On nonlinear interpolation

In a case study on asymptotics of spectral quantities of Schr\"odinger operators we show how the Riesz-Thorin theorem on the interpolation of linear operators can be extended to nonlinear maps.


Introduction
Instead of stating a general abstract theorem on nonlinear interpolation we prefer to focus on how such a result can be applied in a fairly simple context. The arguments of the proofs can easily be adapted to different set-ups.
Note that H s ′ C ֒→ H s C for any 0 ≤ s ≤ s ′ . Denote by B s C (R) ⊆ H s C the open ball of radius R, centered at 0. Furthermore, for any 1 ≤ p < ∞ and t ∈ R denote by ℓ p,t C ≡ ℓ p,t (Z, C) the complex sequence space Note that ℓ p,t ′ C ֒→ ℓ p,t C for any 0 ≤ t ≤ t ′ . Finally, for b > a ≥ 0 and α ≥ 0, β > 0, assume that F : B a C (R) → ℓ p,α+βa C is an analytic map, bounded by M a > 0, so that for some M b > 0, By the characterization of analytic maps with sequence spaces as their range (cf. e.g. [5,Appendix A]) and the analyticity of F it follows from (1.1) that is analytic as well. Hence, equivalently, we are given the following commutative diagram where the horizontal maps F and F | B b C (R) are analytic and bounded by the constants M a > 0 and M b > 0 respectively, and the vertical arrows denote the standard inclusions of the corresponding spaces.
The following theorem is an instance of an extension of the Riesz-Thorin theorem (cf. e.g. [1]) to nonlinear maps and is inspired by a special case treated in [6,Appendix 2].
The proof of Theorem 1.1, presented in Section 2, uses in a crucial way the assumption that F is analytic on a ball in a complex (Hilbert) space. The proof does not apply to nonlinear maps which are merely real analytic. In the sequel we would like to discuss a type of problems where nevertheless Theorem 1.1 can be applied to real analytic nonlinear maps. For q in L 2 0 (T, R) = q ∈ L 2 (T, R) x +q. The Dirichlet spectrum of L(q), considered on the interval [0, 1], is real and consists of simple eigenvalues. We list them in increasing order µ 1 < µ 2 < . . .. Furthermore denote by M(x, λ) the fundamental solution of L(q), i.e. the 2 × 2 matrix valued function satisfying L(q)M = λM, λ ∈ C, and M(0, λ) = Id 2×2 , .
When evaluated at λ = µ n , the Floquet matrix M(1, λ) is lower triangular, hence its eigenvalues are given by y 1 (1, µ n ) and y ′ 2 (1, µ n ). By the Wronskian identity, they satisfy y 1 (1, µ n )y ′ 2 (1, µ n ) = 1. By deforming the potential q to the zero potential along the straight line tq, 0 ≤ t ≤ 1, one sees that (−1) n y 1 (1, µ n ) > 0. Hence the Floquet exponents are given by ±κ n where κ n := − log ((−1) n y 1 (1, µ n )) and log denotes the principal branch of the logarithm. It turns out that together the µ n 's and κ n 's form a system of canonical coordinates for L 2 0see [7]. The κ ′ n 's play also an important rôle for proving the property of 1-smoothing of the periodic KdV equation [4]. We want to determine the asymptotics of the κ n 's as n → ∞. To state them, introduce for any s ∈ R ≥0 the Sobolev spaces and denote by ·, · the standard inner product in L 2 0 (T, R), In [3], the following theorem was proved.
Our aim is to extend Theorem 1.2 to any fractional order Sobolev space H s 0 with s ∈ R ≥0 . To this end we need to extend various quantities to the complex Hilbert space In section 3 we prove the following Theorem 1.3 For any R > 0 there exits n R > 0 so that for any n > n R , κ n can be extended analytically to B 0 0,C (R). Moreover, for any N ∈ Z ≥0 the sequence (κ n ) n>n R satisfies the estimate At the end of Section 3 we show that Theorem 1.3 allows to apply Theorem 1.1 to generalize Theorem 1.2 to any fractional order Sobolev space.
