A note on integral Menger curvature for curves

Size: px

Start display at page:

Download "A note on integral Menger curvature for curves"

Horace Wood
5 years ago
Views:

1 A note on integral Menger curvature for curves Simon Blatt August 26, 20 For continuously dierentiable embedded curves γ, we will give a necessary and sucient condition for the boundedness of integral versions of the Menger curvature. We will show that for p > 3 the integral Menger curvature M p (γ) is nite if and only if γ belongs to the Sobolev Slobodeckij space W 2 2 p,p (R/Z, R n ). The quantity J p (γ) - dened by taking the remum of the p-th power of Menger curvature with respect to one variable and then integrating over the remaining two - is nite for p > 2, if and only if γ belongs to the space W 2 p,p. Introduction In [SSvdM09, SSvdM0], Marta Szuma«ska, Pawel Strzelecki, and Heiko von der Mosel observed that in certain parameter ranges integral variants of the Menger curvature might be used as knot energies. They could show that these quantities have self repulsive and regularizing properties [SSvdM09, SSvdM0]. In this article will sharpen these statements by characterizing all curves with nite integral Menger curvatures. The Menger curvature c(x, y, z) of three points x, y, z R n that are not colinear is given by the inverse of the radius of the circle going through these points. If the three points x, y, z are colinear one sets c(x, y, z) = 0. A well known formula for the Menger curvature is given by sin (y x, z x) c(x, y, z) = 2. (.) y z The integral Menger curvature with exponent p [, ] of a Borel measurable set E R n of locally nite one dimensional Hausdor measure is dened by M p (E) := c p (x, y, z)dh (x)dh (y)dh (z), E E E Departement Mathematik, ETH Zürich, Rämistrasse 0, CH-8004 Zürich, Switzerland, simon.blatt@math.ethz.ch

2 where H denotes the one dimensional Hausdor measure. Exchanging one of the three integrals by a remum, we get the following variant J p (E) := {c p (x, y, z)} dh (y)dh (z). x E E E In the two dimensional Euclidean space, the quantity M 2 (E) - also called total Menger curvature - has deep connections to harmonic analysis and geometric measure theory. To mention only two of the most prominent results, note that M. Leger was able to prove that nite Menger curvature implies rectiability of the set E [Lég99] and Guy David showed that M 2 (E) < is enough to show that E has vanishing analytic capacity and was then able so solve the Vitushkin conjecture [Dav98]. Marta Szuma«ska, Pawel Strzelecki, and Heiko von der Mosel studied rectiable curves γ : R/Z R n which are local homeomorphism with M p (γ) := M p (γ(r/z)) < for some p > 3. They showed that reparameterizing γ by arc length one gets an injective curve belonging to C, 3 p (R/Z, R n ) (cf. [SSvdM0]). Moreover, Marta Szumanska was able to show in her PhD thesis that in the case of the integral Menger curvature the Hölder-exponents are sharp [Szu09] in the sense that there are curves with nite energy that do not belong to Hölder spaces with any higher exponent. For curves as above, they found that J p (γ) := J p (γ(r/z)) < for p > 2 implies that reparameterizing γ proportional to arc length one gets an injective curve belonging to C, 2 p (R/Z, R n ). As mentioned before, we want to give a necessary and sucient condition for a curve γ C (R/Z, R n ) to have nite integral Menger curvature for p > 3 or to have nite J p for p > 2. It turns out that as for O'Hara's knot energies [Bla0a] and the tangent point energies [Bla] such a condition for the integral Menger curvature M p can be given using Sobolev Slobodeckij spaces. A detailed introduction to these function spaces can be found for example in [Tri83] and [RS96]. Let us shortly repeat the denition of these spaces in a form suitable for our purpose. For p [, ), s (0, 2) and f L p (R/Z, R n ) we dene the semi norm f W s,p := ( /2 R/Z /2 ) /p f(x + w) 2f(x) + f(x w) p dwdx. w +sp for f L p (R/Z, R n ) and denote by W s,p (R/Z, R n ) the space of all f L p f W s,p <. We dene a norm on W s,p by setting with f W s,p := f L p + f W s,p for all f W s,p (R/Z, R n ). It is well known that for s (, 2) we can exchange the semi norm W s,p semi norm ( /2 ) /p f W := f (x + w) + f (x) p dwdx s,p R/Z /2 w +(s )p by the 2

3 in the above denition. The rst main result of this article is Theorem.. Let γ C (R/Z, R n ) be an injective curve parameterized by arc length and p > 3. Then the integral Menger curvature M p (γ) is nite if and only if γ W 2 2 p,p (R/Z, R n ). Together with Theorem. in [SSvdM0] and the fact that the W 2 2 p,p C this immediately leads to Corollary.2. Let γ C 0, (R/Z, R n ) be a local homomorphism parametrized by arc length and p > 3. Then M p (γ) < if and only if γ is embedded and belongs to W 2 2 p,p (R/Z, R n ). For the quantity J p we will show the following Theorem.3. Let γ C (R/Z, R n ) be an injective curve parameterized by arc length and p > 2. Then J p (γ) is nite if and only if γ W 2 p,p (R/Z, R n ). Again, an immediate corollary of this theorem, Theorem.2 of [SSvdM0] and the embedding W 2 /p,p (R/Z, R n ) C (R/Z, R n ) for p > 2 is Corollary.4. Let γ C 0, (R/Z, R n ) be a local homeomorphism parameterized by arc length and p > 2. Then J p (γ) < if and only if γ is an embedded curve of class W 2 /p,p (R/Z, R n ). We are convinced that the results and techniques in this article open the door to a deeper understanding of integral Menger curvature for curves. The corresponding results for O'Hara's knot energies paved the way to regularity results for local minimizers and stationary points [BR], and to long time existence results for the L 2 gradient ows of O'Hara's energies [Bla0b]. Throughout this paper, C < and c > 0 denote constants depending on known quantities. These value of these constants are aloud to vary from line to line and even within the same line. 2 Proof of Theorem. Let γ C (R/Z, R n ) be an embedded curve parametrized by arc length. Then for every ε > 0 there is a constant C ε < such that γ(u + w) γ(u) C ε w (2.) for all u R/Z and w [ + ε, ε]. Let us assume that γ C (R/Z, R n ) is an embedded curve parametrized by arc length with M p (γ) <. We will show that γ can locally be written as a graph of a W 2 2 p,p function and thus prove that γ W 2 2 p,p since reparametrization by arc length does not destroy this regularity. 3

4 So let u R/Z. Since γ C (R/Z, R n ), we can assume that after a suitable translation and rotation of the ambient space R n we have γ(0) = 0 and that there is a δ > 0 and a function f C (R, R n ) with f L and f(0) = 0, such that γ(u) := (u, f(u)) satises γ(b 2δ (0)) γ(r/z). Then γ(u) γ(v) u v γ(u) γ(v) 2 (2.2) γ 2 (2.3) and we observe that ( γ(u + sin w ) γ(u) γ(u + w ) γ(u), γ(u + w 2 ) γ(u + w 2 ) γ(u) ) γ(u + w ) γ(u) 2 sign(w ) γ(u + w ) γ(u) γ(u + w 2 ) γ(u) sign(w 2 ) γ(u + w 2 ) γ(u) for all u, w, w 2 R. We hence get from (.) M p (γ) c + ( (2.2) & (2.3) + w /2 B δ (0) δ w /2 w /2 B δ (0) δ w /2 c ( B δ (0) δ w /2 w /2 B δ (0) δ w /2 γ(u+w ) γ(u) sign(w γ(u+w 2) γ(u) ) γ(u+w ) γ(u) sign(w 2) γ(u+w 2) γ(u) p γ(u + w ) γ(u + w 2 ) p γ (u + w 2 ) γ (u + w ) γ (u) dw 2 dw du γ(u w ) γ(u) sign( w γ(u+w 2) γ(u) ) γ(u w ) γ(u) sign(w 2) γ(u+w 2) γ(u) p w /2 γ(u w ) γ(u + w 2 ) p γ (u + w 2 ) γ (u w ) γ (u) dw 2 dw du γ(u+w ) γ(u) sign(w ) γ(u+w ) γ(u) γ(u w ) γ(u) sign( w ) γ(u w ) γ(u) γ(u+w 2) γ(u) p sign(w 2) γ(u+w 2) γ(u) p w w 2 p dw 2 dw du γ(u+w 2) γ(u) sign(w 2) γ(u+w 2) γ(u) p w w 2 p dw 2 dw du ) ) 4

