michel Willem. "Remarks on a Hardy–Sobolev inequality." Comptes Rendus Mathematique 336, no. 10 (May 2003): 811–15. http://dx.doi.org/10.1016/s1631-073x(03)00202-4.

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9

Yang, Jianfu. "Fractional Sobolev–Hardy inequality in RN." Nonlinear Analysis: Theory, Methods & Applications 119 (June 2015): 179–85. http://dx.doi.org/10.1016/j.na.2014.09.009.

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10

do Ó, João Marcos, Abiel Costa Macedo, and José Francisco de Oliveira. "A Sharp Adams-Type Inequality for Weighted Sobolev Spaces." Quarterly Journal of Mathematics 71, no. 2 (February 7, 2020): 517–38. http://dx.doi.org/10.1093/qmathj/haz051.

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Abstract In a classical work (Ann. Math.128, (1988) 385–398), D. R. Adams proved a sharp Trudinger–Moser inequality for higher-order derivatives. We derive a sharp Adams-type inequality and Sobolev-type inequalities associated with a class of weighted Sobolev spaces that is related to a Hardy-type inequality.

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11

Costa, David G., João Marcos do Ó, and K. Tintarev. "Compactness properties of critical nonlinearities and nonlinear Schrödinger equations." Proceedings of the Edinburgh Mathematical Society 56, no. 2 (April 5, 2013): 427–41. http://dx.doi.org/10.1017/s0013091512000363.

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AbstractWe prove the compactness of critical Sobolev embeddings with applications to nonlinear singular Schrödinger equations and provide a unified treatment in dimensions N > 2 and N = 2, based on a somewhat unexpectedly broad array of parallel properties between spaces $\smash{\mathcal{D}^{1,2}(\mathbb{R}^N)}$ and H10 of the unit disc. These properties include Leray inequality for N = 2 as a counterpart of Hardy inequality for N > 2, pointwise estimates by ground states r(2−N)/2 and $\smash{\sqrt{\log(1/r)}}$ of the respective Hardy-type inequalities, as well as compactness of the limiting Sobolev embeddings once the Sobolev norm is appended by a potential term whose growth at singularities exceeds that of the corresponding Hardy-type potential.

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12

Anoop, T. V. "A Note on Generalized Hardy-Sobolev Inequalities." International Journal of Analysis 2013 (January 8, 2013): 1–9. http://dx.doi.org/10.1155/2013/784398.

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We are concerned with finding a class of weight functions g so that the following generalized Hardy-Sobolev inequality holds: ∫Ωgu2≤C∫Ω|∇u|2, u∈H01(Ω), for some C>0, where Ω is a bounded domain in ℝ2. By making use of Muckenhoupt condition for the one-dimensional weighted Hardy inequalities, we identify a rearrangement invariant Banach function space so that the previous integral inequality holds for all weight functions in it. For weights in a subspace of this space, we show that the best constant in the previous inequality is attained. Our method gives an alternate way of proving the Moser-Trudinger embedding and its refinement due to Hansson.

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13

Dyda, Bartłomiej, and Rupert L. Frank. "Fractional Hardy–Sobolev–Maz'ya inequality for domains." Studia Mathematica 208, no. 2 (2012): 151–66. http://dx.doi.org/10.4064/sm208-2-3.

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14

Cheng, Ze, and Congming Li. "An extended discrete Hardy-Littlewood-Sobolev inequality." Discrete & Continuous Dynamical Systems - A 34, no. 5 (2014): 1951–59. http://dx.doi.org/10.3934/dcds.2014.34.1951.

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15

Fall, Mouhamed Moustapha, and El hadji Abdoulaye Thiam. "Hardy-Sobolev inequality with singularity a curve." Topological Methods in Nonlinear Analysis 49, no. 2 (December 9, 2017): 1. http://dx.doi.org/10.12775/tmna.2017.045.

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16

Thiam, El Hadji Abdoulaye. "Hardy–Sobolev inequality with higher dimensional singularity." Analysis 39, no. 3 (October 1, 2019): 79–96. http://dx.doi.org/10.1515/anly-2018-0006.

