1454 CHAPTER 43. INTERPOLATION IN BANACH SPACE

this is short for J (t,a,A0,A1). Thus

J (t,a,A1,A0)≡max(||a||A1

, t ||a||A0

)but unless indicated otherwise, A0 will come first. Now for θ ∈ (0,1) and q ≥ 1, define aspace, (A0,A1)θ ,q,J as follows. The space, (A0,A1)θ ,q,J will consist of those elements, a, ofA0 +A1 which can be written in the form

a =∫

∞

0u(t)

dtt≡ lim

ε→0+

∫ 1

ε

u(t)dtt+ lim

r→∞

∫ r

1u(t)

dtt

(43.7.36)

the limits taking place in A0 +A1 with the norm

K (1,a)≡ infa=a0+a1

(||a0||A0

+ ||a1||A1

),

where u(t) is strongly measurable with values in A0 ∩A1 and bounded on every compactsubset of (0,∞) such that(∫

∞

0

(t−θ J (t,u(t) ,A0,A1)

)q dtt

)1/q

< ∞. (43.7.37)

For such a ∈ A0 +A1, define

||a||θ ,q,J ≡ inf

u

{(∫∞

0

(t−θ J (t,u(t) ,A0,A1)

)q dtt

)1/q}

(43.7.38)

where the infimum is taken over all u satisfying 43.7.36 and 43.7.37.

Note that a norm on A0×A1 would be

||(a0,a1)|| ≡max(||a0||A0

, t ||a1||A1

)and so J (t, ·) is the restriction of this norm to the subspace of A0×A1 defined by

{(a,a) : a ∈ A0∩A1}

Also for each t > 0 J (t, ·) is a norm on A0 ∩A1 and furthermore, any two of these normsare equivalent. In fact, for 0 < t < s,

J (t,a) = max(||a||A0

, t ||a||A1

)≥ max

(||a||A0

,s ||a||A1

)= J (s,a)

≥ max( s

t||a||A0

,s ||a||A1

)=

st

max(||a||A0

, t ||a||A1

)≥ s

tJ (t,a) .

The following lemma is significant and follows immediately from the above definition.