{"paper":{"title":"Certain Multi(sub)linear square functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Loukas Grafakos, Qingying Xue, Sha He","submitted_at":"2015-04-14T05:47:53Z","abstract_excerpt":"Let $d\\ge 1, \\ell\\in\\Z^d$, $m\\in \\mathbb Z^+$ and $\\theta_i$, $i=1,\\dots,m $ are fixed, distinct and nonzero real numbers. We show that the $m$-(sub)linear version below of the Ratnakumar and Shrivastava \\cite{RS1} Littlewood-Paley square function $$T(f_1,\\dots , f_m)(x)=\\Big(\\sum\\limits_{\\ell\\in\\Z^d}|\\int_{\\mathbb{R}^d}f_1(x-\\theta_1 y)\\cdots f_m(x-\\theta_m y)e^{2\\pi i \\ell \\cdot y}K (y)dy|^2\\Big)^{1/2} $$ is bounded from $L^{p_1}(\\mathbb{R}^d) \\times\\cdots\\times L^{p_m}(\\mathbb{R}^d) $ to $L^p(\\mathbb{R}^d) $ when $2\\le p_i<\\infty$ satisfy $1/p=1/p_1+\\cdots+1/p_m$ and $1\\le p<\\infty$. Our pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03424","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}