Angiuli, Luciana

English

Università del Salento: ESE - Salento University Publishing

Appendix AA brief introduction tointerpolation theoryA.1 Interpolation spacesThis appendix is devoted to present an elementary treatment of the interpolationtheory. This theory has a wide range of applications to partial differential operators andpartial differential equations. We have used interpolation techniques in Chapter 3. Inparticular, Theorem 3.1.2 relies on Theorem A.2.7 and both have been proved in [6].The most known and useful families of interpolation spaces are the real and the complexinterpolation spaces.Let X, Y be two real or complex Banach spaces. By X = Y we mean that X and Yhave the same elements with equivalence of the norms. By Y ↪→ X we mean that Y iscontinuously embedded in X.Suppose that Y ↪→ X; we say that D is an intermediate space between X and Y ifY ↪→ D ↪→ X.An interpolation space between X and Y is any intermediate space such that for everyT ∈ L(X), whose restriction to Y belongs to L(Y ), the restriction to D belongs to L(D).Another important class of intermediate spaces are the space of class Jα.Definition A.1.1. An intermediate space D between X and Y is said to be of class Jαif there exists a constant C > 0 such that‖y‖D ≤ C‖y‖αY ‖y‖1−αX , y ∈ Y.In this case we write D ∈ Jα(X,Y ).113114A.1.1 Some interpolation estimatesIn the next section some important examples of interpolatory inclusion are shown.First we prove a useful interpolation estimate which allows us to estimate the Lp norm ofthe gradient of a function with respect to the Lp norm of the function and of its secondderivatives. For a more general statement see [1, Theorem 4.17].Proposition A.1.2. Let 1 ≤ p < ∞, then W 1,p(Rn) is of class J1/2 between Lp(Rn)and W 2,p(Rn). In other words‖Du‖Lp(Rn) ≤ c‖D2u‖1/2Lp(Rn)‖u‖1/2Lp(Rn) (A.1)for u ∈W 2,p(Rn) and some constant c > 0.Proof. We first consider the one-dimensional case. Let u ∈ C∞c (R) and x ∈ R; thenfor h > 0u(x+ h) = u(x) + hu′(x) +∫ h0(h− s)u′′(s+ x)dshenceu′(x) =u(x+ h)− u(x)h− 1h∫ h0(h− s)u′′(s+ x)ds.Taking the Lp norm we get‖u′‖Lp(R) ≤ 2h‖u‖Lp(R) + 1h∫ h0(h− s)‖u′′(s+ ·)‖Lp(R)ds=2h‖u‖Lp(R) + h2 ‖u′′‖Lp(R)Let ε = 2h then‖u′‖Lp(R) ≤ ε‖u‖Lp(R) + 1ε‖u′′‖Lp(R). (A.2)Now, let u ∈ C∞c (Rn), then by (A.2) we get∫R|Diu|pdxi ≤ 2p−1(εp∫R|Diiu|pdxi + 1εp∫R|u|pdxi),and by Fubini’s Theorem∫Rn|Diu|pdx ≤ 2p−1(εp∫Rn|Diiu|pdx+ 1εp∫Rn|u|pdx)therefore‖Diu‖Lp(Rn) ≤ c(ε‖D2u‖Lp(Rn) + 1ε‖u‖Lp(Rn))(A.3)holds for every ε > 0 and some constant c depending only on p. Minimizing (A.3) on ε,we get‖Du‖Lp(Rn) ≤ 2c‖D2u‖1/2Lp(Rn)‖u‖1/2Lp(Rn)for every u ∈ C∞c (Rn). Finally the estimate can be extended by density toW 2,p(Rn).115A.2 Real interpolation spacesLet X,Y be Banach spaces, with Y ↪→ X (in general this is not required; however,this simplifying assumption is satisfied in the case we are investigating). We describebriefly the K-method used to construct a family of intermediate spaces between X andY , called real interpolation spaces and denoted by (X,Y )θ,p, where 0 < θ ≤ 1 and1 ≤ p ≤ ∞. Let I be any interval contained in (0,+∞), 1 ≤ p <∞. We denote by Lp∗(I)the Lebesgue space Lp with respect to the measure dtt in I. If p =∞, L∞∗ (I) = L∞(I).We set 1/∞ = 0.Definition A.2.1. For every x ∈ X and t > 0, setK(t, x,X, Y ) = infx=a+b,a∈X,b∈Y(‖a‖X + t‖b‖Y ).Now we define a family of intermediate spaces by means of the function K.Definition A.2.2. Let 0 < θ ≤ 1, 1 ≤ p ≤ ∞, set(X,Y )θ,p = {x ∈ X : t 7→ t−θK(t, x,X, Y ) ∈ Lp∗(0,+∞)}with‖x‖θ,p = ‖t−θK(t, x,X, Y )‖Lp∗((0,+∞))and(X,Y )θ = {x ∈ X : limt→0+t−θK(t, x,X, Y ) = 0}Definition (A.2.2) concerns only the behavior of t−θK(t, x,X, Y ) as t → 0, sinceK(·, x,X, Y ) is bounded. Moreover since K(t, x,X, Y ) ≥ min{1, t}K(1, x,X, Y ), forθ = 1 we deduce that(X,Y )1,p = {0}, p <∞.Therefore, henceforth we consider the cases (θ, p) ∈ (0, 1)× [1,+∞] and (θ, p) = (1,∞).Such spaces are called real interpolation spaces. One can prove that ‖x‖(X,Y )θ,p is a normin (X,Y )θ,p and that the following results hold (see [31] for their proof).Proposition A.2.3. For all (θ, p) ∈ (0, 1)× [1,+∞] and (θ, p) = (1,∞), (X,Y )θ,p is aBanach space. For all θ ∈ (0, 1), (X,Y )θ is a Banach space, endowed with the norm of(X,Y )θ,∞.The spaces (X,Y )θ,p and (X,Y )θ are of class Jθ(X,Y ) for every p ∈ [1,∞]. They areactually interpolation spaces, as they enjoy the following property.Theorem A.2.4. Let Xi, Yi be Banach spaces such that Yi ↪→ Xi for i = 1, 2. LetT ∈ L(X1, X2) ∩ L(Y1, Y2). Then for every θ ∈ (0, 1) and p ∈ [1,∞], we haveT ∈ L((X1, Y1)θ,p, (X2, Y2)θ,p) ∩ L((X1, Y1)θ, (X2, Y2)θ)and‖T‖L((X1,Y1)θ,p,(X2,Y2)θ,p) ≤ (‖T‖L(X1,X2)θ,p)1−θ(‖T‖L(Y1,Y2)θ,p)θ116Finally we state without proof the duality theorem for the real method. A proof ofit can be found in [46, Section 1.11.2].Theorem A.2.5. (Dual space) Let