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  "body": "\n## Statement\n\nLet $d \\ge 1$.\nFor $f \\in L^2(\\R^d)$, let $e^{it\\Delta} f$ be the solution at time $t$ of the free [[Schrödinger equation|EPZR6H2WJN]] with data $f$.\nIt is given by the [[Fourier transform|DPVM34ADZV]] as $\\widehat{e^{it\\Delta} f}(\\xi) = e^{-it\\abs{\\xi}^2} \\hat f(\\xi)$.\n\nCall a pair $(q, r)$ admissible if $2 \\le q \\le \\infty$, $2 \\le r \u003c \\infty$, $(q, r, d) \\ne (2, \\infty, 2)$ and\n$$\\frac{2}{q} + \\frac{d}{r} = \\frac{d}{2}.$$\nFor $1 \\le s \\le \\infty$, let $s'$ be the conjugate exponent, with $\\frac{1}{s} + \\frac{1}{s'} = 1$.\nFor an interval $I \\subseteq \\R$, let $\\norm{u}_{L^q_t L^{r,2}_x(I \\times \\R^d)}$ be the $L^q$ norm over $t \\in I$ of the function $t \\mapsto \\norm{u(t)}_{L^{r,2}(\\R^d)}$.\nHere $L^{r,2}$ is the [[Lorentz space|B7X9M15Q92]] with Lebesgue measure, and $L^q$ is a [[Lebesgue space|SASP14MA1Z]].\n\nThen the following two estimates hold, where the implied constants depend only on $d$ and the exponents.\n\n1. Let $(q, r)$ be admissible. Then for all $f \\in L^2(\\R^d)$,\n$$\\norm{e^{it\\Delta} f}_{L^q_t L^{r,2}_x(\\R \\times \\R^d)} \\lesssim \\norm{f}_{L^2(\\R^d)}.$$\n\n2. Let $(q_1, r_1)$ and $(q_2, r_2)$ be admissible. Let $I \\subseteq \\R$ be an interval and let $t_0 \\in I$. Then for all $F \\in L^{q_2'}_t L^{r_2',2}_x(I \\times \\R^d)$,\n$$\\Big\\| \\int_{t_0}^t e^{i(t-\\tau)\\Delta} F(\\tau) \\, d\\tau \\Big\\|_{L^{q_1}_t L^{r_1,2}_x(I \\times \\R^d)} \\lesssim \\norm{F}_{L^{q_2'}_t L^{r_2',2}_x(I \\times \\R^d)}.$$\n\n## Remarks\n\nThe classical [[Strichartz estimate|4PDKJ8N1S9]] is the same bound with $L^r$ in place of $L^{r,2}$.\nSince $L^{r,2}$ is contained in $L^{r,r} = L^r$ for $r \\ge 2$, the first estimate is stronger than the classical one.\nThe paper allows $r = \\infty$ in its admissible pairs. This node requires $r \u003c \\infty$, since the linked Lorentz space needs a finite first index.\n",
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    "history": {
      "notes": "The source is arXiv:2609.28454v1 (2026-09), by Zuyu Ma, Changxing Miao, Matthew Rosenzweig, Jiqiang Zheng. A paper also restates older results, so this is not the first statement until a person or a later pass confirms it."
    },
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    "name": "Strichartz estimates in Lorentz space",
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    "sources": [
      {
        "arxiv": "2609.28454v1",
        "licence": "cc-by",
        "locator": "Lemma 2.15",
        "mention": "ax://paper/2609.28454#BKD0",
        "note": "the paper that states the item (corpus arxiv AX9.2)"
      }
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  "markdown": "https://tamnd.github.io/math2/node/5GX76EM5HS.md",
  "msc": [
    "35-XX"
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  "name": "Strichartz estimates in Lorentz space",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/5GX/5GX76EM5HS.md",
  "status": null,
  "tag": "5GX76EM5HS",
  "uri": "m2://5GX76EM5HS",
  "url": "https://tamnd.github.io/math2/wiki/strichartz-estimates-in-lorentz-space/",
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    "tag": "5GX76EM5HS",
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