On the regularity of critical points for O'Hara's knot energies: From smoothness to analyticity
aa r X i v : . [ m a t h . A P ] A p r On the regularity of critical points for O’Hara’s knotenergies: From smoothness to analyticity.
Nicole Vorderobermeier
Abstract.
We prove the analyticity of smooth critical points for O’Hara’sknot energies E α,p , with p = 1 and < α < , subject to a fixed lengthconstraint. This implies, together with the main result in [BR13], that boundedenergy critical points of E α, subject to a fixed length constraint are not only C ∞ but also analytic. Our approach is based on Cauchy’s method of majorantsand a decomposition of the gradient that was adapted from the Möbius energycase E , in [BV19]. Introduction
Knots have always played an important role in arts and crafts, commerce andtrade, as well as in our everyday life. Therefore they naturally became a topic ofinterest for mathematicians. During the 19th century the study of knots strongly in-fluenced the development of topology [TvdG96]. Today the theory of knots appearsin several branches of mathematics such as in calculus of variations, geometric anal-ysis, topology as well as in applications to modern quantum physics (e.g. [Kau05])and biochemistry (e.g. protein molecules [KS98] or DNA [CKS98, GM99]).A knot in mathematical terms is a Jordan curve in the three-dimensional Eu-clidean space (i.e. a continuous embedding of the circle R / Z into R ). We say thattwo given knots belong to the same knot class if one knot can be deformed intothe other without any ‘cuttings and gluings’ nor any self-intersections. Within thiscontext, the following two questions arise: Is it possible to determine ‘nicely’ shapedrepresentatives of each knot class? And if so, how ‘nice’ are they?The first question was originally addressed by Fukuhara [Fuk88] in the context ofpolygonal knots. In order to detect optimal shapes of a polygonal knot, an energymodelling a form of self-avoidance on the space of polygonal knots was introduced—from which optimal shapes can be identified as energy minimizers. Subsequently,O’Hara [O’H91, O’H92] extended Fukuhara’s approach to geometric knots. For anyJordan curve γ : R / Z → R , he introduced the potential energies E α,p ( γ ) = Z Z ( R / Z ) (cid:18) | γ ( x ) − γ ( y ) | α − D ( γ ( x ) , γ ( y )) α (cid:19) p | ˙ γ ( x ) || ˙ γ ( y ) | d x d y. (1.1) Date : May 1, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Analyticity, knot energy, O’Hara’s knot energies, method of majorants,fractional Leibniz rule.The author acknowledges support by the Austrian Science Fund (FWF), Grant P 29487.
1n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 2
Here the quantity D ( γ ( x ) , γ ( y )) denotes the intrinsic distance between the points γ ( x ) and γ ( y ) along the curve, i.e. for | x − y | ≤ we have D ( γ ( x ) , γ ( y )) = min {L ( γ | [ x,y ] ) , L ( γ ) − L ( γ | [ x,y ] ) } , where L ( γ ) = R | ˙ γ ( t ) | d t denotes the length of the curve γ . All values of E α,p are non-negative due to the fact that the intrinsic distance between two points ofthe curve is always greater than the Euclidean distance. The factor | γ ′ ( x ) || γ ′ ( y ) | guarantees the invariance of E α,p under reparametrization of the curve. In addition, E = E , is called Möbius energy since it is invariant under Möbius transformations(cf. [FHW94, Theor. 2.1]). For p ≥ and < αp < p + 1 the energies E α,p areglobally minimized by circles, whereas for αp ≥ p + 1 their values become infinitefor every closed regular curve (cf. [ACF+03, Corol. 3]).To distinguish between knot classes, it is desirable for a knot energy to be self-repulsive (i.e. to penalize self-intersections) and tight (i.e. to blow up on a sequenceof small knots that pull tight). O’Hara showed that the knot energies E α,p areindeed self-repulsive for the cases p ≥ α , < α ≤ and α − > p ≥ α , < α ≤ (cf. [O’H94, Theor. 1.1]) and tight if and only if αp > (cf. [O’H92, Theor. 3.1]).Moreover, [O’H94, Theor. 3.2] showed that for αp > there exist minimizers ofthe energies within every knot class among all curves with fixed length. In case ofthe Möbius energy, we also have self-repulsiveness but not tightness (cf. [O’H94,Theor. 3.1]) and we only know that there exist minimizers in prime knot classes(cf. [FHW94, Theor. 4.3]). We see that due to the well-definedness and existence ofminimizers of a certain range of O’Hara’s energies E α,p , this approach for answeringour first question looks promising.In the following we want to focus on the second question by determining theregularity properties of minimizers of the knot energies. For that reason we restrictourselves to examining O’Hara’s energies E α = E α, for ≤ α < as they arewell-defined in a knot-theoretic sense and have a non-degenerate first variation (incontrast to the p > case). For the Möbius energy, one of the first regularity resultsgoes back to He [He00, Chap. 5] who states, based on Freedman, He and Wang’swork [FHW94, Theor. 5.4], that any local minimizer γ of E with respect to the L ∞ -topology belongs to C ∞ ( R / Z , R ) . Reiter [Rei12] generalized this result to theclass of O’Hara’s knot energies E α for ≤ α < and n ≥ by showing that anycritical point γ of E α in the class H α ( R / Z , R n ) with γ ′′ ∈ L ( R / Z , R n ) is smooth.A subsequent improvement on the energy space conditions furnishes the following C ∞ -result. Theorem 1.1 (Blatt and Reiter [BR13]) . Let γ : R / Z → R n be a simple closedLipschitz-continuous arc-length parametrized curve with γ ∈ H α +12 , ( R / Z , R n ) forany < α < . If γ is a critical point of E α + λ L , i.e. δ E α ( γ ; h ) + λ Z R / Z h ˙ γ, ˙ h i d x = 0 for all h ∈ H α +12 , ( R / Z , R n ) , then γ belongs to C ∞ ( R / Z , R n ) . An important characterization of curves with finite O’Hara energy goes back toBlatt [Bla12] who showed that curves γ have finite energy E α ( γ ) < ∞ if and only ifthey belong to the fractional Sobolev space H α +12 , ( R / Z , R n ) . This result, combinedwith Theorem 1.1, raises the question whether critical points of O’Hara’s energies E α of finite energy are not only smooth but also analytic. For the Möbius energy n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 3 E this question was solved in the affirmative by [BV19]. One might conjecturethat it is possible to transfer this result to the energy classes E α for < α < .Unlike the Möbius energy case, the first variation of E α for < α < leads to theappearance of a fractional derivative of the product of two functions. The latteris subsequently analyzed via a new fractional Leibniz rule instead of the bilinearHilbert transform, which was used in the Möbius energy case.The main result of this article confirms the above analyticity conjecture, namely: Theorem 1.2.
Let γ : R / Z → R n be a closed simple arc-length parametrized curvein C ∞ ( R / Z , R n ) . If γ is a critical point of O’Hara’s energy E α , < α < , with alength term L , i.e. E α + λ L , then the curve γ is analytic. This theorem, together with the characterization of the energy spaces in [Bla12]and Theorem 1.1, implies the following:
Corollary 1.3.
Let γ : R / Z → R n be a closed simple arc-length parametrized curvewith E α ( γ ) < ∞ for < α < . If γ is a critical point of E α + λ L , then the curve γ is analytic. Note that we study critical points of O’Hara’s energy E α subject to a Lagrangemultiplier length term since E α for < α < is not scale-invariant in contrast tothe Möbius energy case. Exposé of the present work.
The main goal of this article is to give a rigorousproof of Theorem 1.2. We adapt the methods from the Möbius energy case in[BV19] to generate a proof which is as elementary as possible. For the convenienceof the reader we will provide detailed proofs in this article on the one hand toemphasize the differences to [BV19] and, on the other hand, to make the articlecomprehensible without the need for a detailed reading of [BV19].In Section 2 we recall some basic definitions and properties of fractional Sobolevspaces and Fourier series. We also characterize analytic functions and state an or-dinary differential equation (ODE)-version of the theorem of Cauchy-Kovalevskaya.We close the preliminaries with recapitulating Faà di Bruno’s formula. In Section3 we decompose the first variation of O’Hara’s energies into the orthogonal projec-tion of a main part Q α and two remaining parts R α and R α of lower order. Themain part Q α and its derivatives are estimated in Section 4. Since the orthogonalprojection of Q α appears in the first variation of O’Hara’s energies, we estimatethe tangential part of Q α using new estimates for a kind of fractional derivative ofa product of two functions that, in Section 5, that behave like a fractional Leibnizrule. In Section 5 and Section 6 we cut-off the singularities of the singular integralsinvolved and derive uniform estimates. More precisely, in Section 6 we rewrite theorthogonal projection of the truncated remaining terms so that they may be ex-pressed by integrals over analytic functions. In Section 7 we realize that estimatesdo not depend on the cut-off parameter and hold for the orthogonal projectionsof Q α , R α , R α , and its derivatives. Finally, we will use the estimates to proveTheorem 1.2 using Cauchy’s method of majorants.2. Preliminaries
Fractional Sobolev spaces.