5 c + ( inequality c w /2 B δ (0) δ w /2 w /2 B δ (0) δ w /2 c B δ (0) δ w /2 B δ (0) δ w /2 γ(u+w ) γ(u) sign(w ) γ(u+w ) γ(u) γ(u w ) γ(u) sign( w ) γ(u w ) γ(u) γ(u+w) γ(u) γ(u+w + γ(u w) γ(u) ) γ(u) γ(u w ) γ(u) p γ(u+w) γ(u) γ(u+w + γ(u w) γ(u) ) γ(u) γ(u w ) γ(u) p γ(u+w 2) γ(u) p sign(w 2) γ(u+w 2) γ(u) p w p dw 2 dw du γ(u+w 2) γ(u) sign(w 2) γ(u+w 2) γ(u) p w p dw 2 dw du w p dw 2 dw du w p dw du, (2.4) where c > 0 is a constant that is allowed to change from line to line. We thus get for a constant C < M p (γ) c c B δ (0) δ C B δ (0) δ B δ (0) δ C Observing that γ(u + w ) 2 γ(u) + γ(u w ) p w 2p dw du B δ (0) δ ( γ(u + w ) γ(u)) ( p ) w p dw du γ(u+w ) γ(u) γ(u w ) γ(u) γ(u + w ) 2 γ(u) + γ(u w ) p w 2p dw du ( w p ) w p dw du. (2.5) γ(u+w ) γ(u) γ(u w ) γ(u) w γ(u + w ) γ(u) w γ(u w ) γ(u) (w, f(u + w ) f(u) + ( w, f(u w ) + f(u)) γ(u + w ) γ(u) γ(u w ) γ(u) γ(u + w ) γ(u) γ(u + w ) γ(u) + γ(u w ) γ(u) γ(u w ) γ(u) (2.6) ) 5

6 we get from (2.4) ( w B δ (0) δ γ(u+w ) γ(u) γ(u w ) γ(u) B δ (0) δ ) p w p dw du Hence, Estimate (2.6) implies B δ (0) δ γ(u+w) γ(u) γ(u+w + γ(u w) γ(u) ) γ(u) γ(u w ) γ(u) p w p dw du γ(u + w ) 2 γ(u) + γ(u w ) p w 2p dw du < CM p (γ) <. and thus γ W 2 2 p,p (B δ (0)). The other implication will be shown in two steps. Let γ W 2 2/p,p (R/Z, R n ) be a curve parametrized by arc length. First, we will see that ( ) p c p (u, u + w, u + w 2 )dw dw 2 du C γ (). (2.7) W 2 p 2,p (u,w,w 2) G where G is the subset of all triple (u, w, w 2 ) [ 2, 2] [ /2, /2] [ /2, /2] which have the property that w and w 2 have dierent sign and the norm of at least one of the w i is less than 3 8. Then we will use a decomposition of [ /2, /2]3 and the symmetry of c(x, y, z) with respect to permutations of the arguments to show that M p (γ) C c p (u, u + w, u + w 2 )dw dw 2 du (2.8) (u,w,w 2) G and thus nish the proof of the Theorem.. Let us introduce some notation. For a set Q R 3 we denote by Q ± the set of all triples (u, w, w 2 ) Q with w positive and w 2 negative, by Q the subset with w negative and w 2 positive, and by Q +, Q the subset where w and w 2 are positive / negative, respectively. Furthermore, we use Q, Q, Q >, Q < to denote the subset of Q with w w 2, w w 2, etc. Of course, we allow combinations of these er- and scripts. So, Q < + denotes for example the set of all (x, w, w 2 ) Q with 0 < w < w 2. For (u, w, w 2 ) G we have and hence by Esimate (2.) we have w w w w 2 C γ(w ) γ(w 2 ). 6

7 Furthermore, max{ w, w 2 } w w 2 2 max{ w, w 2 }. We estimate for u, w, w 2 G sin c(u, u + w, u + w 2 ) = 2 2 C ( ) γ(u+w) γ(u) γ(u+w, γ(u+w2) γ(u) ) γ(u) γ(u+w 2) γ(u) γ(u + w ) γ(u + w 2 ) γ(u+w) γ(u) γ(u+w + γ(u+w2) γ(u) ) γ(u) γ(u+w 2) γ(u) γ(u + w ) γ(u + w 2 ) γ(u+w) γ(u) w γ(u+w2) γ(u) w 2 w w 2 where we have used the estimate (2.) and the fact that v v v is locally Lipschitz on R n {0} to get the last inequality. Using the fundamental theorem of calculus, we get for (u, w, w 2 ) G 0 c(u, u + w, u + w 2 ) C γ (u + τw ) γ (u + τw 2 ) dτ w w 2 and hence G c p (u, u + w, u + w 2 )dw dw 2 du G Jensen 2 G ( ) p 0 γ (u + τw ) γ (u + τw 2 ) dτ /2 2 /2 w 2 = 2 C 2 /2 w w 2 p dw 2 dw 0 γ (u + τw ) γ (u + τw 2 ) p dτ w w 2 p dw 2 dw /2 w 0 γ (u) γ (u + τ w) p dτ w p dw d wdu 0 γ (u) γ (u + τ w) p dτ w p d wdu ( γ () W 2 2/p,p ) p. To get from the third to the forth line, we rst substitute u by u+τw and in a second substitution exchange w 2 by w := w 2 w. Furthermore, we use that w w for (u, w, w 2 ) G. Exchanging the role of w and w 2 in the argument above, we also get G c p (u, u + w, u + w 2 )dw dw 2 du C ( γ () W 2 2/p,p ) p 7

8 and thus (2.7). To show (2.8), let A := [ /2, /2] 3 and B := [, ] [ /2, /2] 2. With the notation introduced above we get c p (u, u + w, u + w 2 )dw dw 2 du A + = c p (u, u + w, u + w 2 )dw dw 2 du + c p (u, u + w, u + w 2 )dw dw 2 du A > + A < + c p (ũ w 2, ũ + w, ũ)d w d w 2 du + B ± B c p (ũ w 2, ũ + w, ũ)d w d w 2 du. c p (u, u + w, u + w 2 )dw dw 2 du + The same estimate is true for A instead of A +. Decomposing B ± into the set W ± where w, w and the rest G ± = B ± W ± and substituting ũ := u + w, w := + (w 2 w ) [0, 4 ] and w 2 := w [ /2, 0], we get after integrating over B ± c p (u, u + w, u + w 2 )dw dw 2 du B ± c p (u, u + w, u + w 2 )dw dw 2 du B ± G ± c p (ũ + w 2, ũ, ũ + w )d w d w 2 dũ + c p (u, u + w, u + w 2 )dw dw 2 du G 2 c p (u, u + w, u + w 2 )dw dw 2 du. G G ± Analogous of course we have c p (u, u + w, u + w 2 )dw dw 2 du 2 B G c p (u, u + w, u + w 2 )dw dw 2 du. 8

9 Summing up, we get /2 /2 c p (u, u + w, u + w 2 )dw dw 2 du R/Z /2 /2 = c p (u, u + w, u + w 2 )dw dw 2 du + c p (u, u + w, u + w 2 )dw dw 2 du A ± A + c p (u, u + w, u + w 2 )dw dw 2 du + c p (u, u + w, u + w 2 )dw dw 2 du A + A 3 c p (u, u + w, u + w 2 )dw dw 2 du + 3 c p (u, u + w, u + w 2 )dw dw 2 du B ± B C c p (u, u + w, u + w 2 )dw dw 2 du. G Together with Estimate 2.7 this shows that M p (γ) is nite if γ W 2 2 p,p (R/Z, R n ). 3 Proof of Theorem.3 Again, let us assume that γ C (R/Z, R n ) is an embedded curve parametrized by arc length with J p (γ) <. We will show that γ can locally be written as a graph of a W 2 p,p -function and thus prove that γ W 2 p,p. As before, it is enough to show that if f C (R, R n ) with such that γ(u) := (u, f(u)) satises then f W 2 /p,p (R/Z, R n ). We have f L, γ(b 2δ (0)) γ(r/z). γ(u) γ(v) u v γ(u) γ(v) 2 γ 2 and ( γ(u + sin w ) γ(u) γ(u + w ) γ(u), γ(u + w 2 ) γ(u + w 2 ) γ(u) ) γ(u + w ) γ(u) 2 sign(w ) γ(u + w ) γ(u) γ(u + w 2 ) γ(u) sign(w 2 ) γ(u + w 2 ) γ(u) 9

