In this paper, we consider a trilinear weighted Hardy-Littlewood-Sobolev inequality as follows:
$ \left|\displaystyle {\int}_{\mathbb{R}^n}\displaystyle {\int}_{\mathbb{R}^n}\displaystyle {\int}_{\mathbb{R}^n}\frac{\prod\nolimits_{i = 1}^3f_i(x_i)|x_i|^{-\alpha_{ii}}} {\prod\nolimits_{1\leqslant j<k\leqslant 3}|x_j - x_k|^{\alpha_{jk}}}dx_1dx_2dx_3\right|\leqslant C\|f_1\|_{p_1}\|f_2\|_{p_2}\|f_3\|_{p_3}, $
which can be regarded as the natural trilinear form of the Stein-Weiss inequality. For $1\leqslant p_1, p_2, p_3\leqslant\infty$, we systematically establish the trilinear Stein-Weiss inequality. In particular, when $1 < p_1, p_2, p_3 < \infty$, by means of appropriate space decomposition, we give the necessary and sufficient conditions to characterize the trilinear Stein-Weiss inequality.
Citation: Yongliang Zhou, Xiaohua Xu, Di Wu. Necessary and sufficient conditions of the trilinear Stein-Weiss inequality[J]. Electronic Research Archive, 2026, 34(3): 1585-1608. doi: 10.3934/era.2026072
In this paper, we consider a trilinear weighted Hardy-Littlewood-Sobolev inequality as follows:
$ \left|\displaystyle {\int}_{\mathbb{R}^n}\displaystyle {\int}_{\mathbb{R}^n}\displaystyle {\int}_{\mathbb{R}^n}\frac{\prod\nolimits_{i = 1}^3f_i(x_i)|x_i|^{-\alpha_{ii}}} {\prod\nolimits_{1\leqslant j<k\leqslant 3}|x_j - x_k|^{\alpha_{jk}}}dx_1dx_2dx_3\right|\leqslant C\|f_1\|_{p_1}\|f_2\|_{p_2}\|f_3\|_{p_3}, $
which can be regarded as the natural trilinear form of the Stein-Weiss inequality. For $1\leqslant p_1, p_2, p_3\leqslant\infty$, we systematically establish the trilinear Stein-Weiss inequality. In particular, when $1 < p_1, p_2, p_3 < \infty$, by means of appropriate space decomposition, we give the necessary and sufficient conditions to characterize the trilinear Stein-Weiss inequality.
| [1] |
E. Stein, G. Weiss, Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503–514. https://doi.org/10.1512/iumj.1958.7.57030 doi: 10.1512/iumj.1958.7.57030
|
| [2] |
Z. Shi, D. Wu, D. Yan, Necessary and sufficient conditions of doubly weighted Hardy-Littlewood-Sobolev inequality, Anal. Theory Appl., 30 (2014), 193–204. https://doi.org/10.4208/ata.2014.v30.n2.5 doi: 10.4208/ata.2014.v30.n2.5
|
| [3] |
B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Am. Math. Soc., 192 (1974), 261–274. https://doi.org/10.1090/S0002-9947-1974-0340523-6 doi: 10.1090/S0002-9947-1974-0340523-6
|
| [4] |
E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349–374. https://doi.org/10.2307/2007032 doi: 10.2307/2007032
|
| [5] |
W. Chen, C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Am. Math. Soc., 136 (2008), 955–962. https://doi.org/10.1090/S0002-9939-07-09232-5 doi: 10.1090/S0002-9939-07-09232-5
|
| [6] |
W. Beckner, Pitt's inequality with sharp convolution estimates, Proc. Am. Math. Soc., 136 (2008), 1871–1885. https://doi.org/10.1090/S0002-9939-07-09216-7 doi: 10.1090/S0002-9939-07-09216-7
|
| [7] |
P. L. De Nápoli, I. Drelichman, R. G. Durán, On weighted inequalities for fractional integrals of radial functions, Ill. J. Math., 55 (2011), 575–587. https://doi.org/10.1215/ijm/1359762403 doi: 10.1215/ijm/1359762403
|
| [8] |
T. Chen, W. C. Sun, Geometric inequalities in harmonic analysis, Sci. China Math., 48 (2018), 1–32. https://doi.org/10.1360/N012018-00081 doi: 10.1360/N012018-00081
|
| [9] |
L. Chen, G. Lu, C. Tao, Existence of extremal functions for the Stein–Weiss inequalities on the Heisenberg group, J. Funct. Anal., 277 (2019), 1112–1138. https://doi.org/10.1016/j.jfa.2019.01.002 doi: 10.1016/j.jfa.2019.01.002
|
| [10] |
P. L. De Nápoli, T. Picon, Stein-Weiss inequality in $L^1$ norm for vector fields, Proc. Am. Math. Soc., 151 (2023), 1663–1679. https://doi.org/10.1090/proc/16241 doi: 10.1090/proc/16241
|
| [11] |
Z. Shi, D. Wu, D. Yan, On the multilinear fractional integral operators with correlation kernels, J. Fourier Anal. Appl., 25 (2019), 538–587. https://doi.org/10.1007/s00041-017-9591-1 doi: 10.1007/s00041-017-9591-1
|
| [12] |
Y. Zhou, Y. Deng, D. Wu, D. Yan, Necessary and sufficient conditions on weighted multilinear fractional integral inequality, Commun. Pure Appl. Anal., 21 (2022), 727–747. https://doi.org/10.3934/cpaa.2021196 doi: 10.3934/cpaa.2021196
|
| [13] |
L. Grafakos, C. Morpurgo, A Selberg integral formula and applications, Pac. J. Math., 191 (1999), 85–94. https://doi.org/10.2140/pjm.1999.191.85 doi: 10.2140/pjm.1999.191.85
|
| [14] | L. Grafakos, Classical Fourier Analysis, 3rd edition, Springer, New York, 2014. https://doi.org/10.1007/978-1-4939-1194-3 |
| [15] | D. Chalishajar, M. Somasundaram, P. Sethuraman, Analyticity of weighted composition semigroups on the space of holomorphic functions, Bull. Iran. Math. Soc., 51 (2025). https://doi.org/10.1007/s41980-024-00923-7 |
| [16] | D. Chalishajar, K. Karthikeyan, P. Karthikeyan, D. S. Raja, P. Sundararajan, Qualitative analysis of $\Psi$-Hilfer fractional impulsive differential equations involving almost sectorial operators with nonlocal multi-point condition, Eur. J. Math. Appl., 2 (2022). https://doi.org/10.28919/ejma.2022.2.13 |
Figures(1) / Tables(1)
Yongliang Zhou, Xiaohua Xu, Di Wu. Necessary and sufficient conditions of the trilinear Stein-Weiss inequality[J]. Electronic Research Archive, 2026, 34(3): 1585-1608. doi: 10.3934/era.2026072
DownLoad: