INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS. = u (ξ), ξ (0, t).

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1 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS 0. Background and history The mean value theorem gives, for u C 0 (R + ), u(t) t = u(t) u(0) t 0 Hence you might suspect that an inequality of the type 0 u p dt c p 0 = u (ξ), ξ (0, t). t p dt, c p > 0 would by likely to hold for these u. This is indeed true and this is of course the famous original Hardy inequality. The story of this inequality goes back to the 920 s and to the person who it is named after, G.Hardy. Hardy himself proved the corresponding integral version of the inequality i.e. ( ) p p ( x p f(t) p dt f(t)dt) dx p x 0 for p >, f(t) 0, where f is integrable over every finite interval (0, A) and the constant in front of the integral is the best possible. In this thesis, however, we will henceforth only consider the differential versions of this inequality and it s generalizations. It is hard to give a precise definition of what a Hardy (or Hardy-type) inequality is, because of the large number of possibilities of generalizing it. A sufficently precise definition, covering many cases, is given for example in [29] : Definition (Hardy s inequality in R n ) ( ) ( u(x) p p v(x)dx C with R n and where w(x), v(x) are weight functions and 0 < q, p. 0 0 ) u(x) q q w(x)dx Of course, not all such inequalities are true for some C > 0. Neccesary and sufficent conditions for such inequalities to hold in one dimension has been extensively studied by many different authors during the last fifty years. Such conditions usually involves integral criteria for the weight functions w and v.

2 2 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS One of the main reasons to study L 2 Hardy inequalities is their close connections to the Schrödinger equation u = u + V (x)u, = Planck s constant i t and especially the special case that descibes the stationary states u V (x)u = 0 which is one of the cornerstones of classical quantum mechanics. Also, the L p inequalities are of great importance in the study of the p-laplacian and the p-schrödinger equation div ( u p 2 u ) V (x, u) = 0 and it s generalizations. In this thesis we will give examples of applications of Hardy inequalities (from now on abbreviated H.I.) to Schrödinger operators describing many particle systems. Also, in article 4, we will see how improved H.I. may be a valuable tool in the theory of viscous incompressible flow. This is a slight generalization and improvement of previous work by P. Sobolevskii (see [4]). Except for that, the main focus in this thesis will not be on applications, but more on the mathematical aspects of the theory. I will below give a short summary over recent achivements in the theory of Hardy inequalities, which are similar in fashion to my results. The text will not give a complete picture of the field, but it will exemplify with some of the highlights. The interest in Hardy-type inequalities seems to have grown enormously in the last decades, especially in the last ten years lots of important papers have been published. The total material available concerning Hardy, and related, inequalities are extensive. Some references, for the interested reader, may be found in the bibliography at the end of this introduction. 0.. Different aspects and generalizations of Hardy s inequality. The classical generalization of H.I. to higher dimensions is p (0.) u p dx n p Rn R p x dx, u n p C 0 (R n \ {0}), where the constant is the best possible. For p = 2 this inequality is also called the uncertanity principle. The proof of (0.) can be found for example in the, nowadays almost classical, book Hardy type Inequalities by B.Opic and A.Kufner [24]. Another really classical book, which has been a source of inspiration for numerous people through the years, worth mentioning here is Inequalities by G.Hardy, J.E.Littlewood and G.Pólya [24]. A more modern book which has been very influential and rewarding for many researchers in the theory of Hardy-type inequalities, including myself, is Sobolev Spaces by V.Maz ya [37]. Also, for a modern introduction to the theory of H.I., especially in the one dimensional case, the book [30] is recommendable.