10 for all u, w, w 2 R. Hence, we get J p (γ) c δ δ c δ δ w 2 B δ (0) γ(u+w ) γ(u) sign(w γ(u+w 2) γ(u) ) γ(u+w ) γ(u) sign(w 2) γ(u+w 2) γ(u) p γ(u+w) γ(u) γ(u+w + γ(u w) γ(u) ) γ(u) γ(u w ) γ(u) p and again there is a constant C < with J p (γ) c δ δ C c δ δ δ δ C δ δ w w 2 p dw du w p dw du (3.) γ(u + w ) 2 γ(u) + γ(u w ) p w 2p dw du δ δ ( γ(u + w ) γ(u)) ( p ) w p dw du γ(u+w ) γ(u) γ(u w ) γ(u) γ(u + w ) 2 γ(u) + γ(u w ) p w 2p dw du ( w p ) w p dw du. γ(u+w ) γ(u) γ(u w ) γ(u) As Estimate (2.6) and Estimate (3.) imply ( w γ(u+w ) γ(u) γ(u w ) γ(u) p ) w p dw du CJ p (γ) <, we get δ δ γ(u + w ) 2 γ(u) + γ(u w ) p w 2p dw du < and hence γ W 2 p,p (B δ (0)). To show the other implication, as in the proof of Theorem. we rst derive some estimates for good parameters and then show that the rest can be reduced to that. For w [ /2, /2], we denote by G () w the subset of all w 2 [ /2, /2] such that () w 2 2 w and w and w 2 have the same sign. Let G w be the subset of all w 2 [ /2, /2] such that w 2 2 w, w and w 2 have opposite signs and w 2 w 7/8. 0

11 By G (2) w we denote the subset of all w 2 [ /2, /2] such that w 2 /2 w and (2) w 2 and w have the same sign, and by G w the set of all w 2 [ /2, /2] such that w 2 2 w, w and w 2 have opposite signs and w 2 w 7/8. Furthermore, we denote by B w () subset of all w 2 [ /2, /2] such that w w 2 7/8, by B w (2) the subset of all w 2 [ /2, /2] such that /2 w w 2 w and w 2 and w have the same sign, and by B w (3) the subset of all w 2 [ /2, /2] such that w w 2 2 w and w 2 and w have the same sign. Let us rst deal with the sets G (j) (j) w and G w, j =, 2. By (2.), we get for u R/Z, w [ /2, /2], and w 2 G (j) (j) w or w 2 G w, j =, 2, that w w 2 C γ(w ) γ(w 2 ). Furthermore, we deduce in these cases that ( ) sin γ(u+w) γ(u) γ(u+w, γ(u+w2) γ(u) ) γ(u) γ(u+w 2) γ(u) c(u, u + w, u + w 2 ) = 2 γ(u + w ) γ(u + w 2 ) γ(u+w) γ(u) γ(u+w + γ(u+w2) γ(u) ) γ(u) γ(u+w 2) γ(u) 2 C γ(u + w ) γ(u + w 2 ) γ(u+w) γ(u) w γ(u+w2) γ(u) w 2 w w 2 Here we have used the estimate (2.) and the fact that v v v is locally Lipschitz on R n {0} to get the last inequality. Using the fundamental theorem of calculus, we then get 0 c(u, u + w, u + w 2 ) C γ (u + τw ) γ (u + τw 2 ) dτ. w w 2.

12 Since for w 2 G () w we have w 2 2 w w 2, we deduce that /2 u R/Z /2 C u R/Z /2 + C C R/Z w 2 G () w c p (u, u + w, u + w 2 )dw du /2 w 2 G () w ( ) 0 γ (u) γ (u + τw 2 ) p w 2 p dw du 0 γ (u + τw ) γ (u) w p dw du /2 j= u R/Z /2 2 j w w 2 2 j+ w w 2 /2 ( ) 0 γ (u) γ (u + τw 2 ) p dτ w 2 p dw du + C γ p W /p,p C j= u R/Z w 2 j 0 γ (u) γ (u + τ2 j+ w ) p dτ 2 j w p dw du + C γ p W /p,p C 2 j j= u R/Z C γ p W /p,p. 0 γ (u) γ (u + τ w) p dτ w p d wdu + C γ p W /p,p Using that w w w 2 for w 2 G () w, we get by the same arguments that /2 () w 2 G u R/Z /2 w c p (u, u + w, u + w 2 )dw du C γ p W /p,p. 2