We denote by | · | the Euclidean norm on C n for any integer n ≥ . In this article we work with closed curves on the periodicdomain R / Z so that a curve f : R → R n is periodic with unit periodicity. n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 4 We know that for any f ∈ L ( R / Z , C n ) the Fourier series of f in x ∈ R / Z isgiven by P k ∈ Z b f ( k ) e πikx , where the k -th Fourier coefficient of f are given by b f ( k ) = Z f ( x ) e − πikx d x. The fractional Sobolev space of order s ≥ (i.e. the Bessel potential space oforder s ≥ ) is defined as H s ( R / Z , C n ) := { f ∈ L ( R / Z , C n ) | k f k H s := k f k H s ( R / Z , C n ) := p ( f, f ) H s < ∞} , equipped with the scalar product ( f, g ) H s := ( f, g ) H s ( R / Z , C n ) := X k ∈ Z (1 + k ) s h b f ( k ) , b g ( k ) i C n . Furthermore, we will need the embeddings H s ( R / Z , C n ) ⊆ H t ( R / Z , C n ) forany ≤ t < s (cf. [DNPV12, Prop. 2.1, Corol. 2.3]) as well as H s ( R / Z , C n ) ⊆ C ( R / Z , C n ) for any s > (cf. [DNPV12, Theor. 8.2]), from which we can deducethat the Fourier series of any f ∈ H s ( R / Z , C ) with s > converges absolutely anduniformly to f . In fact the space H m , for any integer m ≥ , coincides with theclassical Sobolev space and the norms k · k H m and k f k W m := ( P mν =0 k ∂ ν f k L ) are equivalent (cf. [Rei09, Lem. 1.2] or [AF03, 7.62]). Furthermore, let us alsomention the Banach algebra property of H m for any integer m ≥ , i.e. there existsa positive constant C m such that k f g k H m ≤ C m k f k H m k g k H m (2.1)for all f, g ∈ H m ( R / Z , R ) (cf. [AF03, Theor. 4.39]).2.2. Properties of analytic functions.
We shortly recall some basic character-istics of analytic functions. More information on analytic functions can be foundfor instance in [Eva10, Chap. 6.4] and [Fol95, Chap. 1, D].It is well-known that analytic functions can be characterized by [KP02, Prop. 2.2.10]and a standard covering argument as follows:
Theorem 2.1.
A function f ∈ C ∞ (Ω , R n ) on Ω ⊆ R m open, n, m ≥ , is analyticon Ω if and only if for every compact set K ⊆ Ω there are positive constants r K and C K such that k ∂ α f k L ∞ ( K ) ≤ C K | α | ! r | α | K holds for every multiindex α ∈ N m . The previous theorem, together with the embedding of the classical Sobolevspace W k = W k, into C for k ∈ N with k > m , yields the following: Corollary 2.2.
A function f ∈ C ∞ (Ω , R n ) on Ω ⊆ R m open, n, m ≥ , is analyticon Ω if for every compact set K ⊂ Ω there are positive constants r K and C K suchthat k ∂ α f k W ( K ) ≤ C K | α | ! r | α | K holds for every multiindex α ∈ N m . n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 5 By the equivalence of the W - and the H -norm on R / Z as well as the embedding H s ⊆ H t for any t < s we obtain: Corollary 2.3.
Let f ∈ C ∞ ( R / Z , R n ) and s ∈ (0 , . Then the function f isanalytic on R / Z if there are positive constants r and C such that k ∂ k f k H s ≤ C k ! r k holds for all integers k ≥ . Furthermore, we will need a ODE version of the theorem of Cauchy-Kovalevskaya,which is originally an existence and uniqueness theorem for analytic nonlinear par-tial differential equations associated with Cauchy initial value problems.
Theorem 2.4 (Cauchy-Kovalevskaya - ODE case) . Suppose the function g : R n → R n is real analytic around and the function f ∈ C ∞ (( − ε, ε ) , R n ) of the form f ( x ) = ( f ( x ) , . . . , f n ( x )) , for any ε > , is a solution of the initial value problem ˙ f ( x ) = g ( f ( x )) for x ∈ ( − ε, ε ) with f (0) = 0 . Then the function f is real analytic around . One possible method to prove the theorem of Cauchy-Kovalevskaya is the methodof majorants (c.f. [Fol95, Chap. 1, D] or [Eva10, Chap. 4.6.3, Theor. 2]). Thismethod turns out to be useful in proving the main result of this article.2.3.
Faà di Bruno’s formula.
The k -th derivative of the composition of twofunctions f, g ∈ C k ( R , R ) can be expressed by Faà di Bruno’s formula which isgiven by (cid:18) ddt (cid:19) k g ( f ( t )) = X m +2 m + ··· km k = k,m ,...,m k ∈ N k ! m !1! m m !2! m . . . m k ! k ! m k g ( m + ··· + m k ) ( f ( t )) k Y j =1 ( f ( j ) ( t )) m j . This formula can be generalized to the multivariate case by a result of [Mis00]. Inour case the precise generalized Faà di Bruno’s formula is not required. Nonethelesswe will make use of the the following that can be easily proven from scratch byinduction:For any functions f ∈ C k ( R , R n ) and g ∈ C k ( R n , R ) , for integers n ≥ and k ≥ , there exists a universial polynomial p ( n ) k with non-negative coefficients,which are independent of f and g , such that ∂ k g ( f ( x )) = p ( n ) k ( { ∂ α g } | α |≤ k , { ∂ j f i } i =1 ,...,n,j =1 ,...,k ) . (2.2)In addition, p ( n ) k is one-homogeneous in the first entries. n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 6 Decomposition of the first variation of E α Results concerning the differentiability of O’Hara’s knot energies go back tothe paper [FHW94] where the Gâteaux differentiability and the L -gradient of theMöbius energy E was derived. Subsequently, a linearized version for the gradientof the Möbius energy was given by He [He00, Lem. 2.2] in the context of the heatflow. An application of He’s linearization trick to a range of O’Hara’s knot energies E α , for ≤ α < , was given by Reiter [Rei12, Chap. 2].In the following we will use the linearization trick of He and Reiter to decomposethe first variation of the energies E α , for < α < , into a highest order quasilinearpart Q α and two remaining R α , R α parts of lower order (as it was done in [BR13]and [Bla18, Theor. 2.3]). A similar decomposition of the first variation proved aptfor the analyticity proof of critical points of the Möbius energy that was given in[BV19].Let γ ∈ C ∞ ( R / Z , R n ) be a simple closed arc-length parametrized curve and x ∈ R / Z . We recall that the orthogonal projection P ⊥ ˙ γ ( x ) : R n → R n at the point γ ( x ) onto the normal space of the curve γ is given by ( P ⊥ ˙ γ v )( x ) = P ⊥ ˙ γ ( x ) ( v ) = v − h v, ˙ γ ( x ) i R n ˙ γ ( x ) for any v ∈ R n . Furthermore, the tangential projection P T ˙ γ : R n → R n at the point γ ( x ) onto the tangent space of the curve γ is given by ( P T ˙ γ v )( x ) = P T ˙ γ ( x ) ( v ) = h v, ˙ γ ( x ) i R n ˙ γ ( x ) for any v ∈ R n . We remark also that the curvature vector of the curve γ is denotedby κ i.e. κ = ( dds ) γ where dds is the derivative with respect to the arc-length. Werecall the following: Theorem 3.1 (Reiter [Rei12, Theor. 2.24]) . The first variation of E α , for < α < , at a simple regular curve γ ∈ C ∞ ( R / Z , R n ) in direction h ∈ H ( R / Z , R n ) canbe expressed as δ E α ( γ, h ) = Z h ( H α γ )( x ) , h ( x ) i R n | ˙ γ ( x ) | d x, where ( H α γ )( x ) := lim ǫ ↓ Z | w |∈ [ ǫ, ] P ⊥ ˙ γ ( x ) { α γ ( x + w ) − γ ( x ) | γ ( x + w ) − γ ( x ) | α − ( α − κ ( x ) D ( γ ( x + w ) , γ ( x )) α − κ ( x ) | γ ( x + w ) − γ ( x ) | α }| ˙ γ ( x + w ) | d w. Due to the the fact that P ⊥ ˙ γ ( x ) ( ˙ γ ( x )) = 0 , h ˙ γ ( x ) , ¨ γ ( x ) i R n = 0 , the curve γ isparametrized by arc-length and the orthogonal projection P ⊥ ˙ γ ( x ) is linear, one caneasily rewrite H α γ as ( H α γ )( x ) = ( P ⊥ ˙ γ e H α γ )( x ) = P ⊥ ˙ γ ( x ) (( e H α γ )( x )) , (3.1)where ( e H α γ )( x ) := lim ε ↓ Z | w |∈ [ ε, ] { α γ ( x + w ) − γ ( x ) − w ˙ γ ( x ) | γ ( x + w ) − γ ( x ) | α − ( α −
2) ¨ γ ( x ) D ( γ ( x + w ) , γ ( x )) α − γ ( x ) | γ ( x + w ) − γ ( x ) | α } d w. n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 7 Now we want to apply the strategy of He and Reiter indicated as in the above.We decompose e H α γ pointwise for all x ∈ R / Z into three functionals by e H α γ = αQ α γ + 2 αR α γ − R α γ, (3.2)where Q α γ ( x ) = lim ε ↓ Q α,ε γ ( x ) R α γ ( x ) = lim ε ↓ R α,ε γ ( x ) R α γ ( x ) = lim ε ↓ R α,ε γ ( x ) are given, for any < ε ≤ , by ( Q α,ε γ )( x ) = Z | w |∈ [ ε, ] (cid:18) γ ( x + w ) − γ ( x ) − w ˙ γ ( x ) w − ¨ γ ( x ) (cid:19) d w | w | α , ( R α,ε γ )( x ) = Z | w |∈ [ ε, ] ( γ ( x + w ) − γ ( x ) − w ˙ γ ( x )) (cid:18) | γ ( x + w ) − γ ( x ) | α +2 − | w | α +2 (cid:19) d w, ( R α,ε γ )( x ) = Z | w |∈ [ ε, ] ¨ γ ( x ) (cid:18) | γ ( x + w ) − γ ( x ) | α − | w | α (cid:19) d w Blatt and Reiter [BR13] have already used this partitioning for studying the reg-ularity of stationary points of O’Hara’s energies E α , < α < , and furthermore,Blatt (cf. [Bla18, Theor. 2.3]) used it in the study of the gradient flow for the samerange of O’Hara’s knot energies.In the next section we will see that Q α contains the highest order part of thefirst variation of E α . Thereafter the challenge will be to attain sufficient estimatesof the tangential part of Q α , whereas R α and R α are of lower order and easier toget under control. 4. An estimate for the Q α term Let the quantities q ( α ) k := ( π | k | α +1 R π kτ ) − kτ ) τ α d τ k = 0 , k = 0 and set λ ( α ) k := Z kπ sin( τ ) τ α − d τ. We remark that the λ k are well-defined for all k ∈ Z and λ ( α ) ∞ := lim k →∞ λ ( α ) k < ∞ (cf. [Rei12, Lem. 2.4]). We deduce from [Rei12, Prop. 2.3], by switching from theweak formulation of the Euler-Lagrange equation to the strong one, the following: Theorem 4.1.
For every curve γ ∈ C ∞ ( R / Z , R n ) the term Q α γ is a C ∞ -functionand its Fourier coefficients, for all k ∈ Z , are given by d Q α γ ( k ) = − q ( α ) k | k | α +1 ˆ γ ( k ) . (4.1) The constants q ( α ) k are bounded and satisfy ≤ q ( α ) k = 8 πλ ( α ) k α ( α + 1)( α −
1) + O ( k − α ) n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 8 as k → ∞ .Proof. By Taylor’s expansion up to third order we obtain Q α γ ( x ) = lim ε ↓ Z | w |∈ [ ε, ] w | w | α Z (1 − t ) ... γ ( x + tw )d t d w = lim ε ↓ Z | w |∈ [ ε, ] w | w | α Z (1 − t ) ( ... γ ( x + tw ) − ... γ ( x )) d t d w, (4.2)from which directly follows that Q α maps C ,β to L ∞ for any < β ≤ . From thelinearity of Q α we get ∂ l Q α γ ( x ) = Q α ∂ l γ ( x ) , so Q α maps C l +3 ,β to C l − , for allintegers l ≥ , which demonstrates the first part of the Theorem’s statement.Now we define the bilinear functional e Q α ( γ, η ) := lim ε ↓ Z R / Z Z | w |∈ [ ε, ] (cid:16) h γ ( x + w ) − γ ( x ) , η ( x + w ) − η ( x ) i R n w − h ˙ γ ( x ) , ˙ η ( x ) i R n (cid:17) d w d x | w | α . for any γ, η ∈ H α +12 ( R / Z , R n ) . Applying continuous and discrete integration byparts, gives e Q α ( γ, η ) = lim ε ↓ Z R / Z Z | w |∈ [ ε, ] (cid:18) − h γ ( x + w ) − γ ( x ) , η ( x ) i R n w − h ˙ γ ( x ) , ˙ η ( x ) i R n (cid:19) d w d x | w | α = − lim ε ↓ Z R / Z h Q α,ε γ ( x ) , η ( x ) i R n d x = − Z R / Z h Q α γ ( x ) , η ( x ) i R n d x (4.3)because Q α,ε γ converges to Q α γ in L ∞ ( R / Z , R n ) as ǫ ↓ by (4.2). Furthermore,by [Rei12, Prop. 2.3], we have that e Q α ( γ, η ) = X k ∈ Z q ( α ) k | k | α +1 ˆ γ ( k )ˆ η ( k ) and by Plancherel’s identity we get Z R / Z h Q α γ, η i dx = X k ∈ Z d Q α γ ( k ) b η ( k ) . So by comparing the Fourier coefficients in (4.3) we have d Q α f ( k ) = − q ( α ) k | k | α +1 ˆ γ ( k ) for all k ∈ Z . The properties of the coefficients q ( α ) k directly follow from the proofof [Rei12, Prop. 2.3]. (cid:3) By Theorem 4.1 we obtain the following essential corollary.
Corollary 4.2.
There exists a positive constant e C such that for all curves γ ∈ C ∞ ( R / Z , R n ) and integers l ≥ k ∂ l +3 γ k H m + α − ≤ e C k ∂ l Q α γ k H m holds for any real numbers m ≥ and < α < .Proof. By considering Theorem 4.1, in particular the form of the Fourier coefficients(4.1), the definition of the fractional Sobolev-norm, elementary properties of Fourier n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 9 coefficients and the simple estimate (1 + k ) α − ≤ (2 | k | ) α − for any k ∈ Z \ ,we get k ∂ l Q α γ k H m = X k ∈ Z (1 + | k | ) m | \ ∂ l Q α γ ( k ) | = X k ∈ Z (1 + | k | ) m (2 π | k | ) l | d Q α γ ( k ) | = X k ∈ Z (1 + | k | ) m (2 π | k | ) l ( q ( α ) k ) | k | α +1) | ˆ γ ( k ) | ≥ inf k ∈ Z \ { ( q ( α ) k ) (2 π ) − − α } X k ∈ Z (1 + | k | ) m + α − (2 π | k | ) l +3) | ˆ γ ( k ) | = e C − X k ∈ Z (1 + | k | ) m + α − | \ ∂ ( l +3) γ ( k ) | = e C − k ∂ ( l +3) γ k H m + α − , where e C := inf k ∈ Z \ { ( q ( α ) k ) (2 π ) − − α } − is a positive constant. (cid:3) A fractional leibniz rule and the form of P T ˙ γ Q α γ Let γ ∈ C ∞ ( R / Z , R n ) be a closed simple curve parametrized by arc-length.Recall that in the decomposed first variation of O’Hara’s range of energies E α , for < α < , there appears the orthogonal projection of the main part Q α γ , whichis given by P ⊥ ˙ γ Q α γ = Q α γ − P T ˙ γ Q α γ . Since we have worked out an estimate for Q α γ in the previous section, it remains to study P T ˙ γ Q α γ , i.e. the tangential part of Q α γ .In this section we will see that a type of fractional derivative of a product oftwo functions can help to estimate the tangential part of Q α γ . We interpret theresulting estimate as a fractional Leibniz rule .In order to avoid problems coming from the singularities of the integrand, we willwork with the truncated functional Q α,ε for < ε ≤ . By using Taylor’s approxi-mation up to second order with remainder term in integral from and h ˙ γ ( x ) , ¨ γ ( x ) i = 0 for all x ∈ R / Z together with the bilinearity of the scalar product and α > , wecan write h Q α,ε γ ( x ) , ˙ γ ( x ) i R n = Z | w |∈ [ ε, ] (cid:28) γ ( x + w ) − γ ( x ) − w ˙ γ ( x ) w − ¨ γ ( x ) , ˙ γ ( x ) (cid:29) R n d w | w | α = 2 Z | w |∈ [ ε, ] Z (1 − t ) h ¨ γ ( x + tw ) , ˙ γ ( x ) i R n d t d w | w | α = 2 Z | w |∈ [ ε, ] Z (1 − t ) (cid:28) ¨ γ ( x + tw ) , ˙ γ ( x ) − ˙ γ ( x + tw ) w (cid:29) R n d t d ww | w | α − = 2 Z | w |∈ [ ε, ] Z Z (1 − t )( − t ) h ¨ γ ( x + tw ) , ¨ γ ( x + stw ) i R n w d s d t d w | w | α − . (5.1) n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 10 With ε ↓ we also get ( P T ˙ γ Qγ )( x )= 4 lim ε ↓ Z | w |∈ [ ε, ] Z Z [0 , (1 − t )( − t ) h ¨ γ ( x + tw ) , ¨ γ ( x + stw ) i R n w | w | α − ˙ γ ( x )d s d t d w. (5.2)The last terms of (5.1) and (5.2) motivate us to introduce the following: Definition 5.1.
