1. Introduction
1.10. Interpolation spaces
Calder´on [11].
LetpE0,} ¨ }0qandpE1,} ¨ }1qbe two (real or complex) Banach spaces that are both
linear subspaces of a Banach space pH,} ¨ }q, the ambient space, and suppose that the inclusion maps frompEj,} ¨ }jqintopH,} ¨ }qare both continuous. Then the pair
tpE0,} ¨ }0q,pE1,} ¨ }1qu
is acompatible couple (of Banach spaces). It is straightforward to show that, in this case, the spaces E0XE1 and E0`E1 are then Banach spaces under the respective norms
defined by: }x}E
0XE1 “maxt}x}0,}x}1u pxPE0XE1q;
}x}E
0`E1 “inft}x0}0` }x1}1:x“x0`x1, x0PE0, x1PE1u pxPE0`E1q.
A Banach spacepG,} ¨ }qthat containsE0XE1and is contained inE0`E1and is such
that the two inclusions
pE0XE1,} ¨ }E0XE1q Ñ pG,} ¨ }q Ñ pE0`E1,} ¨ }E0`E1q
are continuous is then anintermediate space.
For details of the following remarks, see [6, Chapter 4], for example. For the remainder of this section, all our Banach spaces are complex Banach spaces.
Suppose that tpE0,} ¨ }0q,pE1,} ¨ }1qu is a compatible couple of Banach spaces. Let
L0 and L1 be the lines tiy : y P Ru and t1`iy : y P Ru, respectively, in C, and set S “ p0,1q ˆR ĂC, an open strip in C. Take F to be the linear space of all functions F onS taking values inpE0`E1,} ¨ }E0`E1qsuch thatF is bounded and continuous on
S, such thatF is analytic onS, and such thatF |Lj is a bounded and continuous map intopEj,} ¨ }jqforj “0,1.
We define a norm onF by setting }F}F “max
j“0,1tsupt}Fpzq}j:zPLjuu pF PFq.
By the Phragm´en–Lindel¨of theorem, }Fpzq}E
0`E1 ď }F}F pzPS, F PFq.
FurtherpF,} ¨ }Fqis a Banach space.
Next takeθP p0,1q, and identifyθwith the pointpθ,0qofS. Then the mapF ÞÑFpθq is a contractive linear map fromF intopE0`E1,} ¨ }E0`E1q, and the image of this map
is denoted by
pE0, E1qθ“Erθs;
Erθs is a Banach space with respect to the quotient norm defined by }x}rθs “inft}F}F :F PF, Fpθq “xu pxPErθsq,
so that} ¨ }rθs is theinterpolation norm. FurtherpErθs,} ¨ }rθsqis an intermediate space. We now note that, in the definition of the familyF, we may suppose that Fpiyqand Fp1`iyqtend to 0 inE0 andE1, respectively, as|y| Ñ 8. Indeed, we can multiply each
original function in the familyF by the function
zÞÑexppδpz2´θ2qq, SÑC,
for suitable δą 0 to obtain this without changing the spacepErθs,} ¨ }rθsq; for this, see [13, p. 1007]. This extra property ofFwas assumed by Calder´on when he introduced this theory in [11]. We shall suppose throughout that functions inF have this extra property. We note that, if we move to norms on E0 and E1 that are equivalent to} ¨ }0 and
} ¨ }1 onE0andE1, respectively, we do not change the intermediate spaceErθs (and the interpolation norm is equivalent to the original interpolation norm).
We also note that, in the above situation, the spaceE0XE1is dense inpErθs,} ¨ }rθsq; this is [6, Theorem 4.2.2(a)].
A Banach-space-valued form of the famous Riesz–Thorin interpolation theorem is the following; full details are given in [6, Theorem 5.1.2].
Theorem 1.45. Let Ω be a measure space, and let tE0, E1u be a compatible couple of
complex Banach spaces. TakeθP p0,1qandp0, p1 with1ďp0, p1ă 8, and definepby
1 p “ 1´θ p0 ` θ p1 .
Set E “ pE0, E1qθ. Then tLp0pΩ;E0q, Lp1pΩ;E1qu is a compatible couple of Banach
spaces, and
pLp0pΩ;E
0q, Lp1pΩ;E1qqθ“LppΩ;Eq
with}f}rθs“ }f}LppΩ;Eq pf PLppΩ;Eqq.
In particular, with the above notation,t`p0pE
0q, `p1pE1quis also a compatible couple
of Banach spaces, and
p`p0pE
0q, `p1pE1qqθ“`ppEq, (1.10.1)
whereE“ pE0, E1qθ.
Taken PN. By [6, Theorem 5.1.2], it is also true that t`p0
n pE0q, `n8pE1qu is a com-
patible couple of Banach spaces and that p`p0
n pE0q, `n8pE1qqθ“`nppEq, (1.10.2) where 1{p“ p1´θq{p0 andE“ pE0, E1qθ.
The fundamental theorem in this context is the following [6, Theorem 4.1.4].
Theorem 1.46. Let tpE0,} ¨ }0q,pE1,} ¨ }1qu andtpF0,} ¨ }0q,pF1,} ¨ }1qu be two compati-
ble couples of complex Banach spaces, and suppose thatT :E0`E1ÑF0`F1is a linear
map such thatTpEjq ĂFj andT |Ej:EjÑFj is bounded, with normMj, forj“0,1. TakeθP p0,1q. ThenTpErθsq ĂFrθs and
› ›T |Erθs
›
›ďM01´θM1θ.
Proposition 1.47. Let tE0, E1ube a compatible couple of complex Banach spaces, and
take θ P p0,1q. Suppose that 1 ď pă 8 and that E0 and E1 are both p–spaces. Then
pE0, E1qθ is also ap–space.
Proof. SetE“ pE0, E1qθ. By (1.10.1),p`nppE0q, `nppE1qqθ“`nppEq pnPNq. Takem, nPNandT PBp`p
m, `npq, and considerT as a map defined on the spacesEm0
and onEm
1 , say
Mj“ }T :`mppEjq Ñ`nppEjq} pj“0,1q.
Since E0 and E1 are both p–spaces, in fact Mj ď }T} pj “ 0,1q. By Theorem 1.46,
Tp`mppEqq Ă`nppEqand
}T :`mppEq Ñ`nppEq} ďM1´θ 0 M θ 1 ď }T} 1´θ }T}θ“ }T}, and soE is a p–space by Theorem 1.36, (b)ñ(a).
We shall see in Example 2.16, to be given below, that an apparent generalization of the above result to the case whereE0 and E1 arep0– and p1–spaces, respectively, and