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Short Lecture Notes: Interpolation Theory and Function Spaces

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Short Lecture Notes: Interpolation Theory and Function Spaces Short Lecture Notes: Interpolation Theory and Function Spaces Helmut Abels July 27, 2011 1 Introduction In the following let K = R or K = C. Denition 1.1 Let be Banach spaces over . Then the pair X0;X1 K (X0;X1) is called admissible, compatible or an interpolation couple if there is a Hausdor topological vector space Z such that X0;X1 ,! Z with continuous embeddings. Lemma 1.2 Let (X0;X1) be an admissible pair of Banach spaces. Then X0 \ X1 normed by kxkX0\X1 = max(kxkX0 ; kxkX1 ); x 2 X0 \ X1; and X0 + X1 normed by kxkX0+X1 = inf kx0kX0 + kx1kX1 ; x 2 X0 + X1; x=x0+x1;x02X0;x12X1 are Banach spaces. Notation: In the following we write for simplicity T 2 L(Xj;Yj), j = 0; 1, for the condition that T : X0 + X1 ! Y0 + Y1 is a linear operator with for and . T jXj 2 L(Xj;Yj) j = 0 j = 1 Denition 1.3 Let (X0;X1) and (Y0;Y1) be admissible pairs of Banach spaces and let X; Y be Banach spaces. 1. X is called intermediate space with respect to (X0;X1) if X0 \ X1 ,! X,! X0 + X1 with continuous embeddings. 1 2. X and Y are called interpolation spaces with respect to (X0;X1) and (Y0;Y1) if X and Y are intermediate spaces with respect to (X0;X1) and (Y0;Y1), respectively, and if T 2 L(Xj;Yj); j = 0; 1; ) T jX 2 L(X; Y ) for all T : X0 + X1 ! Y0 + Y1 linear. 3. X is called interpolation space with respect to (X0;X1) if the previous conditions hold with X = Y , (X0;X1) = (Y0;Y1). 4. Interpolation spaces X and Y with respect to (X0;X1) and (Y0;Y1) are called of exponent θ 2 [0; 1], if there is some C > 0 such that kT k ≤ CkT k1−θ kT kθ (1.1) L(X;Y ) L(X0;Y0) L(X1;Y1) for all T 2 L(Xj;Yj); j = 0; 1. If (1.1) holds with C = 1, then X and Y are called exact of exponent θ. Analogously, X is an (exact) interpolation space of exponent θ if the previous conditions hold for X = Y , (X0;X1) = (Y0;Y1). Examples 1.4 1. Let , and let 0 < α < 1 k 2 N0 0 n n C (R ) = ff : R ! K continuous and boundedg kfkC0(Rn) = sup jf(x)j x2Rn k n n -times cont. di. α 0 n C (R ) = f : R ! K k : @x f 2 C (R )8jαj ≤ k α kfkCk(Rn) = sup sup j@x f(x)j jα|≤k x2Rn α n n C (R ) = f : R ! K : kfkCα(Rn) < 1 jf(x) − f(y)j α n kfkC (R ) = sup jf(x)j + sup α x2Rn x;y2Rn;x6=y jx − yj Then Cα(Rn) is an intermediate space of C0(Rn) and C1(Rn). More precisely, there is some C > 0 such that 1−α α 1 n kfk α n ≤ Ckfk kfk 1 n for all f 2 C ( ): C (R ) C0(Rn) C (R ) R 2 2. For 1 ≤ p ≤ 1 let Lp(U; µ) denote the usual Lebesgue space on a measure 1 1−θ θ space (U; µ). Then for any 1 ≤ p0; p1; p ≤ 1 such that = + for p p0 p1 some θ 2 [0; 1] the space Lp(U; µ) is an intermediate space of Lp0 (U; µ) and Lp1 (U; µ). This follows from the generalized Hölder inequality 1−θ θ p0 p1 p for all kfkL (U,µ) ≤ kfkLp0 (U;µ)kfkLp1 (U,µ) f 2 L (U; µ) \ L (U; µ); which itself follows easily from the standard Hölder inequality. Remark 1.5 The last two inequalities are special cases of (1.1) if one chooses . The following theorem shows that p in the latter example is Yj = K L (U; µ) an exact interpolation space with respect to (Lp0 (U; µ);Lp1 (U; µ)) of expo- nent θ. THEOREM 1.6 (Riesz-Thorin) Let 1 ≤ p0; p1; q0; q1 ≤ 1, θ 2 [0; 1], let (U; µ) and (V; ν) be σ-nite measure 1 spaces and let K = C. Then pj qj p q T 2 L(L (U; µ);L (V; ν)); j = 0; 1 ) T jLp(U,µ) 2 L(L (U; µ);L (V; ν)); where 1 1 − θ θ 1 1 − θ θ = + ; = + : p p0 p1 q q0 q1 Moreover, for every T 2 L(Lpj (U; µ);Lqj (V; ν)); j = 0; 1; 1−θ θ p q kT kL(L ;L ) ≤ kT kL(Lp0 ;Lq0 )kT kL(Lp1 ;Lq1 ): The theorem will follow from the results of Section 2 below. Remark 1.7 The proof of the Riesz-Thorin interpolation theorem provided the fundamental ideas for the so-called complex interpolation method. Gener- ally an interpolation method is an functor F from the cathegory of admissible couples of Banach spaces to the cathegory of Banach spaces such that any pair of admissible couples X = (X0;X1), Y = (Y0;Y1) is mapped to an interpolation space F(X; Y ) with respect to X; Y .2 1Meaning that all functions will be complex valued in the following. 