Y dense in X. If 0 < θ < 1 then for 1 ≤ p <∞(X,Y )′θ,p = (Y′, X ′)1−θ,p′ ,1p+1p′= 1,and for p =∞(X,Y )′θ,∞ = (Y′, X ′)1−θ,1. (A.4)A.2.1 ExamplesWe close this section with concrete examples of some interpolation spaces. For θ ∈(0, 1), p ∈ [1,∞), W θ,p(Rn) is the space of all f ∈ Lp(Rn) such that[f ]W θ,p =(∫Rn×Rn|f(x)− f(y)|p|x− y|θp+n dxdy)1/p<∞.It is endowed with the norm ‖ · ‖Lp + [·]W θ,p . When θ > 1 is not integer, let [θ] and{θ} be the integral and fractional parts of θ. Then W θ,p(Rn) consists of the functionsf ∈W [θ],p(Rn) such that ∑|α|=[θ][Dαf ]W{θ},pis finite. Analogously in this case we consider the space W θ,p normed by‖ · ‖W [θ],p +∑|α|=[θ][Dα·]W{θ},p .Example 2. For 0 < θ < 1, 1 ≤ p <∞ we have(Cb(Rn), C1b (Rn))θ,∞ = Cθb (Rn)(Lp(Rn),W 1,p(Rn))θ,p =W θ,p(Rn),with equivalence of the respective norms.Example 3. Let 0 ≤ θ1 < θ2 ≤ 1, 0 < θ < 1, 1 ≤ p <∞. Then(W θ1,p(Rn),W θ2,p(Rn))θ,p =W (1−θ)θ1+θθ2,p(Rn).If Ω is an open set in Rn with uniformly C1 boundary, then(W θ1,p(Ω),W θ2,p(Ω))θ,∞ =W (1−θ)θ1+θθ2,p(Ω). (A.5)Example 4. For 0 < θ < 1, 1 ≤ p, q <∞, m ∈ N,(Lp(Rn),Wm,p(Rn))θ,q = Bmθp,q (Rn).117Here Bsp,q(Rn) is the Besov space defined as follows: if s is not an integer, let [s] and{s} be the integer and the fractional parts of s, respectively. Then Bsp,q(Rn) consists ofthe functions f ∈W [s],p(Rn) such that[f ]Bsp,q =∑|α|=[s](∫Rndh|h|n+{s}q(∫Rn|Dαf(x+ h)−Dαf(x)|pdx)q/p)1/qis finite. In particular, for p = q we have Bsp,p(Rn) = W s,p(Rn). If s = k ∈ N, thenBkp,q(Rn) consists of the functions f ∈W k−1,p(Rn) such that[f ]Bkp,q =∑|α|=k−1(∫Rndh|h|n+q(∫Rn|Dαf(x+ 2h)− 2Dαf(x+ h) +Dαf(x)|pdx)q/pis finite.For a complete proof of Examples above see [46, Sections 2.3, 2.4].Corollary A.2.6. For 0 < θ < 1/2, 1 ≤ p <∞, we have(Lp(Rn),W 2,p(Rn))θ,p =W 2θ,p(Rn)with equivalence of the respective norms.In the following result we characterize the interpolation space between L1(Ω) and asubspace of W 1,1(Ω) which takes into account in a suitable way the boundary conditionsthat are to be imposed in the parabolic of our interest.Theorem A.2.7. Let Ω be a subset of Rn with uniformly C2 boundary; then for everyθ ∈ (0, 1/2) we have(L1(Ω),W 2,1(Ω) ∩W 1,1A,ν(Ω))θ,1 =W 2θ,1(Ω) (A.6)where ν(x) denotes the external normal to ∂Ω at x, A is the matrix in (2.106) andW 1,1A,ν(Ω) is the closure of {u ∈ C1(Ω) | 〈A(x) · ∇u, ν(x)〉 = 0 for x ∈ ∂Ω} with respect tothe topology of W 1,1(Ω).Proof. We