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AbstractFor {N\geq 4}, we let Ω be a smooth bounded domain of {\mathbb{R}^{N}}, Γ a smooth closed submanifold of Ω of dimension k, with {1\leq k\leq N-2}, and h a continuous function defined on Ω. We denote by {\rho_{\Gamma}(\,\cdot\,):=\operatorname{dist}(\,\cdot\,,\Gamma)} the distance function to Γ. For {\sigma\in(0,2)}, we study the existence of positive solutions {u\in H^{1}_{0}(\Omega)} to the nonlinear equation-\Delta u+hu=\rho_{\Gamma}^{-\sigma}u^{2^{*}(\sigma)-1}\quad\text{in }\Omega,where {2^{*}(\sigma):=\frac{2(N-\sigma)}{N-2}} is the critical Hardy–Sobolev exponent. In particular, we prove the existence of solution under the influence of the local geometry of Γ and the potential h.

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17

Adimurthi, P. K. Ratnakumar, and Vijay Kumar Sohani. "A Hardy–Sobolev inequality for the twisted Laplacian." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 147, no. 1 (January 11, 2017): 1–23. http://dx.doi.org/10.1017/s0308210516000081.

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We prove a strong optimal Hardy–Sobolev inequality for the twisted Laplacian on ℂn. The twisted Laplacian is the magnetic Laplacian for a system of n particles in the plane, corresponding to the constant magnetic field. The inequality we obtain is strong optimal in the sense that the weight cannot be improved. We also show that our result extends to a one-parameter family of weighted Sobolev spaces.

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18

Adimurthi and Anusha Sekar. "Role of the fundamental solution in Hardy—Sobolev-type inequalities." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 6 (December 2006): 1111–30. http://dx.doi.org/10.1017/s030821050000490x.

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Let n ≥ 3, Ω ⊂ Rn be a domain with 0 ∈ Ω, then, for all the Hardy–Sobolev inequality says that and equality holds if and only if u = 0 and ((n − 2)/2)2 is the best constant which is never achieved. In view of this, there is scope for improving this inequality further. In this paper we have investigated this problem by using the fundamental solutions and have obtained the optimal estimates. Furthermore, we have shown that this technique is used to obtain the Hardy–Sobolev type inequalities on manifolds and also on the Heisenberg group.

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19

Adimurthi and K. Sandeep. "Existence and non-existence of the first eigenvalue of the perturbed Hardy–Sobolev operator." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 5 (October 2002): 1021–43. http://dx.doi.org/10.1017/s0308210500001992.

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In this paper we study the existence, non-existence and simplicity of the first eigenvalue of the perturbed Hardy-Sobolev operator under various assumptions on the perturbation q. We study the asymptotic behaviour of the first eigenfunction near the origin when the perturbation q is q = s, 0 < s < 1. We will also establish the best constant in a Hardy-Sobolev inequality proved by Adimurthi et al.

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20

Adimurthi and K. Sandeep. "Existence and non-existence of the first eigenvalue of the perturbed Hardy–Sobolev operator." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 5 (October 2002): 1021–43. http://dx.doi.org/10.1017/s0308210502000501.

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In this paper we study the existence, non-existence and simplicity of the first eigenvalue of the perturbed Hardy-Sobolev operator under various assumptions on the perturbation q. We study the asymptotic behaviour of the first eigenfunction near the origin when the perturbation q is q = s, 0 < s < 1. We will also establish the best constant in a Hardy-Sobolev inequality proved by Adimurthi et al.

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21

Francisco de Oliveira, José, João Marcos do Ó, and Pedro Ubilla. "Hardy-Sobolev type inequality and supercritical extremal problem." Discrete & Continuous Dynamical Systems - A 39, no. 6 (2019): 3345–64. http://dx.doi.org/10.3934/dcds.2019138.

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22

Ben Salem, Néjib, and Sami Mustapha. "Hardy–Littlewood–Sobolev inequality on the parabolic biangle." Ramanujan Journal 56, no. 1 (July 1, 2021): 387–409. http://dx.doi.org/10.1007/s11139-021-00451-6.

