INTERPOLATION BETWEEN 77" SPACES: THE REAL METHOD (2 ...

TRANSACTIONSOF THE AMERICANMATHEMATICALSOCIETY Volume 191, 1974

INTERPOLATIONBETWEEN77" SPACES: THE REAL METHOD BY C. FEFFERMAN, N. M. RIVIERE ANDY. SAGHER ABSTRACT. The interpolation 77*' spaces,

0< p 0. (2)

distance

(Rn - fl, Q.) - diameter

and y. a point in R" - fl satisfying (2')

f/S. is supported

(£).) ■ d..

Let x. be the center of Qj

\y. - x.\ < 10^..

in the cube Q. expanded

Thus / iy)
c> 0 for x £ Q..

(2") \\da./dxa\\Jiy)dy< Ca -1

V

'

'

,

\\x\J

so that

'a-.\N+1

Í-" f

Jd«

Now from Cases

R(yX/(y)-P.(y))c/S.(y)¿y< Caf4 7

x

7

1—3, we know that

b*(je)< C/*(x) if x eQ*, «, then the last integral

dominated

by the first integral

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dx,

Vlx-: \\x-xj

on the right is

on the right.

Thus

C. FEFFERMAN, N. M. RIVIERE AND Y. SAGHER

80

f

JRn

ib\x)f ° dx < C C i¡*ix))Qdx. l

~

Now it is easy to piece our estimates

Jq*

for b. together

into an estimate

for b . For,

b = l.b., so b+ < 2.bJ, so that (b+)P0< lib*)"0 (recall that pQ< 1), which implies

f

ib\x))P°dx< £

f

p0) or by a modification

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of it (for a < pQ, see [2])

INTERPOLATION BETWEEN Hp SPACES: THE REAL METHOD

81

s: «-8wH*frci. •*"*'v* f. c.i/if< so that if™it-eKit,f)¥dt/t)Uq