define for an open and regular set ω ⊂ Rn the spaceXAθ (ω) = (L1(ω),W 2,1(ω) ∩W 1,1A,ν(ω))θ,1endowed with the norm‖u‖XAθ (ω) :=∫ +∞0K(t, u)t1+θdt, K(t, u) := infa+b=ua∈L1(ω)b∈W2,1(ω)∩W1,1A,ν(ω)(‖a‖L1(ω) + t‖b‖W 2,1(ω)) (A.7)We want to prove that XAθ (Ω) = W2θ,1(Ω) for θ ∈ (0, 1/2). For the result in the casewhen ω = Rn we refer to [9, Theorem 4.3.6].118We divide this proof in two steps, in the first one we prove thatXIθ (Rn+) =W2θ,1(Rn+),where I is the identity matrix. In the second one, we use a local change of coordinatesand the regularity of the domain Ω to conclude.First stepWe want to prove that(L1(Rn+),W2,1(Rn+) ∩W 1,1N (Rn+))θ,1 =W 2θ,1(Rn+) (A.8)where W 1,1N (Rn+) denote the space W1,1I,−en(Rn+).Fix θ ∈ (0, 1/2) and consider T the operator that to any function u : Rn+ → R associatesTu = u˜(x) :={u(x1, . . . , xn) if xn ≥ 0u(x1, . . . ,−xn) if xn < 0.(A.9)As it is easily seen T ∈ L(L1(Rn+), L1(Rn)) ∩ L(W 1,1N (Rn+) ∩ W 2,1(Rn+),W 2,1(Rn));therefore applying Theorem A.2.4 we getT ∈ L((L1(Rn+),W 1,1N (Rn+) ∩W 2,1(Rn+))θ,1, (L1(Rn),W 2,1(Rn))θ,1).As a consequence we deduce that if u ∈ (L1(Rn+),W 1,1N (Rn+) ∩W 2,1(Rn+))θ,1 then Tu ∈W 2θ,1(Rn) con 2‖u‖W 2θ,1(Rn+) = ‖u˜‖W 2θ,1(Rn+) ≤ ‖u‖XIθ (Rn+), hence u ∈W 2θ,1(Rn+).Conversely let u ∈ W 2θ,1(Rn+); then the function u˜ defined in the same way of (A.9)belongs to W 2θ,1(Rn); indeed[u˜]W 2θ,1(Rn) =∫Rndx∫Rn|u˜(x)− u˜(y)||x− y|n+2θ dy= 2[u]W 2θ,1(Rn+) + 2∫Rn−1×R+dx∫Rn−1×R−|u˜(x)− u˜(y)||x− y|n+2θ dy≤ 4[u]W 2θ,1(Rn+).Thus, since Xθ(Rn) = W 2θ,1(Rn) for θ ∈ (0, 1/2), there exist v1 ∈ L1(Rn) and v2 ∈W 2,1(Rn) such that u˜ = v1 + v2 and t−θK(t, u˜) ∈ L1∗(0,+∞). Now, let g ∈ C∞c (Rn)with Dng = 0 in xn = 0, then u˜ can be represented as the sum (v1+ v2− g)+ g =: w+ gwith w ∈ L1(Rn), g ∈W 2,1(Rn). If we consider the restriction of w and g in Rn+ we getthat u = u˜|Rn+ = w|Rn+ + g|Rn+ with w|Rn+ ∈ L1(Rn+), g|Rn+ ∈ W 2,1(Rn+) ∩W1,1N (Rn+) andt−θK(t, u) ∈ L1∗(0,+∞) since K(t, u) ≤ K(t, u˜) for all t ∈ (0,∞). Thus (A.8) is proved.Second stepNow we consider the same partition of unity {ηh}h associated with the covering {Uh}hof Ω considered in the proof of Proposition 3.1.1. Then, for a given function u defined inΩ, writing u as∑+∞h=0 uηh, we can prove that uη0 ∈ Xθ(Ω) if and only if uη0 ∈W 2θ,1(Ω).For every h ≥ 1 we can find ψh : B+(0) → Uh ∩ Ω such that d(ψh)x(a(x)ν(x)) = −en,and prove that vh := uηh ◦ ψh belongs to Xθ(Rn+) if