BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 80, Number 2, March 1974
INTERPOLATION OF OPERATORS FOR A SPACES
BY ROBERT SHARPLEY
Communicated by Alberto Calderón, September 22, 1973
Lorentz and Shimogaki [2] have characterized those pairs of Lorentz
A spaces which satisfy the interpolation property with respect to two
other pairs of A spaces. Their proof is long and technical and does not
easily admit to generalization. In this paper we present a short proof of
this result whose spirit may be traced to Lemma 4.3 of [4] or perhaps more
accurately to the theorem of Marcinkiewicz [5, p. 112]. The proof involves
only elementary properties of these spaces and does allow for generalization to interpolation for n pairs and for M spaces, but these topics
will be reported on elsewhere.
The Banach space A^ [1, p. 65] is the space of all Lebesgue measurable
functions ƒ on the interval (0, /) for which the norm
is finite, where </> is an integrable, positive, decreasing function on (0, /)
and/* (the decreasing rearrangement of | / | ) is the almost-everywhere
unique, positive, decreasing function which is equimeasurable with \f\.
A pair of spaces (A^, Av) is called an interpolation pair for the two
pairs (A^, AVl) and (A^2, AV2) if each linear operator which is bounded
from A^ to Av (both /== 1, 2) has a unique extension to a bounded operator from A^ to Av.
THEOREM (LORENTZ-SHIMOGAKI). A necessary and sufficient condition
that (A^, Aw) be an interpolation pair for (A^, AVi) and (A^2, AV2) is
that there exist a constant A independent of s and t so that
(*)
^(0/0(5) ^ A max(TO/^(a))
t=1.2
holds, where O 00=ƒS <j>{r) dr,-"
PROOF.
, VaC'Wo Y a (r) dr.
We only sketch the proof of the necessity since it is standard.
AMS (MOS) subject classifications (1970). Primary 46E30, 46E35, 47A30.
Key words and phrases. Lorentz A<f> space, decreasing rearrangement, interpolation
of operators.
Copyright © American Mathematical Society 1974
259
260
ROBERT SHARPLEY
[March
Suppose there are numbers sn and tn in (0, /) such that lF(^w)/0(5,J>
nz max^i.sCT^rJ/O^.yJ). Define the positive operator
TJ(t) = ( C w £ >(s)ds/s n ^ ( M n ) (0,
where C w =min, = 1 . 2 (0,(0/^(0)For each ƒ in A^., Tn ƒ belongs to Av. and r w has operator norm less
than or equal to 1, but as an operator from A^ to Av, Tn has operator
norm larger than n3. Hence the operator T= 2J° TJn2 is a bounded operator from A^ to A^ (/=1, 2), but T is not a bounded operator from A^
to Av.
To show that condition (*) is sufficient, we prove that
(i)
\\7f^£2AM\\fu
where M is the maximum of the operator norms of T acting from A^
to AVi (/= 1,2). We can assume that ƒ is an arbitrary simple function
with finite support since these functions are dense in A^. We can also
require/to be positive since ||/||^=|| \f\ ||^. Each function of this type
can be written as ƒ = 21* O^XE* where the a/s are positive and En <= • • - <= Ev
Hence/* = 2i ^iXio.a ) where a^mE^ But then
(2)
IITxE\\v ^ 2AMd>(mE)9 all measurable E c (0, /)
is equivalent to relation (1), since
\\Tf ||„ = J a, HT^IU = 2 ^ M 2 a^a,) = 2AM ||/1|, .
1
1
Hence, if we let g~(T%E)*, the proof is reduced to the following
LEMMA. Suppose condition (*) holds and g is a positive decreasing
function that satisfies
(3)
HglU = MO/a)
(i = l,2),
then
(4)
||g||v = 2AM«D(a).
PROOF. Firsu assume g is a step function with finite support, i.e.,
g=X?hXio.t,)- Set J={y|maxt.=1,2(T<(^)/Oi(a))=T1(^)/«D1(a)} and then
l et gi=2*e.r PjXw.tj) an< i g2=S—gv Notice that both functions are
positive, decreasing, step functions and
(5)
Uftll = lis!
1974]
INTERPOLATION OF OPERATORS FOR A SPACES
261
in any A space. Now using condition (*), relations (5) and (3), we have
llgilWa) = 2 /W,)/<D(a)
i s J
«=1.2
= A2 /W,)/*^) = A hAJ^ia)
jej
Similarly
Hence, we obtain relation (4) for positive, decreasing, step functions.
Now suppose g is an arbitrary positive decreasing function and let
{gn} be a monotone increasing sequence of positive decreasing step functions converging pointwise to g. By (3) and (5)
WgXt^MQjLa)
(* = 1,2)
so
||gj| v ^ 2M(D(fl).
Applying the monotone convergence theorem to {gnip}, we obtain
relation (4).
The author wishes to thank Professor S. D. Riemenschneider for many
helpful conversations regarding this work.
REFERENCES
1. G. G. Lorentz, Bernstein polynomials, Mathematical Expositions, no. 8, Univ.
of Toronto Press, Toronto, 1953. MR 15, 217.
2. G. G. Lorentz and T. Shimogaki, Interpolation theorems for operators in function
spaces, J. Functional Analysis 2 (1968), 31-51. MR 41 #2424.
3.
, Interpolation theorems for spaces A, Abstract Spaces and Approximation
(Proc. Conf., Oberwolfach, 1968), Birkhauser, Basel, 1969, pp. 94-98. MR 41 #2423.
4. R. C. Sharpley, Spaces Aa(X) and interpolation, J. Functional Analysis 11 (1972),
479-513.
5. A. Zygmund, Trigonometric series. Vol. II, 2nd ed. reprinted with corrections
and some additions, Cambridge Univ. Press, New York, 1968. MR 38 #4882.
DEPARTMENT OF MATHEMATICS, OAKLAND UNIVERSITY, ROCHESTER,
48063
MICHIGAN