3 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS 3 In two dimensions the uncertainity principle does not hold, but if we weaken the singularity a bit by adding a logarithmic term or/and some extra conditions to the functions u, one can get some results. For example, it is true that R2 u 2 u 2 dx C R 2 x 2 ( + ln 2 x ) dx, if u(x)dx = 0 x = and R2 u 2 u 2 dx C R x dx, if u(x)dx = 0 r > x =r (see [43]). For bounded domains and n 3 we have that p u p dx n p,p p dx u W x p 0 () for n > p, where the constant is optimal. That is, the boundedness does not change the value of the best constant. There also exists higher order analogues of the above inequalites, which are applicable to the corresponding operators of higher order, for example the bi-harmonic operator. Examples of such inequalities are the Rellich s inequality [42]: Theorem[Rellich] Let n > 4 and be a bounded domain in R n, then u 2 dx n2 (n 4) 4 u 2 2,2 dx, u W 6 x 4 0 (). We also have a generalization of Rellich inequality to the L p case due to E.B.Davies and A.M.Hinz (see [5]): Theorem[Davies,Hinz] ( ) p (n 2p)(p )n (0.2) u p dx p 2 2,p dx, u W x 2p 0 (), n > 2p. For even higher derivatives, the below theorem was proved in the same article: Theorem[Davies,Hinz] Let m N 0, d N. i) If < p < and n > 2mp, then (0.3) m u p dx C (m, p, n) dx, x 2mp 2m,p u W0 (), where ( C (m, p, n) = m p (n 2kp)(2(k )p + (p )n)). p 2mp k= ii) If p < and n > (2m + )p, then (0.4) ( m u) p dx C 2 (m, p, n) dx, x u (2m+)p W 2m+,p (),

4 4 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS where C 2 (m, p, n) = ( m p (n p)p (n (2k + )p)((2k )p + (p )d)). p (2m+)p k= 0.2. Fractional derivatives. Another way to generalize the H.I. is to consider fractional(non-integer) derivatives also. These are much harder to handle than the ordinary derivatives because the non-local definition of them. In L 2, these may be defined using the Fourier transform. In general one can define the fractional Sobolev space of functions, W s,p 0 (), in some domain, by the completion of C0 () in the seminorm ( ) u(x) u(y) p /p dxdy (p ) (0 < s < ). x y n+sp One of the first multidimensional Hardy inequalities involvning fractional norms was proven by T.Kato [27]: Theorem[Kato] (u, u) 2 R3 u 2 π x dx, u C 0 (R 3 ). The constant 2/π here cannot be improved. This inequality has important consequences in the theory of stability of matter. In a paper [48] from 2000, D.Yafaev obtained the best constants in the higher order and fractional order generalizations of (0.) for functions from W l,2 0 () : Theorem[Yafaev] Let u W l,2 (R n ), then C l ξ R 2l u(ξ) 2 dξ x 2l û(x) 2 dx if l < n/2 n R n and C l ξ R 2l u(ξ) 2 dξ x 2l u(x) (α!) ( α u)(0)x α 2 dx n R n if l > n/2, The best constants here are given by If l is an integer this gives α m l n/2 / Z and m = [l n/2]. C l = 2 2l max{ Γ2 ((n/2 l)/2) Γ 2 ((n/2 + l)/2), Γ2 ((n/2 l + )/2) Γ 2 ((n/2 + l + )/2) }. C l = 2 2l (n 2l) 2 (n 2l + 4) 2... (n + 2l 4) 2. Yafaev used decomposition of the functions u into spherical harmonics to reduce the problem to a one dimensional inequality which is proved by diagonalizing a certain integral operator with the help of the Mellin transform.