2Readers not familiar with the basic notion of cathegory do not have to understand this precisely since no cathegory theory will be needed in the following. 3 In some situations the following theorem due to Marcinkiewicz can be applied, when an application of the Riesz-Thorin theorem fails. THEOREM 1.8 (Marcinkiewicz) Let 1 ≤ p0; p1; q0; q1 ≤ 1 with q0 6= q1, let θ 2 (0; 1), and let (U; µ) and (V; ν) be measure spaces. Moreover, let 1 1 − θ θ 1 1 − θ θ = + ; = + p p0 p1 q q0 q1 and assume that p ≤ q. Then pj qj p q T 2 L(L (U; µ);L∗ (V; ν)); j = 0; 1 ) T jLp(U,µ) 2 L(L (U; µ);L (V; ν)); pj qj Moreover, there is some Cθ such that for every T 2 L(L (U; µ);L∗ (V; ν)); j = 0; 1; 1−θ θ p q q kT kL(L ;L ) ≤ CθkT k p q0 kT k p1 1 : L(L 0 ;L∗ ) L(L ;L∗ ) Here q is the weak q-space, dened by L∗(V; ν) L C Lq(V; ν) = f : V ! measurable : m(t; f) ≤ for all t > 0 and some C > 0 ∗ K tq if 1 ≤ q < 1, where m(t; f) = ν(fx 2 V : jf(x)j > tg); t > 0; denotes the distribution function of . Moreover, if , then 1 f q = 1 L∗ (V; ν) = 1 . We note that q is quasi-normed space with quasi-norm by L (V; ν) L∗(V; ν) 1 q q kfkL∗ = sup t m(t; f) : t>0 Here a quasi-norm k:k on a vector space X satises the same conditions as a norm except that the triangle inequality is replaced by kx + yk ≤ C(kxk + kyk) for all x; y 2 X for some C ≥ 1. The bounded operators and the operator norm is dened in the same way for quasi-normed spaces as for normed spaces. Remark: One also denotes q;1 q . L (V; ν) ≡ L∗(V; ν) 4 A proof of the Marcinkiewicz interpolation theorem in the special case , n n can be found in the appendix. The p0 = q0 = 1; p1 = q1 = r (U; µ) = (R ; λ ) proof provided the basic ideas for the so-called real interpolation method. The real and the complex interpolation methods are the most important methods to construct interpolation spaces and will be one of the central topic of this lecture series. We will start with the complex method since it is easier to understand in the beginning, although the real interpolation method is in some sense more fexible and universal. 2 Complex Interpolation Method 2.1 Holomorphic Functions on the Strip For the following let X be a complex Banach space and let S = fz 2 C : 0 ≤ , . A mapping is called Re z ≤ 1g S0 = fz 2 C : 0 < Re z < 1g f : S0 ! X 0 holomorphic if z 7! hf(z); x i, z 2 S0, is holomorphic in the usual sense for all x0 2 X0. THEOREM 2.1 (Maximum Principle/Phragmen-Lindelöf Theorem) Let f : S ! X be continuous, bounded, and holomorphic in the interior of S. Then sup kf(z)kX ≤ max sup kf(it)kX ; sup kf(1 + it)kX : (2.1) z2S t2R t2R Proof: First let and assume that . Consider the X = C f(z) !j Im zj!1 0 mapping h: S ! C with eiπz − i h(z) = ; z 2 S: (2.2) eiπz + i Then h is a bijective mapping from S onto U = fz 2 C : jzj ≤ 1g n {±1g, that is holomorphic in S0 and maps @S onto fjzj = 1g n {±1g. Therefore g(z) := f(h−1(z)) is bounded and continuous on U and analytic in the interior of U. Moreover, because of limj Im zj!1 f(z) = 0, limz→±1 g(z) = 0 and we can extend g to a continuous function on fjzj ≤ 1g. Hence jg(z)j ≤ max jg(w)j = max sup jf(it)j; sup jf(1 + it)j ; jwj=1 t2R t2R 5 which implies the statement in this case. Next, if X = C and f is a general 2 function as in the assumptions, then we consider δ(z−z0) , fδ;z0 (z) = e f(z) 2 2 2 δ(z−z0) δ(x −y ) δ > 0; z0 2 S0.
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