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  "body": "\n## Statement\n\nLet $(X, \\mu)$ and $(Y, \\nu)$ be [[σ-finite|WEBBT18D4X]] measure spaces, and consider [[Lp spaces|SASP14MA1Z]] of complex-valued functions.\nLet $1 \\le p_0, p_1, q_0, q_1 \\le \\infty$.\nLet $T$ be a linear map from $L^{p_0}(\\mu) + L^{p_1}(\\mu)$ to $L^{q_0}(\\nu) + L^{q_1}(\\nu)$.\nAssume that there are constants $M_0, M_1 \\ge 0$ with\n$$\\norm{T f}_{q_0} \\le M_0 \\norm{f}_{p_0} \\text{ for } f \\in L^{p_0}(\\mu), \\qquad \\norm{T f}_{q_1} \\le M_1 \\norm{f}_{p_1} \\text{ for } f \\in L^{p_1}(\\mu).$$\nLet $0 \u003c \\theta \u003c 1$, and define $p$ and $q$ by\n$$\\frac{1}{p} = \\frac{1 - \\theta}{p_0} + \\frac{\\theta}{p_1}, \\qquad \\frac{1}{q} = \\frac{1 - \\theta}{q_0} + \\frac{\\theta}{q_1}.$$\nThen $T$ maps $L^p(\\mu)$ into $L^q(\\nu)$, and\n$$\\norm{T f}_q \\le M_0^{1 - \\theta} M_1^{\\theta} \\norm{f}_p \\quad \\text{for all } f \\in L^p(\\mu).$$\n\n## Remarks\n\nThe proof is Thorin's complex-variable argument.\nBy [[density of simple functions|B6RYM7QTYZ]] and duality, it suffices to bound $\\int (T f) g \\, d\\nu$ for simple $f$ and $g$.\nOne embeds $f$ and $g$ in analytic families $f_z$, $g_z$ for $z$ in the strip $0 \\le \\operatorname{Re} z \\le 1$, with $f_\\theta = f$ and $g_\\theta = g$.\nThe function $z \\mapsto \\int (T f_z) g_z \\, d\\nu$ is bounded and analytic, and the [[Hadamard three-lines theorem|WHS58R43N0]] interpolates the bounds on the two edges.\n\nExample: the Fourier transform satisfies $\\norm{\\hat f}_\\infty \\le \\norm{f}_1$ and, by Plancherel, $\\norm{\\hat f}_2 = \\norm{f}_2$ on $\\R^n$ with suitable normalisation.\nRiesz–Thorin with $p_0 = 1$, $q_0 = \\infty$, $p_1 = q_1 = 2$ gives the Hausdorff–Young inequality $\\norm{\\hat f}_{p'} \\le \\norm{f}_p$ for $1 \\le p \\le 2$, where $1/p + 1/p' = 1$.\n\nComplex scalars matter for the constant.\nFor spaces of real-valued functions the conclusion still holds, but the constant $M_0^{1-\\theta} M_1^\\theta$ may need an extra factor of at most $2$.\n\n## Explanations\n\n### L1 human\n\nSuppose an operation on functions is under control in two ways of measuring size, say $L^1$ and $L^2$.\nThe theorem says that it is then also under control in every way of measuring size in between, such as $L^{4/3}$.\nThe bound in between is a weighted geometric mean of the two end bounds.\n\n### L2 human\n\nRiesz–Thorin is the basic tool for proving $L^p$ bounds from two easy endpoint bounds.\nTypical uses are the Hausdorff–Young inequality, [[Young's convolution inequality|JXDKJCBXJ8]] and bounds for integral operators.\nIt needs linear operators and strong-type bounds at the endpoints.\nThe [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]] allows sublinear operators and weak-type endpoint bounds, at the cost of a worse constant and, in its basic form, only on the diagonal $p = q$.\nBoth are the starting points of abstract [[interpolation space|VX1R6FN6PS]] theory, the complex and the real method.\n",
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