Kenneth Kuttler

1418 CHAPTER 43. INTERPOLATION IN BANACH SPACE

Definition 43.1.11 LetH1 (0,T,X)

denote the functions f ∈ L2 (0,T,X) whose weak derivative f ′ is also in L2 (0,T,X).

Proposition 43.1.12 Let f ∈ H1 (0,T,X). Then f ∈ C0,(1/2) ([0,T ] ,X) and the inclusionmap is continuous.

Proof: First note that

f (t)− f (s) =∫ t

sf ′ (r)dr

and so

∥ f (t)− f (s)∥X ≤∫ t

∥∥ f ′ (r)∥∥

X dr ≤ ∥ f∥H1 |t− s|1/2

It follows that

sup0≤s<t≤T

∥ f (t)− f (s)∥|t− s|1/2 ≤ ∥ f∥H1

Also

f (t) = f (0)+∫ t

0f ′ (s)ds

∥ f (t)∥ ≤ ∥ f (0)∥+∫ t

∣∣ f ′ (s)∣∣ds≤ ∥ f (0)∥+T 1/2 ∥ f∥H1

Now consider ∥ f (0)∥ . Then integrating by parts yields∫ T

0(T − t) f ′ (t)dt = (T − t) f (t) |T0 +

∫ t

0f (t)dt

and so

T ∥ f (0)∥ ≤∫ T

0∥ f (t)∥dt +T

∫ T

∥∥ f ′ (t)∥∥dt ≤C (T )∥ f∥H1 .

Hencesup

t∈[0,T ]∥ f (t)∥ ≤C (T )∥ f∥H1

Therefore, this has shown that

∥ f∥C0,(1/2)([0,T ],X) ≡ supt∈[0,T ]

∥ f (t)∥+ sup0≤s<t≤T

∥ f (t)− f (s)∥|t− s|1/2 ≤C (T )∥ f∥H1

You could imagine that other interesting versions of this are available with similar prooffor the case where the function and its weak derivative are in Lp (0,T,X) for p > 1.

With this integration by parts formula, the following interesting lemma is obtained.This lemma shows why it was appropriate to define f as in Definition 43.1.2.

1418 CHAPTER 43. INTERPOLATION IN BANACH SPACEDefinition 43.1.11 LetH' (0,T,X)denote the functions f € L? (0,T,X) whose weak derivative f' is also in L? (0,T,X).Proposition 43.1.12 Let f ¢ H'(0,T,X). Then f € C®'/?) ((0,T],X) and the inclusionmap is continuous.Proof: First note thatro-s) = [ sarand soFO-FOMx < [I Ollydr< lla esl"?It follows thatIf -F0IO0<s<t<T |r —s|!/ < WaleAlsof()=F(0)+ f f'(s)ds0NX)IFS vot If’ (s) [ds < [FON + 7? WlNow consider || f (0) ||. Then integrating by parts yieldsT t[ @-or a= r-rel + [ roa0 0and so , ,Tif Ols [ Wrola+r [Ie laser) ifn:Hencesup ||f (|| <C(7) \Fllinte [0,7]Therefore, this has shown thatt)—filsWillow or = sup IFO+, sup OO <c(")\ifln 0tE|O, <S<TS —_You could imagine that other interesting versions of this are available with similar prooffor the case where the function and its weak derivative are in L? (0,T,X) for p > 1.With this integration by parts formula, the following interesting lemma is obtained.This lemma shows why it was appropriate to define f as in Definition 43.1.2.