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23

Anoop, V. P., and Sanjay Parui. "Hardy–Littlewood–Sobolev Inequality for Upper Half Space." Annales Mathématiques Blaise Pascal 28, no. 2 (April 14, 2022): 117–40. http://dx.doi.org/10.5802/ambp.401.

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24

Adimurthi, Nirmalendu Chaudhuri, and Mythily Ramaswamy. "An improved Hardy-Sobolev inequality and its application." Proceedings of the American Mathematical Society 130, no. 2 (June 11, 2001): 489–505. http://dx.doi.org/10.1090/s0002-9939-01-06132-9.

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25

Mengesha, Tadele. "Fractional Korn and Hardy-type inequalities for vector fields in half space." Communications in Contemporary Mathematics 21, no. 07 (October 10, 2019): 1850055. http://dx.doi.org/10.1142/s0219199718500554.

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We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to prove a fractional version of the classical Korn’s inequality.

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26

Tao, Mengfei, and Binlin Zhang. "Solutions for nonhomogeneous fractional (p, q)-Laplacian systems with critical nonlinearities." Advances in Nonlinear Analysis 11, no. 1 (January 1, 2022): 1332–51. http://dx.doi.org/10.1515/anona-2022-0248.

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Abstract In this article, we aimed to study a class of nonhomogeneous fractional (p, q)-Laplacian systems with critical nonlinearities as well as critical Hardy nonlinearities in R N {{\mathbb{R}}}^{N} . By appealing to a fixed point result and fractional Hardy-Sobolev inequality, the existence of nontrivial nonnegative solutions is obtained. In particular, we also consider Choquard-type nonlinearities in the second part of this article. More precisely, with the help of Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions for the related systems based on the same approach. Finally, we obtain the corresponding existence results for the fractional (p, q)-Laplacian systems in the case of N = s p = l q N=sp=lq . It is worth pointing out that using fixed point argument to seek solutions for a class of nonhomogeneous fractional (p, q)-Laplacian systems is the main novelty of this article.

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27

ADIMURTHI. "BEST CONSTANTS AND POHOZAEV IDENTITY FOR HARDY–SOBOLEV-TYPE OPERATORS." Communications in Contemporary Mathematics 15, no. 03 (May 19, 2013): 1250050. http://dx.doi.org/10.1142/s0219199712500502.

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This paper is threefold. Firstly, we reformulate the definition of the norm induced by the Hardy inequality (see [J. L. Vázquez and N. B. Zographopoulos, Functional aspects of the Hardy inequality. Appearance of a hidden energy, preprint (2011); http://arxiv. org/abs/1102.5661]) to more general elliptic and sub-elliptic Hardy–Sobolev-type operators. Secondly, we derive optimal inequalities (see [C. Cowan, Optimal inequalities for general elliptic operator with improvement, Commun. Pure Appl. Anal.9(1) (2010) 109–140; N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy–Rellich inequalities, Math. Ann.349(1) (2010) 1–57 (electronic)]) for multiparticle systems in ℝN and Heisenberg group. In particular, we provide a direct proof of an optimal inequality with multipolar singularities shown in [R. Bossi, J. Dolbeault and M. J. Esteban, Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators, Commun. Pure Appl. Anal.7(3) (2008) 533–562]. Finally, we prove an approximation lemma which allows to show that the domain of the Dirichlet–Laplace operator is dense in the domain of the corresponding Hardy operators. As a consequence, in some particular cases, we justify the Pohozaev-type identity for such operators.

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28

Akça, Haydar, Valéry Covachev, and Zlatinka Covacheva. "Improved Stability Estimates for Impulsive Delay Reaction-Diffusion Cohen-Grossberg Neural Networks Via Hardy-Poincaré Inequality." Tatra Mountains Mathematical Publications 54, no. 1 (April 1, 2013): 1–18. http://dx.doi.org/10.2478/tmmp-2013-0001.

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Abstract An impulsive Cohen-Grossberg neural network with time-varying and S-type distributed delays and reaction-diffusion terms is considered. By using Hardy-Poincaré inequality instead of Hardy-Sobolev inequality or just the nonpositivity of the reaction-diffusion operators, under suitable conditions in terms of M-matrices which involve the reaction-diffusion coefficients and the dimension and size of the spatial domain, improved stability estimates for the system with zero Dirichlet boundary conditions are obtained. Examples are given.

