Disuguaglianza di Hölder

In matematica la disuguaglianza di Hölder è un risultato basilare di analisi funzionale. Essa si è impiegata spesso nello studio degli spazi di funzioni noti come spazi Lp.

La disuguaglianza fu provata in una forma leggermente diversa da Leonard James Rogers nel 1888, e riscoperta indipendentemente da Otto Hölder nel 1889, dal quale prende il nome.[1]

La disuguaglianza

Sia uno spazio di misura con misura e . Sia l'esponente coniugato di , ovvero quel numero tale che

o equivalentemente tale che

Si definisce inoltre se .

La disuguaglianza afferma che, date due funzioni misurabili e , si ha che e:[2]

Esplicitando la norma p-esima nel caso si ottiene la scrittura

La disuguaglianza coincide con la disuguaglianza di Cauchy-Schwarz per . Il numero è anche detto coniugato di Hölder di .

Si dimostra che la disuguaglianza diviene un'uguaglianza se e solo se esistono due costanti e , non entrambe nulle, tali che:

quasi ovunque in .

Dimostrazione

Se uno dei due fattori del secondo membro (ad esempio ) è zero, allora vuol dire che quasi ovunque; dunque anche quasi ovunque e quindi e il risultato vale con il segno di uguaglianza. Se uno dei due indici (ad esempio ) è , allora è e:

quindi il risultato viene per monotonia dell'integrale di Lebesgue.

Altrimenti, per la disuguaglianza di Young vale che:

per quasi ogni . Integrando entrambi i membri si ottiene:

Disuguaglianza di Hölder per numeri reali

Nel caso molto particolare dello spazio euclideo , la disuguaglianza prende la seguente forma:

Dimostrazione alternativa

Posti:

e:

la disuguaglianza è:

Dalla concavità della funzione logaritmo si ha:

quindi per monotonia:

Sommando sull'indice poiché e , si ottiene la tesi.

Generalizzazione

Si può generalizzare il risultato con una tecnica dimostrativa simile, prendendo un numero finito qualsiasi di fattori, con indici opportuni: siano tali che , con:

Allora:

e si ha:

Generalizzazione nei numeri reali

Siano m n-uple di numeri reali e siano dei reali tali che:

Allora:

Una conseguenza importante di questa generalizzazione porta ad un primo risultato di immersione tra spazi , la disuguaglianza di interpolazione. Se:

allora per ogni e:

con tale che:

Note

  1. ^ (EN) Leonard James Rogers in The MacTutor History of Mathematics, su www-history.mcs.st-andrews.ac.uk. URL consultato il 19 giugno 2013.
  2. ^ W. Rudin, Pag. 62.

Bibliografia

  • Walter Rudin, Real and Complex Analysis, Mladinska Knjiga, McGraw-Hill, 1970, ISBN 0-07-054234-1.
  • Brezis, Analisi funzionale. Teoria e applicazioni, Liguori Editore, 2006, ISBN 978-88-207-1501-4.

Voci correlate

Collegamenti esterni

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