5 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS 5 Various results have since then been published concerning inequalites involving fractional Sobolev norms and L p generalizations of the above result. One such result is by V.Maz ya and T. Shaposhnikova: Theorem[Maz ya,shaposhnikova] Let n, p, 0 < s < and sp < n. Then, for an arbitrary function u W s,p 0 (R n ), there holds s( s) Rn u(x) u(y) p Rn c(n, p) dx. (n sp) p x y n+sp x sp R n The main point of this theorem was to show a conjecture of J.Bourgain, H.Brezis and P.Mironescu about the asymptotic behaviour of the constant on the RHS as s tends to 0, or n/p (see [7],[8]). Their proof is completely elementary, but very clever. The value of c(n, p) given in the article is not proven to be optimal. I do not know if the optimal value has been found yet or not. It would certainly be interesting to find out. An older influential article about sharp constant in similar Hardy-Sobolev type inequalities is [33] by E.Lieb Improved optimal Hardy inequalities. One step further, when the best possible constant in an inequality has been found, is to try to see if one could add positive reminder terms containing singularities of lower order. This is often possible in the Hardy case. Many different types of remainder terms are possible. Of course, the possibilities depend very much on the weight function in the main term. One way to improve the H.I. (0.) and it s generalizations (0.3),(0.4) is to add remainder terms of the same type as the main term, but with different exponents. This has been done, for example in the article [2], by F.Gazzola, H.C.Grunau and E.Mitidieri. They proved the following result : Theorem[Gazzola,Grunau,Mitidieri] Let R n be a sufficently smooth domain and let k N, 2k n. Then there exist constants c,..., c k depending only on k, n and, such that for every u W 2,k () with j u = 0 for j N 0 and 2j < k there holds: ( m u 2 dx 2m + 4(m j)) 4 j=(n 2) 2m if k = 2m, m N and m u 2 dx if k = 2m +, m N 0. + ( 2m+ 4 2m+ 2m+ l= c l j= u 2 x dx + 2m c 4m l l= (n + 4m + 2 4j) 2 ) u 2 x 4m+2 2l dx, u 2 dx x 4m 2l u 2 dx x 4m+2

6 6 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS The techniques used here are also a based on reduction of dimension. An infinite series improvent of (0.) involving iterated logartimic weights can be found in [9]: Theorem[Filippas,Tertikas] Let X (t) = ( ln t), X k (t) = X (X k (t)), k = 2, 3,... and let D sup x x. Then, the following inequality hold for any u W,2 0 (): ( ) 2 n 2 u 2 u 2 dx n x dx x 2 X2 ( x D )X2 2( x D )... X2 i ( x D ) u 2 dx. i= Morover, for each k =, 2,... the constant /4 is the best constant for the corresponding k-improved Hardy inequality, that is 4 = inf u 2 dx ( ) n 2 2 u 2 dx k n x 2 4 i= X x X Xi 2 u 2 dx u W,2. 0 () X 2 x X Xk 2 u 2 dx Later, this inequality has been generalized to the L p case and more general weights [4]. Another interesting type of remainder terms, which improve inequality (0.) were obtained in [2]: Theorem[Gazzola,Grunau,Mitidieri] Let n > p, then ( ) n p u p C(p, n) dx dx + p x p p n dx, u W,p 0 (). (The special case with p = 2 is due to H.Brezis and J.L.Vázquez [0]). We will see later how this type of remainder term appear naturally in other Hardy type inequalities as well. Another similar L p inequality, which improves (0.2), is Theorem[Gazzola,Grunau,Mitidieri] Let p 2, let R n (n 2p) be a bounded domain and let e n = B(0, ) = V ol(b(0, )). Then for any u W 2,p () W,p 0 () one has: ( ) p (n 2p)(p )n u p a(p, n) b(p, n) dx dx + dx + dx. p 2 x 2p 2/n x 2p 2 2p/n One natural question here is whether the optimal fractional order inequalities, for examples the ones by Yafaev described above (see [48]), allows one to add reminder terms similar to the ones in the integer order case.

