{
  "id": "1604.01610",
  "version": "v1",
  "published": "2016-04-06T13:27:33.000Z",
  "updated": "2016-04-06T13:27:33.000Z",
  "title": "Summability of multilinear forms on classical sequence spaces",
  "authors": [
    "Tony Nogueira",
    "Pilar Rueda"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We present an extension of the Hardy--Littlewood inequality for multilinear forms. More precisely, let $\\mathbb{K}$ be the real or complex scalar field and $m,k$ be positive integers with $m\\geq k\\,$ and $n_{1},\\dots ,n_{k}$ be positive integers such that $n_{1}+\\cdots +n_{k}=m$. ($a$) If $(r,p)\\in (0,\\infty )\\times \\lbrack 2m,\\infty ]$ then there is a constant $D_{m,r,p,k}^{\\mathbb{K}}\\geq 1$ (not depending on $n$) such that $$ \\left( \\sum_{i_{1},\\dots ,i_{k}=1}^{n}\\left| T\\left( e_{i_{1}}^{n_{1}},\\dots ,e_{i_{k}}^{n_{k}}\\right) \\right| ^{r}\\right) ^{% \\frac{1}{r}}\\leq D_{m,r,p,k}^{\\mathbb{K}} \\cdot n^{max\\left\\{ \\frac{% 2kp-kpr-pr+2rm}{2pr},0\\right\\} }\\left| T\\right| $$ for all $m$-linear forms $T:\\ell_{p}^{n}\\times \\cdots \\times \\ell_{p}^{n}\\rightarrow \\mathbb{K}$ and all positive integers $n$. Moreover, the exponent $max\\left\\{ \\frac{2kp-kpr-pr+2rm}{2pr},0\\right\\} $ is optimal. ($b$) If $(r, p) \\in (0, \\infty) \\times (m, 2m]$ then there is a constant $% D_{m,r,p, k}^{\\mathbb{K}}\\geq 1$ (not depending on $n$) such that $$ \\left( \\sum_{i_{1},\\dots ,i_{k}=1}^{n }\\left| T\\left( e_{i_{1}}^{n_{1}},\\dots ,e_{i_{k}}^{n_{k}}\\right) \\right| ^{r }\\right) ^{% \\frac{1}{r }}\\leq D_{m,r,p, k}^{\\mathbb{K}} \\cdot n^{ max \\left\\{\\frac{% p-rp+rm}{pr}, 0\\right\\}}\\left| T\\right| $$ for all $m$-linear forms $T:\\ell_{p}^{n}\\times \\cdots \\times \\ell_{p}^{n}\\rightarrow \\mathbb{K}$ and all positive integers $n$. Moreover, the exponent $max \\left\\{\\frac{p-rp+rm}{pr}, 0\\right\\}$ is optimal. The case $k=m$ recovers a recent result due to G. Araujo and D. Pellegrino.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-06T13:27:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "classical sequence spaces",
      "multilinear forms",
      "positive integers",
      "summability",
      "complex scalar field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}