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29

Osipenko, K. Yu. "The Hardy-Littlewood-Pólya inequality for analytic functions in Hardy-Sobolev spaces." Sbornik: Mathematics 197, no. 3 (April 30, 2006): 315–34. http://dx.doi.org/10.1070/sm2006v197n03abeh003760.

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30

Liu, Min, De an Chen, and Zhe yu Guo. "A fractional magnetic Hardy-Sobolev inequality with two variables." Journal of Mathematical Inequalities, no. 1 (2022): 181–87. http://dx.doi.org/10.7153/jmi-2022-16-14.

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31

Alvino, Angelo, Vincenzo Ferone, and Guido Trombetti. "On the best constant in a Hardy–Sobolev inequality." Applicable Analysis 85, no. 1-3 (January 2006): 171–80. http://dx.doi.org/10.1080/00036810500277405.

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32

Ngô, Quốc Anh, and Van Hoang Nguyen. "Sharp reversed Hardy–Littlewood–Sobolev inequality on R n." Israel Journal of Mathematics 220, no. 1 (May 5, 2017): 189–223. http://dx.doi.org/10.1007/s11856-017-1515-x.

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33

Wang, Yongda. "An improvement of Sobolev inequality involving Hardy potential term." Monatshefte für Mathematik 185, no. 4 (March 5, 2018): 717–31. http://dx.doi.org/10.1007/s00605-018-1172-0.

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34

Frank, Rupert L., and Elliott H. Lieb. "Inversion positivity and the sharp Hardy–Littlewood–Sobolev inequality." Calculus of Variations and Partial Differential Equations 39, no. 1-2 (December 23, 2009): 85–99. http://dx.doi.org/10.1007/s00526-009-0302-x.

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35

Marano, Salvatore A., and Sunra J. N. Mosconi. "Asymptotics for optimizers of the fractional Hardy–Sobolev inequality." Communications in Contemporary Mathematics 21, no. 05 (July 12, 2019): 1850028. http://dx.doi.org/10.1142/s0219199718500281.

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The existence of optimizers [Formula: see text] in the space [Formula: see text], with differentiability order [Formula: see text], for the Hardy–Sobolev inequality is established through concentration-compactness. The asymptotic behavior [Formula: see text] as [Formula: see text] and the summability information [Formula: see text] for all [Formula: see text] are then obtained. Such properties turn out to be optimal when [Formula: see text], in which case optimizers are explicitly known.

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36

Mancini, G., and K. Sandeep. "Cylindrical symmetry of extremals of a Hardy–Sobolev inequality." Annali di Matematica Pura ed Applicata 183, no. 2 (December 23, 2003): 165–72. http://dx.doi.org/10.1007/s10231-003-0084-2.

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37

Zographopoulos, N. B. "Existence of extremal functions for a Hardy–Sobolev inequality." Journal of Functional Analysis 259, no. 1 (July 2010): 308–14. http://dx.doi.org/10.1016/j.jfa.2010.03.020.

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38

Secchi, Paolo. "A Higher-Order Hardy-Type Inequality in Anisotropic Sobolev Spaces." International Journal of Differential Equations 2012 (2012): 1–7. http://dx.doi.org/10.1155/2012/129691.

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We prove a higher-order inequality of Hardy type for functions in anisotropic Sobolev spaces that vanish at the boundary of the space domain. This is an important calculus tool for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary.

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39

Čižmešija, Aleksandra, Ivan Perić, and Predrag Vuković. "INEQUALITIES OF THE HILBERT TYPE IN $\mathbb{R}^{n}$ WITH NON-CONJUGATE EXPONENTS." Proceedings of the Edinburgh Mathematical Society 51, no. 1 (February 2008): 11–26. http://dx.doi.org/10.1017/s0013091505001690.