Let s , s ∈ [0 , , < ε ≤ and β ∈ (0 , . Then the bilinearsingular integral H s ,s ,β : C ( R / Z , R ) × C ( R / Z , R ) → L ∞ ( R / Z , R ) is given by H s ,s ,β ( f, g )( x ) := lim ε ↓ Z | w |∈ [ ε, ] f ( x + s w ) g ( x + s w ) w | w | β d w (5.3)for all x ∈ R / Z and, its truncated version H εs ,s ,β : C ( R / Z , R ) × C ( R / Z , R ) → L ∞ ( R / Z , R ) is given by H εs ,s ,β ( f, g )( x ) := Z | w |∈ [ ε, ] f ( x + s w ) g ( x + s w ) w | w | β d w (5.4)for all x ∈ R / Z .If β was allowed to vanish, the previous definition would yield the bilinear Hilberttransform. Since we have the factor | w | β for β ∈ (0 , in the denominator in thedefinition of the bilinear singular integral (5.3), the formula is reminiscent of afractional derivative. Lemma 5.2.
For every s , s ∈ [0 , , < ε ≤ and β ∈ (0 , the transforms H s ,s ,β and H εs ,s ,β are well-defined, continuous and bilinear from C ( R / Z , R ) × C ( R / Z , R ) to L ∞ ( R / Z , R ) . Furthermore, the truncated bilinear integral H εs ,s ,β is also continuous from C ( R / Z , R ) × C ( R / Z , R ) to L ∞ ( R / Z , R ) .Proof. It is easy to see by the linearity of the integral in (5.3) that H s ,s ,β isindeed linear in both components. The function H s ,s ,β is also well-defined forevery f, g ∈ C ( R / Z , R ) , which we can see by adding a zero to the definition of thebilinear singular integral (5.3) as H s ,s ,β ( f, g )( x ) = lim ε ↓ Z | w |∈ [ ε, ] f ( x + s w ) g ( x + s w ) w | w | β − f ( x ) g ( x ) w | w | β d w, by inserting a second zero and by the Lipschitz-continuity of continuously differen-tiable functions on a compact set such that | H s ,s ,β ( f, g )( x ) |≤ lim ε ↓ Z | w |∈ [ ε, ] | f ( x + s w ) g ( x + s w ) − f ( x ) g ( x + s w ) + f ( x ) g ( x + s w ) − f ( x ) g ( x ) || w | β d w ≤ lim ε ↓ Z | w |∈ [ ε, ] | f ( x + s w ) − f ( x ) || g ( x + s w ) | + | f ( x ) || g ( x + s w ) − g ( x ) || w | β d w ≤ lim ε ↓ Z | w |∈ [ ε, ] k f ′ k ∞ s | w |k g k ∞ + k f k ∞ k g ′ k ∞ s | w || w | β d w ≤ C ( s , s , β ) k f k C k g k C < ∞ (5.5)for some positive finite constant C ( s , s , β ) . Hence we can deduce from (5.5) that H s ,s ,β is continuous. We get the same properties for H εs ,s ,β for any < ε ≤ by n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 11 transfering the previous arguments. Additionally, since we cut off the singularityin the definition of H εs ,s ,β , the continuity of H εs ,s ,β from C ( R / Z , R ) × C ( R / Z , R ) to L ∞ ( R / Z , R ) follows by the following elementary estimates (cid:12)(cid:12) H εs ,s ,β ( f, g )( x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z | w |∈ [ ε, ] f ( x + s w ) g ( x + s w ) w | w | β − f ( x ) g ( x ) w | w | β d w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | w |∈ [ ε, ] | f ( x + s w ) || g ( x + s w ) | + | f ( x ) || g ( x ) || w | β d w ≤ Z | w |∈ [ ε, ] k f k ∞ k g k ∞ + k f k ∞ k g k ∞ | w | β d w ≤ C ( β, ε ) k f k L ∞ k g k L ∞ < ∞ for all x ∈ [0 , and some constant < C ( β, ε ) < ∞ . (cid:3) We next estimate the truncated bilinear integral (5.4) that leads to a new kindof fractional Leibniz rule.
Theorem 5.3.
Let m > , < ε ≤ , s , s ∈ [0 , and β ∈ (0 , . Then thereexists a positive constant C H = C H ( m, β ) < ∞ independent of ε such that k H εs ,s ,β ( f, g ) k H m ≤ C H k f k H m + β k g k H m + β (5.6) for all f, g ∈ C ∞ ( R / Z , R n ) . For the proof we will identify, for any p ∈ [1 , ∞ ) , the sequence space ℓ p by ℓ p := { x = ( x k ) k ∈ Z ∈ C Z | k x k ℓ p := ( P k ∈ Z | x k | p ) p < ∞} and use the following lemmata. Lemma 5.4 (Young’s inequality) . Let p, q, r ≥ such that p + q = r + 1 . Forsequences x ∈ ℓ p and y ∈ ℓ q the convolution ( x ∗ y )( k ) := P n ∈ Z x ( n ) y ( k − n ) issuch that x ∗ y ∈ ℓ r and k x ∗ y k ℓ r ≤ k x k ℓ p k y k ℓ q . Lemma 5.5 (Sobolev-type inequality [BV19, Lem. 5.5]) . Let f ∈ H m ( R / Z , R n ) for m > . Then ˆ f := ( ˆ f ( k )) k ∈ Z ∈ ℓ and there exists a positive constant C < ∞ such that k ˆ f k ℓ ≤ C k f k H m . Proof of Theorem 5.3.
Let f, g ∈ C ∞ ( R / Z , R ) ⊆ H m ( R / Z , R ) . Because of m > and [DNPV12, Theor. 8.2], the partial sums of the Fourier series, i.e. p n ( x ) := P nk = − n ˆ f ( k ) e − πikx and q n ( x ) := P nk = − n ˆ g ( k ) e − πikx for any n ∈ N and x ∈ R / Z ,converge uniformly to the functions f and g , that means k f − p n k ∞ → and k g − q n k ∞ → as n → ∞ . (5.7)Note that since f, g ∈ C ∞ ( R / Z , R ) , also p n , q n ∈ C ∞ ( R / Z , R ) holds.Our approach is to firstly prove the estimate (5.6) for the approximating func-tions p n and q n and secondly derive the actual statement of Theorem 5.3 by passingto the limit n → ∞ . We start by interchanging the integrals two times due to ε > n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 12 and taking account of the fact that the Fourier coefficients of a product of twofunctions are the convolution of their Fourier coefficients, to gain \ H εs ,s ,β ( p n , q n )( k )= Z Z | w |∈ [ ε, ] p n ( x + s w ) q n ( x + s w ) w | w | β e − πikx d w d x = Z | w |∈ [ ε, ] Z p n ( x + s w ) q n ( x + s w ) e − πikx d x d ww | w | β = Z | w |∈ [ ε, ] X l ∈ Z c p n ( l ) e πils w b q n ( k − l ) e πi ( k − l ) s w d ww | w | β = ( if | k | > n R | w |∈ [ ε, ] P nl = − n e πi ( ls +( k − l ) s ) w b f ( l ) b g ( k − l ) d ww | w | β if | k | ≤ n . (5.8)Now we focus on the non-trivial Fourier coeffients of H εs ,s ,β ( p n , q n ) , that meanson the indices | k | ≤ n . By defining φ l,k := 2 π ( ls + ( k − l ) s ) ∈ R , substitutionand interchanging integral and sum, we get Z | w |∈ [ ε, ] n X l = − n e iwφ l,k b f ( l ) b g ( k − l ) d ww | w | β = Z ε n X l = − n e iwφ l,k − e − iwφ l,k w | w | β b f ( l ) b g ( k − l )d w = Z ε n X l = − n i sin( wφ l,k ) w β b f ( l ) b g ( k − l )d w = n X l = − n b f ( l ) b g ( k − l )2 i Z ε sin( wφ l,k ) w β d w = n X l = − n b f ( l ) b g ( k − l )2 iφ βl,k Z φl,k φ l,k ε sin( e w ) e w β d e w = n X l = − n b f ( l ) b g ( k − l )2 iφ βl,k ( Si β (cid:16) φ l,k (cid:17) − Si β ( φ l,k ε )) , (5.9)where Si β ( x ) := R x t ) t β dt . Hence we can deduce from the previous computationsin (5.8) and (5.9) (cid:12)(cid:12)(cid:12) \ H εs ,s ,β ( p n , q n )( k ) (cid:12)(cid:12)(cid:12) ≤ n X l = − n | b f ( l ) || b g ( k − l ) | | φ l,k | β (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) Si β (cid:18) φ l,k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + | Si β ( φ l,k ε ) | (cid:19) ≤ ˜ M n X l = − n | b f ( l ) || b g ( k − l ) || φ l,k | β , (5.10)where ˜ M := 4 sup φ ∈ [0 , ∞ [ Si β ( φ ) < ∞ due to [Rei12, Lem. 2.4]. By (5.10) and | φ k,l | β = | π ( ls + ( k − l ) s ) | β ≤ π | l | β | k − l | β ≤ π ( l + 1) β (( k − l ) + 1) β , n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 13 we conclude (cid:12)(cid:12)(cid:12) \ H εs ,s ,β ( p n , q n )( k ) (cid:12)(cid:12)(cid:12) ≤ ˜ M n X l = − n | b f ( l ) || b g ( k − l ) || φ l,k | β ≤ M n X l = − n ( l + 1) β | b f ( l ) | (( k − l ) + 1) β | b g ( k − l ) | = M n X l = − n ( l + 1) β | c p n ( l ) | (( k − l ) + 1) β | b q n ( k − l ) | , (5.11)where M := 2 π ˜ M . In addition, we obtain by (5.11) and the elementary estimates ( k + 1) m ≤ m (cid:0) ( l + 1) m + (( k − l ) + 1) m (cid:1) , which are an immediate consequence of | k | ≤ {| l | , | k − l |} , that (cid:12)(cid:12)(cid:12) ( k + 1) m \ H εs ,s ,β ( p n , q n )( k ) (cid:12)(cid:12)(cid:12) ≤ M C ( m ) X l ∈ Z ( l + 1) m + β | c p n ( l ) | (( k − l ) + 1) β | b q n ( k − l ) | + M C ( m ) X l ∈ Z ( l + 1) β | c p n ( l ) | (( k − l ) + 1) m + β | b q n ( k − l ) | , (5.12)where the positive constant C ( m ) < ∞ only depends on the parameter m . Thisleads us to the idea to set the following series component-wise for all k ∈ Z andany λ > p λn ( k ) := ( k + 1) m + λ | c p n ( k ) | and q λn ( k ) := ( k + 1) m + λ | b q n ( k ) | such that we can rewrite (5.12) as (cid:12)(cid:12)(cid:12) ( k + 1) m \ H εs ,s ,β ( p n , q n )( k ) (cid:12)(cid:12)(cid:12) ≤ M C ( m ) (cid:0) ( p βn ∗ q n )( k ) + ( p n ∗ q βn )( k ) (cid:1) . Finally, we obtain the desired estimate (5.6) for p n and q n by applying Lemma 5.4,Lemma 5.5, and Sobolev’s embedding theorem to estimate k H εs ,s ,β ( p n , q n ) k H m = k (( k + 1) m \ H εs ,s ( p n , q n )( k )) k ∈ Z k ℓ ≤ M C ( m ) (cid:0) k (( p βn ∗ q n )( k )) k ∈ Z k ℓ + k (( p n ∗ q βn )( k )) k ∈ Z k ℓ (cid:1) ≤ M C ( m ) (cid:0) k p βn k ℓ k q n k ℓ + k p n k ℓ k q βn k ℓ (cid:1) ≤ M C ( m ) C ( k p n k H m + β k q n k H m + k p n k H m k q n k H m + β )= C H k p n k H m + β k q n k H m + β , (5.13)where C H := 2 M C ( m ) C is a constant only depending on m and β .Finally, we get the statement (5.6) for f and g by passing to the limit n → ∞ .In particular, by using (5.13), we find for any integer n ≥ that k H εs ,s ,β ( p n , q n ) k H m ≤ C H k p n k H m + β k q n k H m + β ≤ C H k f k H m + β k g k H m + β (5.14) n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 14 By uniform convergence and by Lemma 5.2 we find that k H εs ,s ,β ( f, g ) − H εs ,s ,β ( p n , q n ) k ∞ ≤ k H εs ,s ,β ( f, g ) − H εs ,s ,β ( p n , g ) k ∞ + k H εs ,s ,β ( p n , g ) − H εs ,s ,β ( p n , q n ) k ∞ = k H εs ,s ,β ( f − p n , g ) k ∞ + k H εs ,s ,β ( p n , g − q n ) k ∞ ≤ C ( β, ε ) k f − p n k ∞ k g k ∞ + C ( β, ε ) k p n k ∞ k g − q n k ∞ → as n → ∞ , since k f − p n k ∞ → and k g − q n k ∞ → as n → ∞ by our assumption(5.7). Moreover, by the uniqueness of the Fourier coefficients of H εs ,s ,β ( p n , q n ) ∈ H m ( R / Z , R ) ⊆ L ( R / Z , R ) we have, for all k ∈ Z , that (cid:12)(cid:12)(cid:12) \ H εs ,s ,β ( f, g )( k ) − \ H εs ,s ,β ( p n , q n )( k ) (cid:12)(cid:12)(cid:12) → as n → ∞ . Thus for all positive integers N ≥ we find that (cid:12)(cid:12)(cid:12)(cid:16) X | k |≤ N (1 + | k | ) m | \ H εs ,s ,β ( p n , q n )( k ) | (cid:17) − (cid:16) X | k |≤ N (1 + | k | ) m | \ H εs ,s ,β ( f, g )( k ) | (cid:17) (cid:12)(cid:12)(cid:12) → as n → ∞ . Then by (5.14) we conclude that (cid:16) X | k |≤ N (1 + | k | ) m | \ H εs ,s ,β ( f, g )( k ) | (cid:17) ≤ C H k f k H m + β k g k H m + β which implies the desired estimate (5.6) for f and g . (cid:3) The form of the lower order remainder terms R α and R α We show that the orthogonal projection of the truncated remainder terms of thedecomposition of δ E α found in Section 3 can be expressed as multiple integrals ofanalytic functions.Let < ε ≤ , x ∈ [0 , and γ ∈ C ∞ ( R / Z , R n ) be a simple closed arc-lengthparametrized curve. We recall that the truncated remainder terms are given by ( R α,ε γ )( x ) = Z | w |∈ [ ε, ] ( γ ( x + w ) − γ ( x ) − w ˙ γ ( x )) (cid:18) | γ ( x + w ) − γ ( x ) | α +2 − | w | α +2 (cid:19) d w, ( R α,ε γ )( x ) = Z | w |∈ [ ε, ] ¨ γ ( x ) (cid:18) | γ ( x + w ) − γ ( x ) | α − | w | α (cid:19) d w. We begin with transforming the more elementary part P ⊥ ˙ γ R α,ε γ . Theorem 6.1.
The term P ⊥ ˙ γ R α,ε γ can be re-written as multiple integral of theform ( P ⊥ ˙ γ R ε γ )( x ) = Z | w |∈ [ ε, ] Z Z Z Z [0 , ( s − s ) G α ( γ )( x )d φ d φ d s d s d w | w | α − , n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 15 where G α ( γ )( x ) = G α ( γ )( x ; s , s , φ , φ , w )= G α (cid:16) Z ˙ γ ( x + tw )d t, ˙ γ ( x ) , ¨ γ ( x + s w + ( s − s ) φ w ) , ¨ γ ( x + s w + ( s − s ) φ w ) , ¨ γ ( x ) (cid:17) for some analytic function G α : R n \ { , } × R n → R n .Proof. We begin by computing the first term in the integrand of R α,ε γ . To do sowe use the fundamental theorem of calculus and the arc-length parametrization ofthe curve γ (as in the proof of [BR13, Prop. 3.2]) to get | γ ( x + w ) − γ ( x ) | α − | w | α = | w | α | γ ( x + w ) − γ ( x ) | α − | γ ( x + w ) − γ ( x ) | α | w | α − | γ ( x + w ) − γ ( x ) | w − | γ ( x + w ) − γ ( x ) | w | w | α = g α (cid:18)Z ˙ γ ( x + tw )d t (cid:19) (cid:16) − | γ ( x + w ) − γ ( x ) | w (cid:17) | w | α = g α (cid:18)Z ˙ γ ( x + tw )d t (cid:19) RR [0 , | ˙ γ ( x + s w ) − ˙ γ ( x + s w ) | d s d s | w | α , (6.1)where g α ( x ) = | x | α −| x | α −| x | for all x ∈ R n \ { , } is analytic away from the origin.Since we have by the fundamental theorem of calculus | ˙ γ ( x + s w ) − ˙ γ ( x + s w ) | w = Z Z ( s − s ) × h ¨ γ ( x + s w + ( s − s ) φ w ) , ¨ γ ( x + s w + ( s − s ) φ w ) i R n d φ d φ , (6.2)we can express R α,ε γ as ( R α,ε γ )( x ) = Z | w |∈ [ ε, ] Z Z Z Z [0 , ( s − s ) × f G α (cid:16) Z ˙ γ ( x + tw )d t, ¨ γ ( x + s w + ( s − s ) φ w ) , ¨ γ ( x + s w + ( s − s ) φ w ) , ¨ γ ( x ) (cid:17) d φ d φ d s d s d w | w | α − , where f G α : R n \ { , } × R n → R n defined by f G α ( a, x, y, z ) := g α ( a ) h x, y i R n z isan analytic function away from the origin in the first variable as well. If we applythe orthogonal projection P ⊥ ˙ γ on R α,ε γ , we finally get the following representation ( P ⊥ ˙ γ R α,ε γ )( x ) = Z | w |∈ [ ε, ] Z Z Z Z [0 , ( s − s ) G α ( γ )( x )d φ d φ d s d s d w | w | α − , n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 16 where G α ( γ )( x ) = G α ( γ )( x ; s , s , φ , φ , w )= G α (cid:16) Z ˙ γ ( x + tw )d t, ˙ γ ( x ) , ¨ γ ( x + s w + ( s − s ) φ w ) , ¨ γ ( x + s w + ( s − s ) φ w ) , ¨ γ ( x ) (cid:17) and G α : R n \ { , } × R n → R n given by G α ( a, v, x, y, z ) := P ⊥ v f G α ( a, x, y, z ) isclearly analytic. (cid:3) We use the previous computations to rewrite the orthogonal projection of thefirst remaining part.
Theorem 6.2.
The term P ⊥ ˙ γ R α,ε γ can be re-written as multiple integral of theform ( P ⊥ ˙ γ R ε γ )( x ) = Z | w |∈ [ ε, ] Z Z Z Z [0 , ( r − r ) G α ( γ )( x )d ψ d ψ d r d r d w | w | α − , where G α ( γ )( x ) = G α ( γ )( x ; r , r , ψ , ψ , w )= G α (cid:16) Z ˙ γ ( x + sw )d s, ˙ γ ( x ) , ¨ γ ( x + r w + ( r − r ) ψ w ) , ¨ γ ( x + r w + ( r − r ) ψ w ) , Z ¨ γ ( x + tw )(1 − t )d t (cid:17) for some analytic function G α : R n \ { , } × R n → R n .Proof. By the integral form of the remainder of a first order Taylor approximation,we compute the integrand of R α,ε γ as ( γ ( x + w ) − γ ( x ) − w ˙ γ ( x )) (cid:18) | γ ( x + w ) − γ ( x ) | α +2 − | w | α +2 (cid:19) = w Z ¨ γ ( x + tw )(1 − t )d t (cid:18) | γ ( x + w ) − γ ( x ) | α +2 − | w | α +2 (cid:19) = Z ¨ γ ( x + tw )(1 − t )d t (cid:18) w | γ ( x + w ) − γ ( x ) | α +2 − | w | α (cid:19) . (6.3)Thus the last term in (6.3) can be expressed as w | γ ( x + w ) − γ ( x ) | α +2 − | w | α = | w | α +2 | γ ( x + w ) − γ ( x ) | α +2 − | γ ( x + w ) − γ ( x ) | α +2 | w | α +2 | w | α = | w | α +2 | γ ( x + w ) − γ ( x ) | α +2 − | γ ( x + w ) − γ ( x ) | α +2 | w | α +2 − | γ ( x + w ) − γ ( x ) | | w | − | γ ( x + w ) − γ ( x ) | | w | | w | α = g α (cid:18)Z ˙ γ ( x + sw )d s (cid:19) − | γ ( x + w ) − γ ( x ) | | w | | w | α , (6.4) n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 17 where g α ( x ) = | x | α +2 −| x | α +2 −| x | for all x ∈ R n \ { , } is analytic away from theorigin. By transfering the computations in (6.1) and (6.2) to the last term in (6.4)we find that ( R α,ε γ )( x )= Z | w |∈ [ ε, ] Z Z Z Z [0 , ( r − r ) × f G α (cid:18) Z ˙ γ ( x + sw )d s, ¨ γ ( x + r w + ( r − r ) ψ w ) , ¨ γ ( x + r w + ( r − r ) ψ w ) , Z ¨ γ ( x + tw )(1 − t )d t (cid:19) d ψ d ψ d r d r d w, where f G α : R n \ { , } × R n → R n defined by f G α ( a, x, y, z ) := g α ( a ) h x, y i R n z isanalytic away from 0 and 1 in the first variable.Finally, by applying the orthogonal projection to R α,ε γ , we obtain ( P ⊥ ˙ γ R α,ε γ )( x ) = Z | w |∈ [ ε, ] Z Z Z Z [0 , ( r − r ) G α ( γ )( x )d ψ d ψ d r d r d w, where G α ( γ )( x ) = G α ( γ )( x ; r , r , ψ , ψ , w )= G α (cid:16) Z ˙ γ ( x + sw )d s, ˙ γ ( x ) , ¨ γ ( x + r w + ( r − r ) ψ w ) , ¨ γ ( x + r w + ( r − r ) ψ w ) , Z ¨ γ ( x + tw )(1 − t )d t (cid:17) and G α : R n \ { , } × R n → R n given by G α ( a, v, x, y, z ) := P ⊥ v f G α ( a, x, y, z ) isobviously analytic. (cid:3) Proof of the main theorem by Cauchy’s method of majorants
We now turn to the proof of Theorem 1.2. Our strategy is to first establish arecursive estimate for k ∂ l γ k H α − from which we can infer, by Cauchy’s method ofmajorants, the analyticity of the curve γ .Let m := 1 > and γ = ( γ , . . . , γ n ) : R / Z → R n be a simple closed arc-lengthparametrized curve that is in the class C ∞ ( R / Z , R n ) . If γ is a critical point of E α + λ L , i.e. if we have δ E α ( γ ; h ) + λ Z R / Z h ˙ γ, ˙ h i d x = 0 for all h ∈ H α +12 , ( R / Z , R n ) , then Theorem 3.1 and integration by parts imply thatthe curve γ solves the Euler-Lagrange equation ( H α γ + λ ¨ γ, h ) L ([0 , , R n ) = 0 for all h ∈ H α +12 , ( R / Z , R n ) . Thus by the decompositions (3.1) and (3.2) of thegradient of O’Hara’s knot energies, we conclude that H α γ = αP ⊥ ˙ γ Q α γ + 2 αP ⊥ ˙ γ R α γ − P ⊥ ˙ γ R α γ + λ ¨ γ ≡ n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 18 on R / Z . Moreover, by Corollary 4.2 with β := α − ∈ (0 , and the triangleinequality, it follows for any integer l ≥ that k ∂ l +3 γ k H β ≤ e C k ∂ l Q α γ k H = e C (cid:13)(cid:13) ∂ l (cid:0) αP T ˙ γ Q α γ − αP ⊥ ˙ γ R α γ + 2 P ⊥ ˙ γ R α γ − λ ¨ γ (cid:1)(cid:13)(cid:13) H ≤ e C ( k ∂ l P T ˙ γ Q α γ k H + k ∂ l P ⊥ ˙ γ R α γ k H + k ∂ l P ⊥ ˙ γ R α γ k H + λ k ∂ l +2 γ k H ) . (7.1)In order to get a recursive estimate on k ∂ l γ k H β we aim to derive suitable estimatesfor the first three terms on the right-hand side of (7.1). To do so for the tangentialpart of Q α , we use the fractional Leibniz rule from Section 5 in the following: Lemma 7.1.
Let l ∈ N , β := α − ∈ (0 , for some < α < and < ε ≤ .Then there exists a positive constant C Q α independent of ε , γ and l such that k ∂ l P T ˙ γ Q α,ε γ k H ≤ C Q α l X k =0 k X k =0 (cid:18) lk (cid:19)(cid:18) k k (cid:19) k ∂ l − k +2 γ k H β k ∂ k − k +2 γ k H β k ∂ k +1 γ k H . Proof.
Using formula (5.2) and the Leibniz rule twice yields ∂ l (cid:16) P T ˙ γ Q α,ε γ ( x ) (cid:17) = ∂ l (cid:16) Z | w |∈ [ ε, ] Z Z [0 , (1 − t )( − t ) h ¨ γ ( x + tw ) , ¨ γ ( x + stw ) i R n w | w | α − ˙ γ ( x )d s d t d w (cid:17) = 2 Z | w |∈ [ ε, ] Z Z [0 , (1 − t )( − t ) ∂ l (cid:18) h ¨ γ ( x + tw ) , ¨ γ ( x + stw ) i R n w | w | β ˙ γ ( x ) (cid:19) d s d t d w = 2 Z | w |∈ [ ε, ] Z Z [0 , (1 − t )( − t ) × l X k =0 k X k =0 (cid:18) lk (cid:19)(cid:18) k k (cid:19) (cid:10) ∂ l − k +2 γ ( x + tw ) , ∂ k − k +2 γ ( x + stw ) (cid:11) R n w | w | β ∂ k +1 γ ( x )d s d t d w = 2 Z Z [0 , (1 − t )( − t ) × l X k =0 k X k =0 (cid:18) lk (cid:19)(cid:18) k k (cid:19) n X k =1 H εt,st,β ( ∂ l − k +2 γ k , ∂ k − k +2 γ k )( x ) ∂ k +1 γ ( x )d s d t, n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 19 where the smoothness of the integrand allows interchanging of integrals and deriv-ative. Then for all ≤ m ≤ n we obtain the component-wise estimate k ∂ l ( P T ˙ γ Q α,ε γ ) m k H ≤ Z Z [0 , | − t || t |× l X k =0 k X k =0 (cid:18) lk (cid:19)(cid:18) k k (cid:19) n X k =1 C k H εt,st,β ( ∂ l − k +2 γ k , ∂ k − k +2 γ k ) k H k ∂ k +1 γ m k H d s d t ≤ C Z Z [0 , | − t || t |× l X k =0 k X k =0 (cid:18) lk (cid:19)(cid:18) k k (cid:19) n X k =1 C H k ∂ l − k +2 γ k k H β k ∂ k − k +2 γ k k H β ! k ∂ k +1 γ m k H d s d t ≤ C C H l X k =0 k X k =0 (cid:18) lk (cid:19)(cid:18) k k (cid:19) k ∂ l − k +2 γ k H β k ∂ k − k +2 γ k H β k ∂ k +1 γ m k H (7.2)via the Banach algebra property (2.1) of H and the fractional Leibniz rule for eachof the components (i.e. Theorem 5.3). The desired statement follows from (7.2) andthe equivalence between the -norm and the -norm with a finite positive constant C Q α := 2 C C H √ n . (cid:3) Next we estimate the orthogonal projection of the truncated remaining parts R α,ε γ and R α,ε γ . The important ingredients for this are the forms of P ⊥ ˙ γ R α,ε γ and P ⊥ ˙ γ R α,ε γ derived in Chapter 6, Faà di Bruno’s formula and the Banach algebraproperty of H . Lemma 7.2.
Let l ∈ N , < α < , and consider the mapping f = ( f , . . . , f n ) : R / Z → R n \ { , } × R n given by f ( x ) := ( ˙ γ ( x ) , ˙ γ ( x ) , ¨ γ ( x ) , ¨ γ ( x ) , ¨ γ ( x )) . Then there exist positive constants C R α and r independent of ε , γ and l such that theuniversal polynomials p (5 n ) l from the multivariate form of Faà di Bruno’s formula. (2.2) satisfy k ∂ l P ⊥ ˙ γ R α,ε γ k H ≤ C R α p (5 n ) l (cid:16)n ( | η | + 1)! r | η | +1 o | η |≤ l , n C k f ( j ) i k H o ≤ i ≤ n ≤ j ≤ l (cid:17) , where C is the constant from the estimate (2.1) with m = 1 .Proof. By Theorem 6.2 we can express P ⊥ ˙ γ R α,ε γ as ( P ⊥ ˙ γ R α,ε γ )( x ) = Z | w |∈ [ ε, ] Z Z Z Z [0 , ( r − r ) G α ( f ( x ))d ψ d ψ d r d r d w | w | α − n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 20 where G α : R n \ { , } × R n → R n is an analytic function and f ( x ) := R ˙ γ ( x + sw )d s ˙ γ ( x )¨ γ ( x + r w + ( r − r ) ψ w )¨ γ ( x + r w + ( r − r ) ψ w )) R ¨ γ ( x + tw )d t . Due to the smoothness of the integrand and the generalized Faà di Bruno’s formula(2.2), we component-wise find for all ≤ k ≤ n that ∂ l ( P ⊥ ˙ γ R α,ε γ ) k ( x )= ∂ l Z | w |∈ [ ε, ] Z Z Z Z [0 , ( r − r ) G α ,k ( f ( x ))d ψ d ψ d r d r d w | w | α − = Z | w |∈ [ ε, ] Z Z Z Z [0 , ( r − r ) ∂ l G α ,k ( f ( x ))d ψ d ψ d r d r d w | w | α − = Z | w |∈ [ ε, ] Z Z Z Z [0 , ( r − r ) × p (5 n ) l (cid:16) { ( ∂ η G α ) k ( f ( x )) } | η |≤ l , { f ( j )1 ,i ( x ) } ≤ i ≤ n ≤ j ≤ l (cid:17) d ψ d ψ d r d r d w | w | α − . Moreover, by applying the H -norm on ∂ l ( P ⊥ ˙ γ R α,ε γ ) component-wise for any ≤ k ≤ n , we find via the Banach algebra property of H that k ∂ l ( P ⊥ ˙ γ R α,ε γ ) k k H ≤ Z | w |∈ [ ε, ] Z Z Z Z [0 , | r − r | × (cid:13)(cid:13)(cid:13) p (5 n ) l (cid:16) { ( ∂ η G α ) k ( f ) } | η |≤ l , { f ( j )1 ,i } ≤ i ≤ n ≤ j ≤ l (cid:17)(cid:13)(cid:13)(cid:13) H d ψ d ψ d r d r d w | w | α − ≤ Z | w |∈ [ ε, ] Z Z Z Z [0 , | r − r | × p (5 n ) l (cid:16) {k ( ∂ η G α ) k ( f ) k H } | η |≤ l , { C k f ( j )1 ,i k H } ≤ i ≤ n ≤ j ≤ l (cid:17) d ψ d ψ d r d r d w | w | α − , where the constant C appears only in front of k f ( j )1 ,i k H because of the one-homogeneity of p (5 n ) l in its first components.For any ≤ j ≤ l we also observe that k f ( j )1 ,k k H ≤ k f ( j ) k k H whenever ≤ k ≤ n , n +1 ≤ k ≤ n and that k f ( j )1 ,k k H = k f ( j ) k k H whenever n +1 ≤ k ≤ n . Moreover, k f ( j ) k k H does not depend on any of the parameters w, ψ i , r i , s, t for all ≤ k ≤ n and ≤ j ≤ l . In addition, since G α is an analytic function, we can deduce fromthe proof of [BV19, Theor. 7.2] that k ( ∂ η G α )( f ) k H ≤ e C R α ( | η | + 1)! r | η | +1 for some positive constant e C R α independent of ε , γ and l . n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 21 Now using the elementary estimate R | w |∈ [ ε, ] d w | w | α − ≤ − α < ∞ since < α < and the fact that p (5 n ) l is one-homogeneous in the first components, we finally obtain k ∂ l ( P ⊥ ˙ γ R ε γ ) k H ≤ C R α p (5 n ) l (cid:16)n ( | η | + 1)! r | η | +1 o | η |≤ l , n k C f ( j ) i k H o ≤ i ≤ n ≤ j ≤ l (cid:17) , where C R α := e C R α − α is finite and the right-hand side is independent of ε . (cid:3) The orthogonal projection of the second remainder term can be estimated alongthe lines of the previous Lemma 7.2 to get the following:
Lemma 7.3.
Let l ∈ N and f = ( ˙ γ, ˙ γ, ¨ γ, ¨ γ, ¨ γ ) as in Lemma 7.2. Then there existpositive constants C R α and s that are independent of ε , γ and l such that k ∂ l P ⊥ ˙ γ R α,ε γ k H ≤ C R α p (5 n ) l (cid:16)n ( | η | + 1)! s | η | +1 o | η |≤ l , n C k f ( j ) i k H o ≤ i ≤ n ≤ j ≤ l (cid:17) , where p (5 n ) l is the same universal polynomial as in Lemma 7.2. As a direct consequence of the proof of [BV19, Theor. 7.4] the estimates fromLemmas 7.1, 7.2 and 7.3 also hold for the limit case ε ↓ . Theorem 7.4.
Let l ∈ N and f = ( ˙ γ, ˙ γ, ¨ γ, ¨ γ, ¨ γ ) as in Lemma 7.2. Then thereexist positive constants r , s , C Q α , C R α and C R α which are independent of γ and l such that k ∂ l P T ˙ γ Q α γ k H ≤ C Q α l X k =0 k X k =0 (cid:18) lk (cid:19)(cid:18) k k (cid:19) k ∂ l − k +2 γ k H β k ∂ k − k +2 γ k H β k ∂ k +1 γ k H k ∂ l P ⊥ ˙ γ R α γ k H ≤ C R α p (5 n ) l (cid:16)n ( | η | + 1)! r | η | +1 o | η |≤ l , n C k f ( j ) i k H o ≤ i ≤ n ≤ j ≤ l (cid:17) k ∂ l P ⊥ ˙ γ R α γ k H ≤ C R α p (5 n ) l (cid:16)n ( | η | + 1)! s | η | +1 o | η |≤ l , n C k f ( j ) i k H o ≤ i ≤ n ≤ j ≤ l (cid:17) , where β := α − for some < α < and p (5 n ) l is the universal polynomial fromLemma 7.2. We are now in a position to give the proof of the main theorem.
Proof of Theorem 1.2.
Let β = α − for < α < . Now suppose γ = ( γ , . . . , γ n ) : R / Z → R n is a closed simple arc-length parametrized curve in the class C ∞ ( R / Z , R n ) which is a stationary point of E α + λ L . Using the smoothness of the curve γ , weintroduce an auxiliary function f : R / Z → R n given by f ( x ) := ( ˙ γ ( x ) , ˙ γ ( x ) , ¨ γ ( x ) , ¨ γ ( x ) , ¨ γ ( x )) and define a l := C k ∂ l f k H β for all integers l ≥ . If we can establish the existence of finite positive constants δ and C γ such that a l ≤ C γ l ! δ l we immediately obtain the analyticity of the curve γ on R / Z by Corollary 2.3. n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 22 In order to obtain the desired bounds on a l we first note that there exists apositive constant ˜ C such that k ∂ l +3 γ k H β ≤ e C (cid:16) k ∂ l P T ˙ γ Q α γ k H + k ∂ l P ⊥ ˙ γ R α γ k H + k ∂ l P ⊥ ˙ γ R α γ k H + λ k ∂ l +2 γ k H (cid:17) by the criticality of the curve γ , the first variation formula of E α of Theorem 3.1,our decomposition of the first variation of E α + λ L and Corollary 4.2. Next wededuce from Theorem 7.4 and the Sobolev embedding of H β ⊆ H that a l +1 = C k ∂ l +1 f k H β ≤ C (3 k ∂ l +3 γ k H β + 2 k ∂ l +2 γ k H β ) ≤ e CC h k ∂ l P T ˙ γ Q α γ k H + k ∂ l P ⊥ ˙ γ R α γ k H + k ∂ l P ⊥ ˙ γ R α γ k H + λ k ∂ l +2 γ k H i + 2 C a l ≤ e CC h C Q α l X k =0 k X k =0 (cid:18) lk (cid:19)(cid:18) k k (cid:19) k ∂ l − k +2 γ k H β k ∂ k − k +2 γ k H β k ∂ k +1 γ k H + C R α p (5 n ) l (cid:16)n ( | η | + 1)! r | η | +1 o | η |≤ l , n C k f ( j ) i k H o ≤ i ≤ n ≤ j ≤ l (cid:17) + C R α p (5 n ) l (cid:16)n ( | η | + 1)! s | η | +1 o | η |≤ l , n C k f ( j ) i k H o ≤ i ≤ n ≤ j ≤ l (cid:17) + λ k ∂ l +2 γ k H i + 2 C a l . Therefore, by the embedding of H β ⊆ H again, there exist positive constants C > and r γ independent of γ and l such that a l +1 ≤ C h l X k =0 k X k =0 (cid:18) lk (cid:19)(cid:18) k k (cid:19) a l − k a k − k a k + p (5 n ) l (cid:16)n ( | η | + 1)! r | η | +1 γ o | η |≤ l , n a j o ≤ j ≤ l (cid:17) + a l i . (7.3)To apply Cauchy’s method of majorants, we define a majorant F : R n → R n component-wise by setting F i ( y ) := C (cid:16) y i + 1(1 + na − ( y + ··· + y n ) r γ ) + y i (cid:17) for any y = ( y , . . . , y n ) ∈ R n and all ≤ i ≤ n . It is clearly analytic around a (1 , . . . , . Moreover, we consider the following initial value problem ˙ c ( t ) = F ( c ( t )) ,c (0) = a (1 , . . . , . (7.4)and note that the ODE has a unique smooth solution c : ( − ε, ε ) → R n that is alsoanalytic around for some ε > by Theorem 2.4. Hence we can write c = c ( t ) as n the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. 23 a Taylor series in a neighbourhood of , i.e. c ( t ) := ∞ X k =0 e a k (1 , . . . , k ! t k where e a k (1 , . . . ,
1) = c ( k ) (0) . Then by Theorem 2.1 we can, for all t ∈ B r c (0) ,bound | c ( l ) ( t ) | ≤ M l ! r lc (7.5)for some finite positive constants r c and M which are independent of any integer l ≥ . By applying the Leibniz rule and Faà di Bruno’s formula (2.2) we have that ∂ l F i ( c ( t ))= C h l X k =0 k X k =0 (cid:18) lk (cid:19)(cid:18) k k (cid:19) c ( l − k ) i ( t ) c ( k − k ) i ( t ) c ( k ) i ( t )+ p (5 n ) l (cid:16)n ( | η | + 1)! r | η | +1 γ (cid:16) (cid:16) na r γ − ( c ( t )+ ··· + c n ( t )) r γ (cid:17)(cid:17) | η | +2 o | η |≤ l , n c ( j ) i ( t ) o ≤ i ≤ n ≤ j ≤ l (cid:17) + c ( l ) i ( t ) i for any ≤ i ≤ n and l ∈ N . Then by considering the initial condition (7.4) wefind that ˜ a l +1 = ∂ l F i ( c (0))= C h l X k =0 k X k =0 (cid:18) lk (cid:19)(cid:18) k k (cid:19) ˜ a l − k ˜ a k − k ˜ a k + p (5 n ) l (cid:16)n ( | η | + 1)! r | η | +1 γ o | η |≤ l , n ˜ a j o ≤ j ≤ l (cid:17) + ˜ a l i . (7.6)Finally an induction argument obtained from comparing (7.3) and (7.6), togetherwith the fact that Faà di Bruno’s polynomials p l (2.2) have non-negative coefficientsand the initial condition a = e a , gives a l ≤ e a l . Then by (7.5) we conclude that k ∂ l +1 γ k H β ≤ a l ≤ M l ! r lC for all l ∈ N which, byCorollary 2.3, implies the analyticity of the curve γ . (cid:3) References [ACF+03] A. Abrams, J. Cantarella, J. H. G. Fu, M. Ghomi, and R. Howard,
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Fachbereich Mathematik, Universität Salzburg, Hellbrunner Strasse 34, 5020 Salzburg,Austria
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