By the same method one can show that similar results hold for many other spectral quantities. We state without proof another such result. For q ∈ L 2 0 , the periodic spectrum of L(q) on the interval [0, 2] is real and discrete. When listed in increasing order and with their multiplicities the eigenvalues satisfy  Related work: In the context of nonlinear PDEs, Tartar obtained a nonlinear interpolation theorem in [11] which later was slightly improved in [2, Theorem 1]. However, for nonlinear maps such as the ones encountered in the analysis of Schrödinger operators the assumptions of these theorems are not satisfied and hence they cannot be applied.
Acknowledgment: The motivation of this paper originated from our observation that Theorem 6.1 in the paper [9] of A. Savchuk and A. Shkalikov, the theorem of Tartar on nonlinear interpolation [11] cannot be applied as stated by the authors. After having pointed this out to A. Shkalikov, he sent us the statement of an unpublished interpolation theorem obtained by them earlier, which can be used to prove their results. Due to unfortunate personal circumstances their paper with a proof of this interpolation theorem was delayed. In order to be able to further progress on our projects, we came up with our own version, tailored to our needs and with an optimal estimate on the bounds of the interpolated maps. After having sent our preprint to A. Shkalikov, he sent us a preprint with the proof of their theorem [10]. As the two papers turned out to be quite different, we agreed to publish them independently.
2 Proof of Theorem 1.1 The proof of Theorem 1.1 uses arguments of the proof of the Riesz-Thorin theorem on the interpolation of linear operators -see e.g. [1]. Key ingredient is Hadamard's three-lines theorem.
Proof of Theorem 1.1. For any k ∈ Z, denote by F k the k'th component of where z = u + iv ∈ C, u, v ∈ R, and k := 1 + |k|. As u and v are real we see that for any ϕ ∈ B s C (R), In particular, for any v ∈ R, Similarly, for any a ≤ u ≤ b and v ∈ R, Hence ϕ u+iv ∈ B a C (R) for any a ≤ u ≤ b and any v ∈ R. In particular, F (ϕ u+iv ) ∈ ℓ p,α+βa C and F (ϕ u+iv ) p,α+βa ≤ M a . (

2.2)
As ϕ s = ϕ for any ϕ ∈ B s C (R), we need to prove the claimed estimate for z = s. Denote by 1 < q ≤ ∞ the number conjugate to p, i.e. 1 p + 1 q = 1, and consider an arbitrary sequence ξ = (ξ k ) k∈Z ⊆ C with finite support so that ξ q,−(α+βs) ≤ 1. Similarly as in the case of H s C , define for z = u + iv ∈ C, u, v ∈ R, ξ z := k −β(s−z) ξ k k∈Z . Then Similarly as above, ξ s = ξ, and for any a ≤ u ≤ b and v ∈ R, Denote by ·, · the ℓ 2 dual pairing as well as its extension to ℓ p,α+βa C × ℓ q,−(α+βa) C and introduce the vertical strip and its closure S a,b . To obtain the claimed estimate we want to apply Hadamard's three lines theorem to the following function As the support of ξ is finite, the latter sum is finite, and hence the function f is well defined and ξ q,−α+βb < ∞. In view of (2.2) and (2.4), f : S a,b → C is bounded as for any z ∈ S a,b , Note that on the strip S a,b , the curves z → ξ z ∈ ℓ q,−(α+βa) C and z → ϕ z ∈ B a C (R) ⊆ H a C are analytic. As F k : B a C (R) → C is analytic for any k ∈ Z it then follows that f , being a finite sum of analytic functions, is analytic on S a,b and continuous on the closure S a,b . Moreover, in view of (2.1) and (2.3), the following estimates hold for any v ∈ R, |f (a + iv)| ≤ F (ϕ a+iv ) p,α+βa ξ a+iv q,−(α+βa) ≤ M a and |f (b + iv)| ≤ F (ϕ b+iv ) p,α+βb ξ b+iv q,−(α+βb) ≤ M b .