13 Futhermore, we can estimate using that w 2 w w 2 for w 2 G (2) w that /2 u R/Z /2 w 2 G (2) w c p (u, u + w, u + w 2 )dw du C /2 u R/Z /2 + C C R/Z w 2 G (2) w ( ) 0 γ (u) γ (u + τw 2 ) p w p dw du 0 γ (u + τw ) γ (u) w p dw du /2 j= u R/Z /2 2 j w w 2 2 j w 0 γ (u) γ (u + τw 2 ) p w p dw du + C γ p W /p,p C j= u R/Z w 2 j 0 γ (u) γ (u + τ2 j w ) p w p dw du + C γ p W /p,p C 2 j( p) j= u R/Z C γ p W /p,p. γ (u) γ (u + τ w) p w p d wdu + C γ p W /p,p In the same way we get using w w w 2 for w 2 G (2) w that /2 u R/Z /2 w 2 G (2) w c p (u, u + w, u + w 2 )dw du C /2 w 2 u R/Z /2 + R/Z G (2) w ( ) 0 γ (u) γ (u + τw 2 ) p w p dw du 0 γ (u + τw ) γ (u) w p dw du C γ p W /p,p. To deal with the sets B (j) w, j =, 2, 3, we observe that substituting w = w, 3

14 ũ = u + w, and w 2 = w 2 w we get /2 u R/Z /2 and w 2 B (3) w c p (u, u + w, u + w 2 )dw du /2 u R/Z /2 (2) w 2 G w c p (ũ + w, ũ, ũ + w 2 )dw du /2 u R/Z /2 w 2 B (2) w c p (u, u + w, u + w 2 )dw du /2 u R/Z /2 w 2 G (2) w c p (ũ + w, ũ, ũ + w 2 )dw du. To deal with B w () we substitute ũ = u + w, w = w and w 2 = + w 2 + w if w [ /2, 0], w 2 = w 2 + w if w [0, /2] to get /2 u R/Z /2 w 2 B () w c p (u, u + w, u + w 2 )dw du 0 u R/Z /2 + u R/Z /2 0 /2 w 2 B () w c p (ũ + w, ũ, ũ + w 2 )dw du w 2 B () w c p (ũ + w, ũ, ũ + w 2 )dw du (2) w 2 G u R/Z /2 Together with the above estimates we hence get /2 R/Z /2 Summing it all up and using w c p (ũ + w, ũ, ũ + w 2 )dw du w 2 B (i) w c p (u, u + w, u + w 2 )dw du C γ W /p,0. [ /2, /2] = G () () w G w G (2) (2) w G w B w () B w (2) B w (3), 4

15 we nally deduce that J p (γ) j=,2 + ( /2 R/Z /2 + j=,2,3 R/Z /2 C γ p W /p,p. w 2 G (j) w c p (u, u + w, u + w 2 )dw du /2 (j) w 2 G R/Z /2 /2 ) c p (u, u + w, u + w 2 )dw du w w 2 G (j) w c p (u, u + w, u + w 2 )dw du This shows that J p (γ) is nite if γ W 2 /p (R/Z, R n ) and thus nishes the proof of Theorem.3. References [Bla0a] Simon Blatt. Boundedness and regularizing eects of O'Hara's knot energies. Accepted for publication by the Journal of Knot Theory and its Ramication, 200. [Bla0b] Simon Blatt. The gradient ow of O'Hara's knot energies. Preprint, 200. [Bla] [BR] Simon Blatt. The energy spaces of the tangent-point energies. Preprint, 20. Simon Blatt and Philipp Reiter. Stationary points of O'Hara's knot energies. Preprint, 20. [Dav98] Guy David. Unrectiable -sets have vanishing analytic capacity. Rev. Mat. Iberoamericana, 4(2):369479, 998. [Lég99] J. C. Léger. Menger curvature and rectiability. Ann. of Math. (2), 49(3):83869, 999. [RS96] Thomas Runst and Winfried Sickel. Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial dierential equations, volume 3 of de Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin, 996. [SSvdM09] Paweª Strzelecki, Marta Szuma«ska, and Heiko von der Mosel. A geometric curvature double integral of Menger type for space curves. Ann. Acad. Sci. Fenn. Math., 34():9524,

16 [SSvdM0] Pawel Strzelecki, Marta Szuma«ska, and Heiko von der Mosel. Regularizing and self-avoidance eects of integral Menger curvature. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9():4587, 200. [Szu09] Marta Szuma«ska. Integral versions of Menger curvature: smoothing potentials for rectiable curves. PhD thesis, Technical University of Warsaw, [Tri83] Hans Triebel. Theory of function spaces, volume 78 of Monographs in Mathematics. Birkhäuser Verlag, Basel,

Boundedness and Regularizing Effects of O Hara s Knot Energies

March, 2 2:24 WSPC/INSTRUCTION FILE OHarasEnergies Boundedness and Regularizing Effects of O Hara s Knot Energies Simon Blatt Departement Mathematik ETH Zürich Rämistrasse CH-84 Zürich Switzerland simon.blatt@math.ethz.ch