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  "body": "\n## Statement\n\nLet $(X, \\mu)$ and $(Y, \\nu)$ be $\\sigma$-finite measure spaces, and let $S = \\set{z \\in \\C : 0 \\le \\operatorname{Re} z \\le 1}$.\nCall a simple function admissible if its support has finite measure.\nFor each $z \\in S$ let $T_z$ be a linear operator from the admissible simple functions on $X$ to the measurable functions on $Y$.\nAssume that $(T_z)$ is an admissible analytic family, which means the following.\nLet $f$ on $X$ and $g$ on $Y$ be admissible simple functions.\nIt is required that $T_z(f) g$ is integrable for each $z$, and that $\\Phi(z) = \\int_Y T_z(f) \\, g \\, d\\nu$ is continuous on $S$ and [[holomorphic|0NAHXR1FRH]] in the interior of $S$.\nMoreover $\\log \\abs{\\Phi(z)} \\le C e^{a \\abs{\\operatorname{Im} z}}$ on $S$ for some constants $C$ and $a \u003c \\pi$.\n\nLet $1 \\le p_0, p_1, q_0, q_1 \\le \\infty$.\nLet $M_0, M_1 \\colon \\R \\to (0, \\infty)$ satisfy $\\log M_j(y) \\le C' e^{b \\abs{y}}$ for some constants $C'$ and $b \u003c \\pi$.\nAssume that for all admissible simple $f$ on $X$ and all $y \\in \\R$,\n$$\\norm{T_{iy} f}_{L^{q_0}} \\le M_0(y) \\norm{f}_{L^{p_0}}, \\qquad \\norm{T_{1 + iy} f}_{L^{q_1}} \\le M_1(y) \\norm{f}_{L^{p_1}}.$$\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 $\\norm{T_\\theta f}_{L^q} \\le M_\\theta \\norm{f}_{L^p}$ for all admissible simple $f$ on $X$, where\n$$\\log M_\\theta = \\frac{\\sin \\pi\\theta}{2} \\int_{\\R} \\left( \\frac{\\log M_0(y)}{\\cosh \\pi y - \\cos \\pi\\theta} + \\frac{\\log M_1(y)}{\\cosh \\pi y + \\cos \\pi\\theta} \\right) dy.$$\n\n## Remarks\n\nThe proof follows the proof of the [[Riesz–Thorin theorem|AGRKRG1TMJ]].\nFor fixed simple $f$ and $g$ one builds families $f_z$, $g_z$ of simple functions whose absolute values are powers of $\\abs{f}$ and $\\abs{g}$ depending on $z$.\nThen one studies the holomorphic function $z \\mapsto \\int_Y T_z(f_z) g_z \\, d\\nu$ on the strip.\nThe hypotheses bound it on the two boundary lines.\nA form of the [[Hadamard three-lines theorem|WHS58R43N0]] for functions of admissible growth, proved with the Poisson kernel of the strip, then bounds it at $z = \\theta$.\nThe two kernels in the formula for $M_\\theta$ are the Poisson kernels of the strip at the point $\\theta$.\n\nIf $T_z = T$ does not depend on $z$ and $M_0$, $M_1$ are constants, the formula gives $M_\\theta = M_0^{1 - \\theta} M_1^{\\theta}$, which is the Riesz–Thorin theorem.\nA typical use is the family of fractional integration or Bochner–Riesz type operators $T_z$, where the exponent of a kernel depends analytically on $z$.\nThe bound $a \u003c \\pi$ in the growth condition cannot be relaxed to $a = \\pi$.\nThe function $h(z) = \\exp(i e^{-i\\pi z})$ has modulus $1$ on both boundary lines of $S$, but $\\abs{h(\\tfrac12 + iy)} = \\exp(e^{\\pi y})$ is unbounded.\n\n## Explanations\n\n### L1 human\n\nSuppose a family of operators depends on a complex parameter in a smooth, holomorphic way.\nIf the operators are bounded between two pairs of $L^p$ spaces on the two edges of a strip, then the operator in the middle is bounded between the intermediate spaces.\nThis is a version of Riesz–Thorin interpolation in which the operator itself is allowed to move.\n\n### L2 human\n\nThe Riesz–Thorin theorem interpolates one fixed operator.\nMany problems in harmonic analysis come instead with a natural family, for example operators whose kernel contains a power $\\abs{x}^{z}$ or a factor $(1 - \\abs{\\xi}^2)^{z}_+$.\nAt one end of the strip the family is easy to bound in $L^2$ by Plancherel's theorem.\nAt the other end it is bounded in $L^\\infty$ by a pointwise estimate of the kernel.\nStein's theorem gives the bounds in between, as in results on Bochner–Riesz means and on Fourier restriction.\nIt belongs to the complex method of interpolation, while the [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]] belongs to the real method.\n",
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