Hence we can apply Hadamard's three-lines theorem to f (cf. e.g. [8, Appendix to IX,.4]) to conclude that, for s = (1 − λ)a + λb and any v ∈ R In particular, for z = s one has where we took into account that ϕ s = ϕ and ξ s = ξ. As the sequences ξ with finite support are dense in ℓ q,−(α+βs) C the claimed estimate F (ϕ) p,α+βs ≤ (M a ) 1−λ (M b ) λ follows from Hahn-Banach theorem and the fact that ℓ q,−(α+βs) C is the dual space of ℓ p,α+βs C .

Proofs of Theorem 1.and Theorem 1.4
Before proving Theorem 1.3 we need to make some preparatory considerations. Note that for q in L 2 0,C the operator L(q) is no longer symmetric with respect to the L 2 -inner product f, g = 1 0 f (x)g(x) dx. The Dirichlet spectrum is still discrete, but the eigenvalues might be complex valued and multiple. We list them according to their algebraic multiplicities and in lexicographic ordering By [7] there exists for any R > 0 an integer m R > 0 so that for any q ∈ L 2 0,C with q := q, q 1/2 < R the Dirichlet eigenvalues µ n ≡ µ n (q), n ≥ 1, satisfy the following estimates |µ n − n 2 π 2 | < π/4 ∀ n > m R and |µ n | < m 2 R π 2 + π/4 ∀1 ≤ n ≤ m R . In particular, for any n > m R , µ n is a simple Dirichlet eigenvalue of L(q) and hence analytic on the complex ball B 0 0,C (R) = {q ∈ L 2 0,C   q < R}. To see that κ n can be analytically extended to B 0 0,C (R) for n sufficiently large we first note that by [7], the Floquet matrix M(1, λ, q) is analytic on C × L 2 0,C and, with λ = ν 2 , y 1 (1, ν 2 , q) satisfies the following estimate As for n > m R , one has |µ n − n 2 π 2 | < π/4, it follows that ν n ≡ ν n (q) = + µ n (q) is well defined. Here + √ z denotes the principal branch of the square root. Then for any n > m R and q ∈ B 0 0,C (R), ν n = nπ + √ 1 + z n , where z n = µn−n 2 π 2 n 2 π 2 satisfies |z n | ≤ 1 4πn 2 ≤ 1 10 and thus These estimates are used in the asymptotics of y 1 (1, ν 2 n , q). As (−1) n cos ν n = 1+(ν n −nπ) 1 0 − sin(t(ν n −nπ))dt one concludes that |(−1) n cos ν n −1| ≤ 1 4n for any n > m R and as |ν n | ≥ nπ − 1 4n ≥ 2n it follows from (3.1) that for any n > m R and q < R. Now choose n R ≥ m R so large that As a consequence, for any q ∈ L 2 0,C with q < R and any n > n R , is well defined. We thus have proved. is well defined and analytic.
Proof of Theorem 1.3. In view of Lemma 3.1 it remains to prove for any R > 0, and any N ∈ Z ≥0 , Going through the arguments of the proof of Theorem 1.1 in [3], one sees that this is indeed the case.
It remains to show Theorem 1.4. As already mentioned in the introduction, we will use the result of Theorem 1.1 on nonlinear interpolation.
Proof of Theorem 1.4. By Theorem 1.2, Theorem 1.4 holds in the case where s is an integer. Let N ∈ Z ≥0 and N < s < N + 1. For any given R > 0 choose n R > 0 as in Theorem 1.3. It follows from Theorem 1.2 that there exists M R > 0 such that for any q ∈ L 2 0 with q < R and any 1 ≤ n ≤ n R   2πnκ n − q, sin 2πnx   ≤ M R and therefore n≤n R n 2(s+1)   2πnκ n − q, sin 2πnx On the other hand, by Theorem 1.3, for any n > n R , κ n extends analytically to B 0 0,C (R) and there exist constants M N,R , M N +1,R so that