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AbstractIn this paper we state and prove a new general Hilbert-type inequality in $\mathbb{R}^{n}$ with $k\geq2$ non-conjugate exponents. Using Selberg's integral formula, this result is then applied to obtain explicit upper bounds for the doubly weighted Hardy–Littlewood–Sobolev inequality and some further Hilbert-type inequalities for $k$ non-negative functions and non-conjugate exponents.

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40

Shen, Yao Tian, and Yang Xin Yao. "Nonlinear elliptic equations with critical potential and critical parameter." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 5 (October 2006): 1041–51. http://dx.doi.org/10.1017/s030821050000487x.

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We give a positive answer to an open problem about Hardy's inequality raised by Brézis and Vázquez, and another result obtained improves that of Vázquez and Zuazua. Furthermore, by this improved inequality and the critical-point theory, in a k-order Sobolev–Hardy space, we obtain the existence of multi-solution to a nonlinear elliptic equation with critical potential and critical parameter.

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41

Zhou, Yongliang, Yangkendi Deng, Di Wu, and Dunyan Yan. "Necessary and sufficient conditions on weighted multilinear fractional integral inequality." Communications on Pure & Applied Analysis 21, no. 2 (2022): 727. http://dx.doi.org/10.3934/cpaa.2021196.

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<p style='text-indent:20px;'>We consider certain kinds of weighted multi-linear fractional integral inequalities which can be regarded as extensions of the Hardy-Littlewood-Sobolev inequality. For a particular case, we characterize the sufficient and necessary conditions which ensure that the corresponding inequality holds. For the general case, we give some sufficient conditions and necessary conditions, respectively.</p>

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42

Kinnunen, Juha, and Pilar Silvestre. "Resistance Conditions and Applications." Analysis and Geometry in Metric Spaces 1 (October 25, 2013): 276–94. http://dx.doi.org/10.2478/agms-2013-0007.

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AbstractThis paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality for compactly supported Lipschitz functions on balls as well as capacitary strong type estimates for the Hardy-Littlewood maximal function. We also consider extensions to Sobolev type inequalities with two different measures and Lorentz type estimates.

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43

Drábek, P., A. Kufner, and V. Mustonen. "Pseudo-monotonicity and degenerated or singular elliptic operators." Bulletin of the Australian Mathematical Society 58, no. 2 (October 1998): 213–21. http://dx.doi.org/10.1017/s0004972700032184.

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Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

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44

Sloane, Craig A. "A fractional Hardy-Sobolev-Maz’ya inequality on the upper halfspace." Proceedings of the American Mathematical Society 139, no. 11 (November 1, 2011): 4003–16. http://dx.doi.org/10.1090/s0002-9939-2011-10818-9.

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45

Chou, Kai Seng, and Chiu Wing Chu. "On the Best Constant for a Weighted Sobolev-Hardy Inequality." Journal of the London Mathematical Society s2-48, no. 1 (August 1993): 137–51. http://dx.doi.org/10.1112/jlms/s2-48.1.137.

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46

Ritorto, Antonella. "A minimization problem involving a fractional Hardy–Sobolev type inequality." Illinois Journal of Mathematics 64, no. 3 (September 2020): 305–17. http://dx.doi.org/10.1215/00192082-8591568.

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47

Chen, Wenxiong, and Congming Li. "The best constant in a weighted Hardy-Littlewood-Sobolev inequality." Proceedings of the American Mathematical Society 136, no. 03 (November 30, 2007): 955–63. http://dx.doi.org/10.1090/s0002-9939-07-09232-5.

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48

Sano, Megumi. "Minimization problem associated with an improved Hardy–Sobolev type inequality." Nonlinear Analysis 200 (November 2020): 111965. http://dx.doi.org/10.1016/j.na.2020.111965.

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49

Wu, Di, ZuoShunHua Shi, and DunYan Yan. "Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality." Science China Mathematics 57, no. 5 (October 1, 2013): 963–70. http://dx.doi.org/10.1007/s11425-013-4681-2.

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50

Dou, Jingbo, and Meijun Zhu. "Sharp Hardy–Littlewood–Sobolev Inequality on the Upper Half Space." International Mathematics Research Notices 2015, no. 3 (October 16, 2013): 651–87. http://dx.doi.org/10.1093/imrn/rnt213.

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