Rees-Thorin Theorem

The Ries – Torin theorem is a statement about the properties of interpolation spaces . It was formulated in 1926 by Marcel Rees ^[1] , and was formulated and proved in operator form by Olof Torin in 1939 ^[2] ^[3] .

According to the theorem, for two spaces $(\Omega _{1},\Sigma _{1},\mu _{1})$ ${\ displaystyle (\ Omega _ {1}, \ Sigma _ {1}, \ mu _ {1})}$ $(\ Omega _ {{1}}, \ Sigma _ {{1}}, \ mu _ {{1}})$ and $(\Omega _{2},\Sigma _{2},\mu _{2})$ ${\ displaystyle (\ Omega _ {2}, \ Sigma _ {2}, \ mu _ {2})}$ $(\ Omega _ {{2}}, \ Sigma _ {{2}}, \ mu _ {{2}})$ with measures $\mu _{1}$ ${\ displaystyle \ mu _ {1}}$ $\ mu _ {{1}}$ and $\mu _{2}$ ${\ displaystyle \ mu _ {2}}$ $\ mu _ {{2}}$ respectively, and two Banach spaces of complex-valued functions $L_{p}(\Omega _{i})$ ${\ displaystyle L_ {p} (\ Omega _ {i})}$ $L _ {{p}} (\ Omega _ {{i}})$ summable with $p$ ${\ displaystyle p}$ $p$ i degree $(p\geqslant 1)$ ${\ displaystyle (p \ geqslant 1)}$ $(p \ geqslant 1)$ by measure $\mu _{i}$ ${\ displaystyle \ mu _ {i}}$ $\ mu _ {{i}}$ $(i=1,2)$ ${\ displaystyle (i = 1,2)}$ $(i = 1,2)$ , triple Banach spaces $(L_{p_{0}}(\Omega _{1}),L_{p_{1}}(\Omega _{1}),L_{p}(\Omega _{1}))$ ${\ displaystyle (L_ {p_ {0}} (\ Omega _ {1}), L_ {p_ {1}} (\ Omega _ {1}), L_ {p} (\ Omega _ {1}))}$ $(L _ {{p _ {{0}}}} (\ Omega _ {{1}}), L _ {{p _ {{1}}}} (\ Omega _ {{1}}), L _ {{p} } (\ Omega _ {{1}}))$ is a normal interpolation type $\alpha$ ${\ displaystyle \ alpha}$ $\ alpha$ regarding troika $(L_{q_{0}}(\Omega _{2}),L_{q_{1}}(\Omega _{2}),L_{q}(\Omega _{1}))$ ${\ displaystyle (L_ {q_ {0}} (\ Omega _ {2}), L_ {q_ {1}} (\ Omega _ {2}), L_ {q} (\ Omega _ {1}))}$ ${\ displaystyle (L_ {q_ {0}} (\ Omega _ {2}), L_ {q_ {1}} (\ Omega _ {2}), L_ {q} (\ Omega _ {1}))}$ , if a:

{\frac {1}{p}}={\frac {1-\alpha }{p_{0}}}+{\frac {\alpha }{p_{1}}}

{\ displaystyle {\ frac {1} {p}} = {\ frac {1- \ alpha} {p_ {0}}} + {\ frac {\ alpha} {p_ {1}}}}

{\ displaystyle {\ frac {1} {p}} = {\ frac {1- \ alpha} {p_ {0}}} + {\ frac {\ alpha} {p_ {1}}}}

and

{\frac {1}{q}}={\frac {1-\alpha }{q_{0}}}+{\frac {\alpha }{q_{1}}}

{\ displaystyle {\ frac {1} {q}} = {\ frac {1- \ alpha} {q_ {0}}} + {\ frac {\ alpha} {q_ {1}}}

{\ displaystyle {\ frac {1} {q}} = {\ frac {1- \ alpha} {q_ {0}}} + {\ frac {\ alpha} {q_ {1}}}

,

Where $0\leqslant \alpha \leqslant 1$ ${\ displaystyle 0 \ leqslant \ alpha \ leqslant 1}$ $0 \ leqslant \ alpha \ leqslant 1$ ^[4] . (Three Banach spaces $(A,B,E)$ ${\ displaystyle (A, B, E)}$ $(A, B, E)$ is an interpolation type $\alpha$ ${\ displaystyle \ alpha}$ $\ alpha$ where $0\leqslant \alpha \leqslant 1$ ${\ displaystyle 0 \ leqslant \ alpha \ leqslant 1}$ $0 \ leqslant \ alpha \ leqslant 1$ relative to troika $(C,D,F)$ ${\ displaystyle (C, D, F)}$ $(C, D, F)$ if it is interpolational and the inequality holds $\|T\|_{E\rightarrow F}\leqslant c\|T\|_{A\rightarrow C}^{1-\alpha }\|T\|_{B\rightarrow D}^{\alpha }$ ${\ displaystyle \ | T \ | _ {E \ rightarrow F} \ leqslant c \ | T \ | _ {A \ rightarrow C} ^ {1- \ alpha} \ | T \ | _ {B \ rightarrow D} ^ {\ alpha}}$ $\ | T \ | _ {{E \ rightarrow F}} \ leqslant c \ | T \ | _ {{A \ rightarrow C}} ^ {{1- \ alpha}} \ | T \ | _ {{B \ rightarrow D}} ^ {{\ alpha}}$ ^[5] .)

The proof of the theorem uses the theorem on three lines from the theory of analytic functions ^[6] .

Notes

↑ Riesz M., Sur les maxima des formes bilineares et sur les fonctionalles linearies, Acta Math., 49 (1926), 465-497
↑ Thorin GO, Anne of convexity Theorem Due to M. Riesz, Comm. Sem. Math Univ. Lund, 4 (1939), 1-5
↑ Thorin GO, Convexity theorems generalizing those of M. Riesz and Hadamard with some applications, Comm. Sem. Math Univ. Lund, 9 (1948), 1-58
↑ Crane, 1978 , p. 37.
↑ Crane, 1978 , p. 36
↑ Sigmund A. Trigonometric series, M., Mir, 1965, Vol. II, p. 144-148

Literature

Kerin S. G. , Petunin Yu. I. , Semenov E. M. Interpolation of linear operators. - M .: Science, 1978. - 400 p.
Berg J., Löfström J. Interpolation spaces. Introduction - M .: Mir, 1980. - 264 p.

[1] Riesz M., Sur les maxima des formes bilineares et sur les fonctionalles linearies, Acta Math., 49 (1926), 465-497

[2] Thorin GO, Anne of convexity Theorem Due to M. Riesz, Comm. Sem. Math Univ. Lund, 4 (1939), 1-5

[3] Thorin GO, Convexity theorems generalizing those of M. Riesz and Hadamard with some applications, Comm. Sem. Math Univ. Lund, 9 (1948), 1-58

[_6fb3f5e2f0eabfa0-4] Crane, 1978 , p. 37.

[_6fb3f5e2f0eabfa1-5] Crane, 1978 , p. 36

[6] Sigmund A. Trigonometric series, M., Mir, 1965, Vol. II, p. 144-148

Rees-Thorin Theorem

Notes

Literature

More articles: