Stable self-similar blowup in the supercritical heat flow of harmonic maps

We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear stability analysis of self-similar shrinkers to the spectral analysis of the associated self-adjoint linearized operators.


Introduction
Let (M, g) and (N, h) be Riemannian manifolds with metrics g and h, respectively. A map U : M → N is called harmonic if it is a critical point of the functional where we employ Einstein's summation convention throughout. Note that S(U) is a natural generalization of the Dirichlet energy. The Euler-Lagrange equations associated to S are ∆ M U a − g jk Γ a bc (U)∂ j U b ∂ k U c = 0, where Γ a bc are the Christoffel symbols on the target manifold N and is the Laplace-Beltrami operator on M. The study of harmonic maps is a classical subject in geometric analysis, see e.g. [20,36,34,37,39,19,18,35,27,28]. The basic mathematical questions concern the existence and, ideally, the classification of harmonic maps. A standard tool in this respect is the associated heat flow, i.e., one considers a one-parameter family {U t : t ≥ 0} of maps from M to N that evolve according to the heat equation The idea then is to take an arbitrary map U 0 : M → N as initial data at t = 0 and due to the regularizing effects of the heat flow, the solution U t is expected to converge to an equilibrium as t → ∞. In other words, the heat flow is supposed to deform arbitrary maps  into harmonic ones. Indeed, this strategy works well under certain curvature assumptions as is demonstrated in the classical paper [20]. In the general case, however, the flow tends to form singularities (or "blow up") in finite time [10,8,7,30,31,21,22,44,43,26,2,32,33,1]. This is a severe obstruction which can only be overcome if one is able to continue the flow past the singularity in a well-defined manner. Such a construction is a challenging endeavor which presupposes a detailed understanding of possible blowup scenarios. Naturally, one is mainly interested in blowup behavior that is stable under small perturbations of the initial data.
In this paper we are interested in singularity formation in the heat flow of harmonic maps U : S d → S d . As it turns out, the blowup is a local phenomenon and the curvature of the base manifold is irrelevant for the asymptotic behavior near the singularity. Consequently, we may equally well consider maps U : R d → S d , cf. [38,22]. Furthermore, we restrict ourselves to the case d = 3 and assume corotational symmetry. That is to say, we choose standard spherical coordinates (r, θ, ϕ) on R 3 , hyperspherical coordinates on S 3 , and make the ansatz U(r, θ, ϕ) = (u(r), θ, ϕ) for the map U : R 3 → S 3 . Under this symmetry reduction, the Euler-Lagrange equations associated to the functional S reduce to a single nonlinear ordinary differential equation for u which reads u ′′ (r) + 2 r u ′ (r) − sin(2u(r)) r 2 = 0, r ≥ 0.
In order to obtain the associated heat flow, we introduce an artificial time dependence and consider the Cauchy problem for the equation ∂ t u(r, t) − ∂ 2 r u(r, t) − 2 r ∂ r u(r, t) + sin(2u(r, t)) r 2 = 0. (1.1) Our main result shows the existence of a stable self-similar blowup scenario for Eq. (1.1).
For the precise formulation we introduce the following function space. The Banach space Y is defined as the completion ofỸ with respect to · Y .
There exists an f 0 ∈ C ∞ ([0, ∞)) ∩ Y with f 0 > 0 on (0, ∞) such that, for any T 0 > 0 and t ∈ [0, T 0 ), has a unique solution u h that blows up at t = T h and converges to u * T h in the sense that In particular, the class {u * T 0 : T 0 > 0} of self-similar solutions is nonlinearly asymptotically stable under small perturbations of the initial data.
Some remarks are in order.
• The map U : R 3 → S 3 has values on the sphere and thus, there is no blowup in L ∞ . However, the self-similar solution u * T 0 blows up in Y . Indeed, a simple scaling argument shows The blowup profile f 0 is constructed in the companion paper [3] by a novel computerassisted (but rigorous) method. It is not known in closed form. Furthermore, f 0 is not the only self-similar profile. In fact, there exist infinitely many self-similar solutions to Eq. (1.1), see [21]. To the knowledge of the authors, Theorem 1.2 is the first result on stable blowup with a nonunique blowup profile that is not known explicitly. • The norm · Y might look odd at first glance since it is based on homogeneous Sobolev spaces on R 5 whereas Eq. (1.1) is posed on R 3 . However, if one sets u(r, t) = rv(r, t), Eq. (1.1) transforms into a radial heat equation on R 5 for the function v. In addition, this transformation regularizes the nonlinearity at the center, see below. In this sense, the effective dimension of the problem is 5 and it is natural to work with radial functions on R 5 . • In the formulation of Theorem 1.2 we do not specify the precise solution concept we are using. We will study Eq. (1.1) in similarity coordinates by semigroup theory which yields a canonical notion of strong solution (which is actually called "mild solution" in semigroup theory). Since Eq. (1.1) is parabolic, smoothing effects will kick in immediately and turn strong solutions into classical ones. • For obvious reasons, self-similar solutions of the form f ( r √ T 0 −t ) are called shrinkers. Since Eq. (1.1) is not time-reversible, there is another, independent class of selfsimilar solutions, so-called expanders, which take the form f ( r √ t−T 0 ). The latter have also attracted considerable interest, in particular in connection with the question of unique continuation beyond blowup [2,24,23], but they play no role in the present paper.
1.1. Related results. The analysis of harmonic maps is a vast subject that is impossible to review in this paper. We restrict ourselves to a brief discussion of recent blowup results that are directly related to our work and refer the reader to the monographs and survey articles [39,19,18,35,27,28] for the general background.
As already indicated, self-similar solutions for the corotational heat flow of harmonic maps U : R d → S d for d ∈ {3, 4, 5, 6} are constructed in [21,22]. Expanding self-similar solutions are studied in [24]. For d ≥ 7, there are no self-similar shrinkers [5] and the blowup is of a more complicated nature [1,4]. The case d = 2 is of special interest since it is energy-critical and blowup takes place via shrinking of a soliton [44,32,33]. The unique continuation beyond blowup is investigated in [2,23]. Needless to say, there are similar results for closely related problems like the Yang-Mills heat flow or the nonlinear heat equation, see the discussion in [17] for a brief overview. Of particular interest in this context is the recent paper [9] which also considers self-similar blowup for a nonlinear heat equation with a blowup profile that is not known in closed form. In contrast to our result, however, the blowup studied in [9] is highly unstable and the necessary spectral properties can be obtained by a perturbative argument.

1.2.
Outline of the proof. The proof of Theorem 1.2 proceeds by a perturbative construction around the blowup solution u * T 0 . We would like to emphasize that this is a robust approach that uses no structure other than the spectral stability of the self-similar profile f 0 which is established in [3]. As a consequence, our method provides a universal framework for studying self-similar blowup in general parabolic evolution equations. We briefly outline the main steps.
• We consider Eq. (1.1) with initial data u(r, 0) = u * T 0 (r, 0) + h(r). By time translation invariance we may assume T 0 = 1 and we introduce similarity coordinates s = − log(T − t) + log T , y = r √ T −t which go back to [25]. Here, T > 0 is a free parameter which will be adjusted later. Then we rescale the dependent variable u in a suitable manner to obtain the evolution equation wherew =w(y, s), with initial dataw(y, 0) = f 0 ( √ T y)/y+h( √ T y)/y. This equation has the static solutionw(y, s) = f 0 (y)/y. To study its stability, we make the ansatz w(y, s) = f 0 (y)/y + w(y, s) which leads to an evolution equation of the form for the perturbation w. The linear operatorL is given bŷ with the potential V 0 (y) = 2 cos(2f 0 (y))−2 y 2 and N denotes the nonlinear remainder. In the spirit of standard local well-posedness theory we now try to solve Eq. (1.2) by treating the nonlinear terms perturbatively. Consequently, we first have to understand the linearized equation that arises from (1.2) by dropping the nonlinear terms.
• The operatorL, interpreted as an operator acting on radial functions on R 5 , has a self-adjoint extension L on L 2 σ (R 5 ) with the weight σ(x) = e −|x| 2 /4 . Here we encounter the fundamental problem in studying self-similar blowup for parabolic equations: In order to apply self-adjoint spectral theory, it seems necessary to study the evolution in Sobolev spaces with exponentially decaying weights. This, however, is impossible since one cannot control nonlinear terms in such spaces.
There are (at least) two ways around this issue. First, one can study the evolution in unweighted Sobolev spaces and rely on nonself-adjoint spectral theory. This approach was chosen in [17] for the study of the Yang-Mills heat flow. In this paper we follow a different strategy which is based on the simple observation that in a certain sense the problem splits into a self-adjoint part on a compact domain, where the exponentially decaying weight is irrelevant, and a nonself-adjoint part on an unbounded domain which, however, is easy since the potential term is negligible there. We remark that this is not a new discovery but a well-known phenomenon in parabolic problems, see e.g. [6,29,40,9]. Somewhat paradoxically, we can therefore study the linearized evolution on unweighted spaces by using self-adjoint spectral theory in a weighted space.
More precisely, we consider the semigroup e sL on L 2 σ (R 5 ) generated by the selfadjoint operator L. From [3] we know that L has precisely one nonnegative eigenvalue λ = 1 with eigenfunction ψ 1 . As usual, this instability is related to the freedom in choosing the parameter T in the similarity coordinates. From self-adjoint spectral theory we obtain the weighted decay estimate for some constant c 0 > 0, provided f ⊥ ψ 1 . Similar bounds hold for higher Sobolev spaces with weights. As a matter of fact, also on unweighted homogeneous Sobolev spaces of sufficiently high degree we have decay, but a priori only for the free operator L 0 = L − V 0 . Indeed, an integration by parts shows on the unweighted L 2 . Similar bounds hold for higher derivatives. Consequently, by combining the unweighted bounds, the weighted decay, and the smallness of V 0 (y) for large y, we derive the unweighted decay • From now on we follow the argument introduced in our earlier works [11,13,14,12,15,16] on self-similar blowup for wave-type equations. We first show that the nonlinearity is locally Lipschitz on X. This is not hard but requires at least some work due to the removable singularity of the nonlinearity at the center. Then we employ Duhamel's principle to rewrite Eq. (1.2) as where φ(s)(y) = w(y, s) and U(h, T ) is an abbreviation for the initial data. In general, Eq. (1.3) does not have a global solution due to the unstable eigenvalue 1 ∈ σ(L). We deal with this issue by employing the Lyapunov-Perron method. That is to say, we first suppress the instability by subtracting a correction term and instead of Eq. (1.3), we consider the modified equation Here, P is the orthogonal projection on the unstable subspace ψ 1 . By a fixed point argument we show that for any small h and T close to 1, Eq. (1.4) has a global (in s) solution φ h,T that decays like the stable linear flow, i.e., φ h,T (s) e −ω 0 s . In the final step we prove that for any small h, there exists a T h close to 1 which makes the correction term vanish. In other words, φ h,T h is a solution to the original equation (1.3).

Preliminary transformations
The basic evolution equation is where r ≥ 0. For any T 0 > 0, we have the self-similar solution [3]. Our goal is to study the evolution of small initial perturbations of u * T 0 . By time translation invariance, we may restrict ourselves to T 0 = 1. Consequently, we consider the Cauchy problem where h is a free function. In order to regularize the nonlinearity, it is useful to change variables according to u(r, t) = rv(r, t). This yields Here, T > 0 is a free parameter that will be needed to account for the time translation invariance of the problem which introduces an artificial instability. Eq. (2.3) transforms into Observe that the only trace of the parameter T is in the initial data. Furthermore, by construction, is a static solution to Eq. (2.4). By making the ansatzw(y, s) = f 0 (y)/y + w(y, s), we rewrite Eq. (2.4) as ∂ s w(y, s) =Lw(y, s) +N (w(y, s)) w(y, with the linear operatorL defined bŷ and the nonlinearitŷ

The linearized evolution
In this section we study the linearized equation, i.e., we drop the nonlinearity in Eq. (2.5) and focus on Furthermore, we do not specify the initial data explicitly because their specific form is irrelevant for the linear theory. Note that the operatorL contains the 5-dimensional radial Laplacian and for the rest of this paper we actually find it convenient to switch to 5-dimensional notation. To this end, we define the operator In the following, the variable x is used to denote an element of R 5 . In this spirit we define the potential V 0 : By [3], f 0 is odd 1 and thus, V 0 ∈ C ∞ (R 5 ), see [45]. Now we define a differential operatorL byL where throughout, ∆ denotes the Laplacian on R 5 . Then we havẽ for all radial functions f : where φ(s)(x) = w(|x|, s). Formally, the solution of Eq. (3.2) is given by φ(s) = e sL φ(0). In the following, we make this rigorous.
