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  "body": "\n## Statement\n\nLet $n \\ge 1$.\nFor $f \\in L^2(\\R^n)$ and $t \\in \\R$, let $e^{it\\Delta} f$ be the solution at time $t$ of the free [[Schrödinger equation|EPZR6H2WJN]] $i u_t + \\Delta u = 0$ with $u(0) = f$.\nIt is given by the [[Fourier transform|DPVM34ADZV]] as $\\widehat{e^{it\\Delta} f}(\\xi) = e^{-it\\abs{\\xi}^2} \\hat f(\\xi)$.\nCall a pair $(q, r)$ admissible if $2 \\le q \\le \\infty$, $2 \\le r \\le \\infty$,\n$$\\frac{2}{q} + \\frac{n}{r} = \\frac{n}{2},$$\nand $(q, r, n) \\ne (2, \\infty, 2)$.\nLet $(q, r)$ be admissible, and let $L^r$ denote the [[Lebesgue space|SASP14MA1Z]].\nThen there is a constant $C = C(n, q, r)$ with\n$$\\Big( \\int_{\\R} \\norm{e^{it\\Delta} f}_{L^r(\\R^n)}^q \\, dt \\Big)^{1/q} \\le C \\norm{f}_{L^2(\\R^n)} \\quad \\text{for all } f \\in L^2(\\R^n).$$\nFor $q = \\infty$ the left side means $\\sup_{t} \\norm{e^{it\\Delta} f}_{L^r(\\R^n)}$.\n\n## Remarks\n\nThe proof combines two facts.\nThe first is the dispersive estimate $\\norm{e^{it\\Delta} f}_{L^\\infty} \\le (4\\pi\\abs{t})^{-n/2} \\norm{f}_{L^1}$, which follows from the explicit kernel of $e^{it\\Delta}$; interpolation with the conservation of the $L^2$ norm gives the $L^{r'} \\to L^r$ bounds.\nThe second is the $TT^*$ argument, which reduces the estimate to a bound for a time convolution handled by the Hardy–Littlewood–Sobolev inequality when $q \u003e 2$.\nThe endpoint $q = 2$, $r = \\frac{2n}{n - 2}$ for $n \\ge 3$ was proved by Keel and Tao in 1998.\nThe relation $\\frac{2}{q} + \\frac{n}{r} = \\frac{n}{2}$ is forced by scaling: if $f_\\lambda(x) = f(\\lambda x)$, then $e^{it\\Delta} f_\\lambda (x) = (e^{i\\lambda^2 t\\Delta} f)(\\lambda x)$, and comparing powers of $\\lambda$ on both sides gives the relation.\nExample: for $n = 1$ the pair $(q, r) = (4, \\infty)$ is admissible, and for $n = 2$ the pair $(4, 4)$ is admissible, the case first proved by Strichartz.\n\n## Explanations\n\n### L1 human\n\nWaves of the Schrödinger equation spread out, so the solution cannot stay concentrated for long.\nThe Strichartz estimate measures this: although the total mass $\\norm{u(t)}_{L^2}$ never changes, the solution is small in other norms for most times.\nAveraged in time, the solution is more integrable in space than the initial data alone would suggest.\nThis extra integrability is what makes nonlinear equations tractable.\n\n### L2 human\n\nStrichartz estimates are the basic tool for the local and global well-posedness of nonlinear [[dispersive equations|AYMJ2NFR4W]], such as the nonlinear Schrödinger equation $i u_t + \\Delta u = \\pm\\abs{u}^{p-1} u$.\nOne writes the equation in Duhamel form and runs a fixed point argument in the spaces $L^q_t L^r_x$, using the inhomogeneous version of the estimate.\nSimilar estimates hold for the wave and Klein–Gordon equations, with different admissibility conditions and a loss of derivatives.\nAbstract versions, due to Keel and Tao, need only an energy bound and a dispersive decay estimate for the evolution.\n",
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