and only if belongs to W 2θ,1(Rn+),by which uηh ∈ Xθ(Ω) if and only if uηh ∈W 2θ,1(Ω). Now in order to conclude we haveto show that u ∈ Xθ(Ω) if and only if u ∈W 2θ,1(Ω). Notice that the result is immediateif Ω is bounded, since in that case the covering {Uh}h is finite.119Suppose first that u ∈ Xθ(Ω). Since Xθ(Ω) continuously embeds in L1(Ω), it issufficient to estimate the seminorm [u]W 2θ,1(Ω). Moreover, since u ∈ Xθ(Ω) we also havethat uηh ∈ Xθ(Ω) for each h ∈ N. Notice that, for fixed x ∈ Uh, y ∈ Uk there existsIhk ⊂ N such thatu(x)− u(y) =∑i∈Ihku(x)ηi(x)− u(y)ηi(y)where either supp (ηi) ∩ Uh 6= ∅ or supp (ηi) ∩ Uk 6= ∅. Since {Uh}h has a boundedoverlapping κ, then #(Ihk) ≤ 2κ. Then∫Ω∫Ω|u(x)− u(y)||x− y|n+2θ dxdy ≤∞∑h,k=1∫Uhdx∫Uk|∑i∈Ihk(u(x)ηi(x)− u(y)ηi(y))||x− y|n+2θ dy≤∞∑h,k=1∑i∈Ihk∫Uhdx∫Uk|u(x)ηh(x)− u(y)ηh(y)||x− y|n+2θ dy . (A.10)Now, we define Vh =⋃{j:Uj∩Uh 6=∅} Uj , then there is a constant cκ > 0 depending onlyon κ, the overlapping of the Ui, such that∑i∈Ihk ‖uηi‖L1(Ui) ≤ cκ‖u‖L1(Vh∪Vk)∑i∈Ihk ‖uηi‖W 2,1(Ui) ≤ cκM‖u‖W 2,1(Vh∪Vk)(A.11)where M := suph∈N ‖ηh‖2,∞. Moreover we can write Ω =⋃κi=1 Ωi where Ωi = {x ∈ Ω :#{j : x ∈ Uj} = i} and Ωi ∩ Ωk = ∅ if i 6= k. Then∑h,k∫Vh∪Vk|u| dx =∑h,kκ∑i=1∫(Vh∪Vk)∩Ωi|u| dx=κ∑i=1∑h,k∫(Vh∪Vk)∩Ωi|u| dx=κ∑i=1i∫Ωi|u| dx ≤ κ‖u‖L1(Ω). (A.12)Analogously, ∑h,k‖u‖W 2,1(Vh∪Vk) ≤ κ‖u‖W 2,1(Ω). (A.13)Since the functions vh := uηh ◦ ψh belong both to (L1(Rn+),W 2,1(Rn+) ∩W 1,1A,ν(Rn+))θ,1and W 2θ,1(Rn+), and in RN+ the norms of W2θ,1(Rn+) and Xθ(Rn+) are equivalent, we geta constant κ0, depending only on the norm of the embedding of Xθ(Rn+) in W2θ,1(Rn+)and ψh, such that∫Uhdx∫Uk|u(x)ηi(x)− u(y)ηi(y)||x− y|n+2θ dy ≤ κ0∫ +∞01t1+θK(t, uηi)dt (A.14)120where K is defined in (A.7). By definition of K(t, ·) and by (A.11) we get∑i∈IhkK(t, uηi) =∑i∈Ihkinfa˜+ b˜ = uηia˜ ∈ L1(Ω), b˜ ∈ W 2,1(Ω)(‖a˜‖L1(Ω) + t‖b˜‖W 2,1(Ω))≤∑i∈Ihkinfa+ b = ua ∈ L1(Ω), b ∈ W 2,1(Ω)(‖aηi‖L1(Ω) + t‖bηi‖W 2,1(Ω))≤ infa+ b = ua ∈ L1(Ω), b ∈ W 2,1(Ω)∑i∈Ihk(‖aηi‖L1(Ω) + t‖bηi‖W 2,1(Ω))≤ κ1 infa+ b = ua ∈ L1(Ω), b ∈ W 2,1(Ω)(‖a‖L1(Vh∪Vk) + t‖b‖W 2,1(Vh∪Vk))where κ1 depends on κ and M . Summing up on h, k we get, by (A.12) and (A.13),+∞∑h,k=1∑i∈IhkK(t, uηi) ≤ κ1K(t, u) .Then by (A.10), (A.14) and using the last estimate we get∫|x−y|<ρ|u(x)− u(y)||x− y|n+2θ dxdy ≤+∞∑h,k=1∑i∈Ihkκ0∫ +∞01t1+θK(t, uηi)dt≤ κ0 κ1∫ +∞01t1+θK(t, u)dt = κ0 κ1‖u‖Xθ(Ω) ,whence Xθ(Ω) ⊂W 2θ,1(Ω). To