1 By this we mean that f 0 can be extended to all of R as a smooth, odd function. In other words, and denote by L 2 w (Ω) the completion of C ∞ c (Ω) with respect to · L 2 w (Ω) . We promoteL to an unbounded linear operator on the Hilbert space has the explicit solutionf For y ∈ (0, 1] we have and thus,f 1 / ∈ L 2 ρ (0, 1). Similarly, for y ≥ 1, ∞). By the Weyl alternative, the Sturm-Liouville operator defined by (3.3) is in the limit-point case at both endpoints and the Kato-Rellich theorem implies that T (and henceL) is essentially self-adjoint, see e.g. [42].
3.2. Estimates in local Sobolev norms. We upgrade the L 2 σ bound on e sL to a local H 4 bound. In the following we use for f ∈ D(L) to denote the graph norm of L. Furthermore, the letter C (possibly with subscripts to indicate dependencies) denotes a positive constant that might change its value at each occurrence and c 0 > 0 is the constant from Proposition 3.1. Finally, for R > 0 we set Proof. Let f ∈ C ∞ c (R 5 ) and R ≥ 1. An integration by parts yields and we infer Now let f ∈ D(L). Since C ∞ c (R 5 ) is a core for L, there exists a sequence (f n ) ⊂ C ∞ c (R 5 ) such that f n → f in the graph norm · G(L) . Consequently, Eq. (3.5) shows that (∂ j f n ) is Cauchy in L 2 (B 5 R ) for any j ∈ {1, 2, . . . , 5}. We set g j := lim n→∞ ∂ j f n ∈ L 2 (B 5 R ). By dominated convergence we infer Consequently, ∂ j f = g j in the weak sense and this shows ∇f ∈ L 2 (B 5 R ) with the bound (3.5).
Let f ∈ C ∞ c (R 5 ). Then we have Lf 2 which yields the bound by Eq. (3.5). Consequently, a density argument as above finishes the proof. 9 In order to control the full Sobolev norm for k = 2, we need two technical results which are completely elementary since we restrict ourselves to radial functions. First, we have a trace lemma.
for all radial f ∈ C 2 (B 5 R ).
Next, by an extension argument and Fourier analysis, we easily get control on mixed derivatives. Here and in the following, F is the Fourier transform for all radial f ∈ C 2 (B 5 R ) and all j, k ∈ {1, 2, . . . , 5}.
Next, we improve the above by two derivatives.

11
Lemma 3.6. Let f ∈ D(L 2 ) and R ≥ 1. Then ∇∆f, ∆ 2 f ∈ L 2 (B 5 R ) and we have the bound Proof. Let f ∈ C ∞ c (R 5 ) and R ≥ 1. From Lemma 3.2 we have the bound . Expanding the square yields and thus,

by Lemmas 3.2 and 3.4.
For ∆ 2 f we expand ∆Lf 2 L 2 (B 5 R ) and use Lemma 3.2 together with the bound on ∇∆f to obtain 3.3. Estimates in unweighted global Sobolev norms. Next, we prove bounds inḢ 2 (R 5 ) andḢ 4 (R 5 ). The intersectionḢ 2 (R 5 ) ∩Ḣ 4 (R 5 ) will be our main space where we study the evolution. First, we have to ensure that unweighted Sobolev spaces are invariant under e sL .  This is easily verified by an explicit computation. Since K s ∈ L 1 (R 5 ) for any s > 0, dominated convergence and Young's inequality immediately imply the invariance of H k (R 5 ) under e sL 0 . By rescaling we infer and Minkowski's inequality yields Since scaling and translation are continuous operations on L 2 (R 5 ), we infer as s → 0+ for any fixed x ′ ∈ R 5 . Consequently, by dominated convergence, we obtain as s → 0+. The same argument yields e sL 0 f − f H k (R 5 ) → 0 as s → 0+. We conclude that e sL 0 is strongly continuous on H k (R 5 ). Evidently, the map f → V 0 f is bounded on H k (R 5 ) and thus, by the bounded perturbation theorem, e sL is a strongly continuous semigroup on H k (R 5 ).
