@article{Zhou2026, 
author = {Yongliang Zhou and Xiaohua Xu and Di Wu},
title = {Necessary and sufficient conditions of the trilinear Stein-Weiss inequality},
year = {2026},
journal = {Electronic Research Archive},
volume = {34},
number = {3},
pages = {1585-1608},
keywords = {Stein-Weiss inequalities, boundedness of trilinear functionals, necessary and sufficient conditions},
url = {https://www.sciopen.com/article/10.3934/era.2026072},
doi = {10.3934/era.2026072},
abstract = {In this paper, we consider a trilinear weighted Hardy-Littlewood-Sobolev inequality as follows:         |                            ∫                                                    R                        n                                                            ∫                                                              R                            n                                                                          ∫                                                                        R                                n                                                                                        ∏                                  i                  =                  1                                3                                            f                i                            (                              x                i                            )                              |                                            x                i                                                              |                                                  −                                      α                                          i                      i                                                                                                                          ∏                                  1                  ⩽                  j                  <                  k                  ⩽                  3                                                            |                                            x                j                            −                              x                k                                                              |                                                                      α                                          j                      k                                                                                                    d                      x            1                    d                      x            2                    d                      x            3                                |    ⩽  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  ⩽      p    1    ,      p    2    ,      p    3    ⩽  ∞, we systematically establish the trilinear Stein-Weiss inequality. In particular, when    1  <      p    1    ,      p    2    ,      p    3    <  ∞, by means of appropriate space decomposition, we give the necessary and sufficient conditions to characterize the trilinear Stein-Weiss inequality.}
}