prove the reverse inclusion, consider {ηh, Uh}h as before.First of all observe that, we can estimate for each ρ > 0[uηh]W 2θ,1(Ω) ≤cρn+2θ‖u‖L1(Uh) +∫|x−y|<ρ|u(x)ηh(x)− u(y)ηh(y)||x− y|n+2θ dxdy .where c = 2|Uh| is a positive constant independent on h since Uh are balls with fixedradius. Adding and subtracting u(x)ηh(y) we can estimate∫|x−y|<ρ|u(x)ηh(x)− u(y)ηh(y)||x− y|n+2θ dxdy≤∫Ω×Ω[Lip(ηh)|u(x)||x− y|n−1+2θ χAh,ρ(x, y) +|u(x)− u(y)||x− y|n+2θ χΩ×Uh(x, y)]dxdywhere Ah,ρ = (Uh×Ω∪Ω×Uh)∩{(x, y) ∈ Ω×Ω : |x− y| < ρ}. Then, choosing ρ smallenough in order that the ρ-enlarged sets Uρh have the same overlapping as the Uh’s andAh,ρ ⊂ Uρh × Uρh , we get‖uηh‖W 2θ,1(Ω) ≤ κ2‖u‖L1(Uρh) +∫Uhdy∫B(y,ρ)|u(x)− u(y)||x− y|n+2θ dxwhere κ2 depends (only) on ‖ηh‖W 1,∞ , θ, ρ, n. Since the overlapping is bounded we canfind two constants κ3, κ4 such that∑h‖uηh‖W 2θ,1(Ω) ≤ κ3[‖u‖L1(Ω) +∫Ωdy∫B(y,ρ)|u(x)− u(y)||x− y|n+2θ dx]≤ κ4‖u‖W 2θ,1(Ω) .121Then for each  > 0 we can find a˜h ∈ L1(Ω), b˜h ∈W 2,1(Ω) such that a˜h + b˜h = uηh and‖a˜h‖L1(Ω) + t‖b˜h‖W 2,1(Ω) ≤ K(t, uηh) + 2−h. Define a =∑h a˜h and b =∑h b˜h. Thena+ b = u andK(t, u)≤‖a‖L1(Ω) + t‖b‖W 2,1(Ω)≤∑h‖a˜h‖L1(Ω) + t‖b˜h‖W 2,1(Ω)≤∑hK(t, uηh) + and thenK(t, u) ≤∑hK(t, uηh). Now, as before, since the functions vh are inW 2θ,1(Rn+)and in Rn+, the norms of W2θ,1(Rn+) and Xθ(Rn+) are equivalent, there exists a constantκ5, depending only on the norm of the embedding of W 2θ,1(Rn+) in Xθ(Rn+) and ψh,such that∫ +∞01t1+θK(t, uηh)dt ≤ κ5∫Ωdx∫Ω|u(x)ηh(x)− u(y)ηh(y)||x− y|n+2θ dy . (A.15)Therefore there is a constant κ6 (depending only on κ4 and κ5)∫ +∞01t1+θK(t, u)dt ≤∫ +∞01t1+θ+∞∑h=1K(t, uηh)dt≤ κ5+∞∑h=1‖uηh‖W 2θ,1(Ω) ≤ κ6‖u‖W 2θ,1(Ω).A.3 Complex interpolation spacesThe complex interpolation methods were introduced by J. L. Lions in [29], A. P.Caldero´n in [11] and [12]. We shall follow the treatment of [46]. Let Y,X be complexBanach spaces with Y ↪→ X and let S be the strip {z = x + iy ∈ C : 0 ≤ x ≤ 1}.By the maximum principle for holomorphic functions defined on a strip, we get that ifF : S → X is holomorphic in the interior of S, continuous and bounded in S, then foreach z ∈ S‖F (z)‖X ≤ max{supt∈R‖F (it)‖X , supt∈R‖F (1 + it)‖X}.Definition A.3.1. Denote by H(X,Y ) the space consisting of all continuous and boundedfunctions F : S → X which are holomorphic in the interior of the strip such that t 7→F (it) ∈ C(R, X), t 7→ F (1 + it) ∈ C(R, Y ) and such that‖F‖H(X,Y ) = max{ supt∈R‖F (it)‖X , supt∈R‖F (1 + it)‖Y } <∞.By using the maximum principle, it is not hard to prove that H(X,Y ) is a Banachspace. The complex interpolation spaces are defined by means of functions in H(X,Y ).Definition A.3.2. For every θ ∈ [0, 1], we define[X,Y ]θ = {F (θ) : F ∈ H(X,Y )},122with norm‖f‖[X,Y ]θ = infF∈H(X,Y ),F (θ)=f‖F‖H(X,Y )That [X,Y ]θ is a Banach space follows from the fact that [X,Y ]θ is isomorphic tothe quotient space H(X,Y )/Nθ where Nθ is the subset of H(X,Y ) consisting of thefunctions which vanish at z = θ. Since Nθ is closed, the quotient space is a Banach spaceand so is [X,Y ]θ. The Banach space [X,Y ]θ is indeed an intermediate space as the nextproposition states.Proposition A.3.3. Let θ ∈ (0, 1); thenY ↪→ [X,Y ]θ ↪→ X.Proof. Let f ∈ Y . The constant function F (z) = f belongs to H(X,Y ) and‖F‖F(X,Y ) = max{‖f‖X , ‖f‖Y } ≤ c‖f‖Yfor some c > 0. Therefore f = F (θ) ∈ [X,Y ]θ and ‖f‖[X,Y ]θ ≤ c‖f‖Y . The otherembedding is a consequence of the maximum principle. Indeed if f = F (θ) with F ∈H(X,Y ) then‖f‖X ≤ max{supt∈R‖F (it)‖X , supt∈R‖F (1 + it)‖X}≤ cmax{supt∈R‖F (it)‖X , supt∈R‖F (1 + it)‖Y }= c‖F‖H(X,Y )so that f ∈ X and ‖f‖X ≤ c‖F‖H(X,Y ).In general [X,Y ]θ does not coincide with any (X,Y )θ,p. If X,Y are Hilbert spacesthen the equality holds for p = 2, that is[X,Y ]θ = (X,Y )θ,2 0 < θ < 1.In the non Hilbertian case there are no general rules.Two other useful facts are recalled here, one concerning the dual space of such complexinterpolation spaces and the last proves that [X,Y ]θ are actually interpolation spaces.Theorem A.3.4. (Dual space) Let θ ∈ (0, 1). If Y is dense in X and one of the twospaces X or Y is reflexive, then[X,Y ]′θ = [Y′, X ′]1−θ. (A.16)This theorem is a consequence of the results in A.P. Caldero´n [12]. For the proof werefer to [12].123Theorem A.3.5. Let (X1, Y1), (X2, Y2) be complex interpolation couples. Assume thatT ∈ L(X1, X2) ∩ L(Y1, Y2), then the restriction of T|[X1,Y1]θ is in L([X1, Y1]θ, [X2, Y2]θ)for every θ ∈ (0, 1). Moreover,‖T‖L([X1,Y1]θ,[X2,Y2]θ) ≤ (‖T‖L[X1,X2])1−θ(‖T‖L[Y1,Y2])θ.For the proof and a complete analysis of these spaces we refer to [46].

Appendix A: a brief introduction to interpolation theory

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Appendix A: a brief introduction to interpolation theory

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