Now we claim the estimate for all R ≥ 1. To prove this, we note that ∆(V 0 f ) = ∆V 0 f + 2∇V 0 ∇f + V 0 ∆f and estimate each of these terms individually. Clearly, where we have used Lemma 3.2 and the decay |V 0 (x)| x −2 . Next, . Thanks to the decay |∇V 0 (x)| x −3 , the last term can be estimated as before. For the first term we use the decay of ∇V 0 , Lemma 3.2, and Hardy's inequality to estimate . In view of the decay |∆V 0 (x)| x −4 , the term (∆V 0 f |∆f ) L 2 (R 5 ) can be estimated analogously. This proves Eq. (3.8).
Having Eq. (3.8) at our disposal, we obtain from Eq. (3.7) the bound By approximation, Eq. (3.9) extends to all f ∈ D(L) satisfying Lf, f ∈ H 2 (R 5 ). From Lemma 3.8 we know that Le sL f, e sL f ∈ H 2 (R 5 ) and Eq. (3.9) yields by choosing R ≥ 1 sufficiently large. From now on R is fixed and hence, C R = C. Upon setting c 1 = 1 2 min{c 0 , 1 8 } > 0, we infer and this inequality may be rewritten as Consequently, integration yields the bound By a density argument, this bound holds for all f ∈ D(L) ∩Ḣ 2 (R 5 ).
It is now straightforward to upgrade toḢ 4 .
14 Proof. Let f ∈ C ∞ c (R 5 ). By applying the commutator relation [∆, Λ]f = ∆f twice, we obtain the estimate Eq. (3.7). Consequently, it suffices to follow the logic in the proof of Lemma 3.9 and apply Lemma 3.6.
3.4. Control of the linearized flow. Finally, we arrive at the main result on the linearized flow. First, we define the main Sobolev space we will be working with and prove an elementary embedding result.
Definition 3.11. The Banach space X is defined as the completion of all radial functions in C ∞ c (R 5 ) with respect to the norm Lemma 3.12. Let s ∈ [0, 3 2 ). Then we have the bound Now we can prove the following simple but useful embedding theorem. Lemma 3.13. We have the continuous embeddings . Then there exists a sequence (f n ) n∈N ⊂ C ∞ c (R 5 ) of radial functions such that f n → f with respect to · H 4 (R 5 ) . This implies that (f n ) n∈N is Cauchy with respect to · X and thus, there exists a limiting elementf ∈ X such that f n →f in X. We define a map ι : H 4 rad (R 5 ) → X by setting ι(f ) :=f . Obviously, ι is linear. We claim that ι is injective. Indeed, if ι(f ) = 0, there exists a sequence (f n ) n∈N ⊂ C ∞ c (R 5 ) that converges to f in H 4 rad (R 5 ) and to 0 in X. By Lemma 3.12 we see that lim n→∞ f n L ∞ (R 5 ) = 0. In particular, f n ⇀ 0 in L 2 (R 5 ). On the other hand, f n → f in H 4 rad (R 5 ) implies f n ⇀ f in L 2 (R 5 ) and the uniqueness of weak limits shows that f = 0. Clearly, we have ι(f ) X f H 4 (R 5 ) and thus, ι : H 4 rad (R 5 ) → X is a continuous embedding. The second assertion is proved similarly. Indeed, given f ∈ X we find a sequence (f n ) n∈N ⊂ C ∞ c (R 5 ) such that f n → f in X. By Lemma 3.12, (f n ) n∈N is Cauchy in W 1,∞ (R 5 ) and therefore converges to a limiting functionf ∈ C 1 (R 5 ) ∩ W 1,∞ (R 5 ). Using this, we define an inclusion map ι : X → C 1 (R 5 ) ∩ W 1,∞ (R 5 ) by setting ι(f ) :=f . It remains to show that ι is injective. If ι(f ) = 0, it follows that there exists a sequence (f n ) n∈N ⊂ C ∞ c (R 5 ) that converges to f in X and to 0 in L ∞ (R 5 ). Consequently, for any ϕ ∈ C ∞ c (R 5 ) and thus, ∆f n ⇀ 0 in L 2 (R 5 ). Analogously, we obtain ∆ 2 f n ⇀ 0 in L 2 (R 5 ). By the uniqueness of weak limits we therefore have lim n→∞ f n X = 0 and this shows f = 0.
Theorem 3.14. The Sobolev space X is invariant under e sL and there exists a constant ω 0 > 0 such that e sL f X e −ω 0 s f X for all s ≥ 0 and all f ∈ X satisfying (f |ψ 1 ) L 2 σ (R 5 ) = 0. Proof. By Lemma 3.13, D(L 2 ) ∩H 4 (R 5 ) ֒→ X. Since the former space is invariant under e sL , see Lemma 3.8, it follows that e sL f ∈ X for all s ≥ 0 and all f ∈ C ∞ c (R 5 ). Consequently, in view of Lemmas 3.9, 3.10, and a density argument, it suffices to prove the bound Thanks to the strong decay of the weight σ(x) = e −|x| 2 /4 , we immediately obtain f X by Hardy's inequality.

The nonlinear evolution
Now we turn to the full nonlinear problem Eq. (2.5). As before with the linear operator, we switch to 5-dimensional notation and define the nonlinearity N , acting on functions f : R 5 → R, by With this convention, Eq. (2.5) can be written as So far, this is purely formal. In what follows we first prove basic embedding theorems and then some Moser-type inequalities. These will allow us to show that the nonlinearity is locally Lipschitz on X. Next, we study mapping properties of the "initial data operator" U and finally, we implement an infinite-dimensional version of the Lyapunov-Perron method to prove global existence for Eq. (4.1).

Further properties of the space X.
Corollary 4.1 (Algebra property). We have the bound As a consequence, X is a Banach algebra.
Proof. This is a straightforward consequence of the Leibniz rule, the Gagliardo-Nirenberg inequality (see e.g. [41]), and Lemma 3.12.
Next, we prove weighted L ∞ bounds outside of balls. As opposed to Lemma 3.12 and Corollary 4.1, the restriction to radial functions is crucial here.
Lemma 4.2. We have the bounds for all radial f ∈ C ∞ c (R 5 ).
Proof. Let f ∈ C ∞ c (R 5 ) be radial and write f (x) =f (|x|). The fundamental theorem of calculus yieldsf and thus, by Cauchy-Schwarz, for all r ≥ 1. This implies the first assertion.
For the second statement we proceed similarly and usẽ to obtain the bound for all r ≥ 1. Now note thatf ′ (|x|) = x j |x| ∂ j f (x) and thus, by Hardy's inequality, we infer for all r ≥ 1, which is the desired result.

Nonlinear estimates.
For δ > 0 we set The goal of this section is to prove that the nonlinearity N is locally Lipschitz on X.
The key results in this respect are the following Moser-type inequalities. First, we focus on large radii where we need to assume a decay property.
Then we have the bound Proof. Let f, g ∈ X 1 ∩ C ∞ c (R 5 ) and set I(f )(x) := Φ(|x|f (x), x). Then we have and Consequently, it suffices to prove where Ω := R 5 \ B 5 . We start with the estimate for I(f ). By the chain rule, where F (x) = |x|f (x). The strategy is to use Lemma 4.2 to absorb the growing weight in F . We consider gh∆I(f ) L 2 (Ω) and estimate g X h X . Next, we estimate gh∆ 2 I(f ) L 2 (Ω) . The easy terms are It remains to control the most delicate term, |∆F | 2 . For this one we use Hardy and Lemma 4.2 to obtain g X h X f 2 X . The above estimates easily imply ∆ 2 [ghI(f )] L 2 (Ω) g X h X . Putting everything together, we arrive at the desired ghI(f ) Ḣ2 (Ω)∩Ḣ 4 (Ω) g X h X . The bound on J (f, g) is proved in the exact same way.
The next bound controls the nonlinearity near the center. Here the issue is to handle powers of | · | −1 that arise by differentiation.
In fact, we need a slightly more general form of Lemma 4.4.
We are now in a position to prove that the nonlinearity N is locally Lipschitz on X.
Lemma 4.6. We have the bound for all f, g ∈ X 1 .