Banach Spaces of Analytic Functions Proceedings of the Pelczynski Co
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Vol. 431: 5eminaire Bourbaki - vol. 1973/74. Exposes 436-452. IV, 347 pages. 1975.
Vol. 401: M. F. Atiyah, Elliptic Operators and Compact Groups. V, 93 pages. 1974.
Vol. 432: R. P. Pflug, Holomorphiegebiete, pseudokonvexe Ge· biete und das Levi-Problem. VI, 210 Seiten. 1975.
Vol. 402: M. Waldschmidt, Nombres Transcendants. VIII, 277 pages. 1974.
Vol. 433: W. G. Faris, Self-Adjoint Operators. VII, 115 pages. 1975.
Vol. 403: Combinatorial Mathematics. Proceedings 1972. Edited by D. A. Holton. VIII, 148 pages. 1974.
Vol. 434: P. Brenner, V. Thomee, and L. B. Wahl bin, Besov Spaces and Applications to Difference Methods for Initial Value Problems. II, 154 pages. 1975.
Vol. 404: Thimrie du Potentiel et Analyse Harmonique. Edite par J. Faraut. V, 245 pages. 1974. Vol. 405: K. J. Devlin and H. Johnsbraten, The Souslin Problem. VIII, 132 pages. 1974. Vol. 406: Graphs and Combinatorics. Proceedings 1973. Edited by R. A. Sari and F. Harary. VIII, 355 pages. 1974. Vol. 407: P. Berthelot, Cohomologie Cristalline des Schemas de Caracteristique p > o.ll, 604 pages. 1974. Vol. 408: J. Wermer, Potential Theory. VIII, 146 pages. 1974. Vol. 409: Fonctions de Plusieurs Variables Complexes, Seminaire Fran p/2 < ~ whenever C:
(where p* = p/(p - i) denotes the conj~ate of p).
~P* into Z I It is n a t ~ a l to suspect that
(7) is obtainable from (5) and (6) simply by "inte~ol~ing". site is true:
theorem:
But just the oppo-
indeed~ the proof of (7) leads to new versions of the Riesz-Thorin
The ~gument given in [i] is indirect ~ d
mate:
if 1 < p < 2~
(8)
begins with the related esti-
~ (~ Ic 12) p/2 < ~ whenever C: k j jk
zP* ~ Z I
itself an easy consequence of Littlewood's inequality.
was to interchange the order of summation in (8).
The heart of the matter, then,
This was achieved by means of the
following inequality, valid for arbitr~y matrices C:
(9)
ICjkI2)p/2
D~
jk
ql" )
(These, in turn,
Given a matrix A we must show that
l-e llall.,q 0 and
(18)
ll~'ll~,lllMl_po- Tldjl
D
JEJ i
D
to satisfy the following three conditions.
JEJ i
1/(l-p)
for j 6 Ji
fdjrp/(1-P) < ~-P
I ~ b j kxk(i) I ->~6"
Condition (37) is the most important one:
the other two are needed only
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