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Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions
Necessary and sufficient conditions on weighted multilinear fractional integral inequality
| 1. | University of Chinese Academy of Sciences, No.19(A) Yuquan Road, Shijingshan District, Beijing, China 100049 |
| 2. | Zhejiang University of Science and Technology, No.318 Liuhe Road, Hangzhou, Zhejiang, China 310023 |
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.
References:
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E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
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W. Rudin, Real and Complex Analysis, 3$^{nd}$ edition, McGraw-Hill Education, New York, 1987. |
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Z. Shi, D. Wu and D. Yan,
Necessary and sufficient conditions of doubly weighted Hardy-Littlewood-Sobolev inequality, Anal. Theor. Appl., 30 (2014), 193-204.
doi: 10.4208/ata.2014.v30.n2.5. |
| [4] |
Z. Shi, D. Wu and D. Yan,
On the Multi-linear Fractional Integral Operators with Correlation Kernels, J. Fourier Anal. Appl., 25 (2019), 538-587.
doi: 10.1007/s00041-017-9591-1. |
| [5] |
E. M. Stein and G. Weiss,
Fractional integrals on $n$-dimensional euclidean space, J. Math. Mech., 7 (1958), 503-514.
doi: 10.1512/iumj.1958.7.57030. |
| [6] |
D. Wu, Z. Shi and D. Yan,
Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality, Sci. Chi. Math., 57 (2014), 963-970.
doi: 10.1007/s11425-013-4681-2. |
| [7] |
D. Wu and D. Yan,
Sharp constants for a class of multi-linear integral operators and some applications, Sci. Chi. Math., 59 (2016), 907-920.
doi: 10.1007/s11425-015-5120-3. |
show all references
References:
| [1] |
E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
| [2] |
W. Rudin, Real and Complex Analysis, 3$^{nd}$ edition, McGraw-Hill Education, New York, 1987. |
| [3] |
Z. Shi, D. Wu and D. Yan,
Necessary and sufficient conditions of doubly weighted Hardy-Littlewood-Sobolev inequality, Anal. Theor. Appl., 30 (2014), 193-204.
doi: 10.4208/ata.2014.v30.n2.5. |
| [4] |
Z. Shi, D. Wu and D. Yan,
On the Multi-linear Fractional Integral Operators with Correlation Kernels, J. Fourier Anal. Appl., 25 (2019), 538-587.
doi: 10.1007/s00041-017-9591-1. |
| [5] |
E. M. Stein and G. Weiss,
Fractional integrals on $n$-dimensional euclidean space, J. Math. Mech., 7 (1958), 503-514.
doi: 10.1512/iumj.1958.7.57030. |
| [6] |
D. Wu, Z. Shi and D. Yan,
Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality, Sci. Chi. Math., 57 (2014), 963-970.
doi: 10.1007/s11425-013-4681-2. |
| [7] |
D. Wu and D. Yan,
Sharp constants for a class of multi-linear integral operators and some applications, Sci. Chi. Math., 59 (2016), 907-920.
doi: 10.1007/s11425-015-5120-3. |
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