7 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS Improved optimal Hardy inequalities involvning distance functions. In my research I have focused on inequalities where the distance function to the origin, appearing in the main term in the classical Hardy inequality, have been replaced by the distance function to the boundary, δ(x). Those inequalities depend very much on the geometrical properties of. A basic question here is to find the largest possible constant, C p, in the inequality u p dx C p δ(x) dx, u p C 0 () for a specific class of domains R n. For simply connected domains there exists no such positive constant, not depending on the dimension [3]. However, in 986, A. Ancona proved that C 2 for 6 R2 []. It is still an open question whether this constant is optimal or not. One class of domains that are particulary nice and bit easier to treat here are the convex ones. For this subclass the best constant is equal to (p ) p /p p regardless of the dimension. This inequality was proven in the -dimensional case by Hardy himself in 925 [22],[23]. The 2-dimensional case was proven in 997 by T. Matskewich and P.E.Sobolevskii and they pointed out that their proof also holds in any dimension [40]. The proof is based on the fact that it is sufficent to prove the inequality for convex polytopes. Another proof of the inequality in it s full generality(for all p > and all dimensions) was given the following year in an appendix of an article by M.Marcus, V.J.Mizel and Y.Pinchover [35]. Since then a lot of results concerning generalizations and improvements of this type of inequalities have been published by many different authors. Much focus has been given to the question of what type of remainder terms one can obtain. Especially the problem of finding the largest possible constant, D p,n (), such that u p dx C p,n () δ(x) p dx + D p,n() where C p,n () is the optimal constant for the main term i.e. C p,n () = inf u p dx u C0 () dx, δ(x) p dx, u C 0 (), R 2, has been given much attention (see for example [4],[5],[6], [9],[20],[26],[37], [44] and [45]). The ideal situation here would be to know the optimal value of D p,n () for every specific choice of. That is in most cases too much to hope for, so one has to satisfy with some lower(and upper) estimates of the constant. Usually such estimates may be expressed in terms of some geometric properties of. Examples of such estimates for convex domains are : where C p,n () α(p, n) (diam()) p, C p,n() β(p, n), C p p,n () n γ(p, n) δ in () p, diam() = sup x y, = Vol() and δ in () = sup δ(x) x,y x The proofs can be found in [9](in the case p = 2), [26] and [44] respectively.

8 8 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS One may note that first type of estimate follows from the second one which in turn follows from the third one. Another important field of study here is about the question of obtaining as large potential, V (x), as possible such that u p dx C p,n () δ(x) dx + V (x) dx p holds. Here significant contributions have been given by several authors. For example G.Barbatis, S.Filippas and A.Tertikas have been given much insights into this problem in their work (see [4],[5], [6],[20]). One of their main theorems will be given below after some preliminaries: Define recursively X (t) = ( ln t), t (0, ), X k (t) = X (X k (t)), k = 2, 3,... Let be a domain in R n and K a piecewise smooth surface of codimension k. Furthermore, let δ(x) = dist(x, K) fulfil the conditions (0.5) p δ p k p and Then we have the following theorem : = div( δ p k p p 2 δ p k p ) 0 sup δ(x) <. x in \ K Theorem[Barbatis,Filippas,Tertikas] There exists a positive constant D 0 = D 0 (k, p) sup x δ(x) such that for any D D 0 and all u W,p 0 ( \ K) there holds p u p dx k p p δ dx p + p k p ( p 2 ) 2p p δ p X2 ( δ D )... X2 i ( δ D )dx If 2 p < k, then we can take D 0 = sup x δ(x) Moreover, the constants in front of each terms are the best possible in each of the following cases : (a) (b) (c) i= k = n and K = {0} k = and K = 2 k n and K One may note (as in Tertikas) that if p = 2, K = and is convex, then condition (0.5) is automatically satisfied. Hardy type inequalities involvning distance functions will be discussed further in the summaries of my articles.

9 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS Further applications of Hardy s inequality in physics. Except for the direct application of Hardy s inequality to the Schrödinger operator, other useful variants has been successfully developed for applications in other areas of physics. I will give some examples of this below. J.Dolbeault, M.J.Esteban and E.Séré proved an optimal Hardy-type inequality involving Spinors and Pauli matrices [7]: Theorem[Dolbeault,Esteban,Séré] Let σ = (σ i ) i=,2,3 be the Pauli-matrices: ( ) ( ) ( ) 0 0 i 0 σ =, σ 0 2 =, σ i 0 3 = 0 Then for all u W,2 (R 3, C 2 ). R 3 ( σ u 2 + x ) R3 + u 2 u 2 dx x dx, Several extensions and generalizations of this result may be found in the article [8]. The proofs in the mentioned article are very nice examples of clever applications of ideas from [4],[5], [6]. A very interesting and popular subject of study is Hardy-type inequalities related to the magnetic Schrödinger operator and its corresponding magnetic form h[u, u] = ( i a) 2 dx u C0(R n ), n 2, R n where a is a vector of real-valued functions belonging to L 2 (R n ) In a very popular article A.Laptev and T.Weidl [32] showed among other things that an inequality of the type R2 (i + a)u 2 u 2 dx C R + x dx, u 2 2 C 0 (R 2 ), holds where the constant C depends strongly on the magnetic field. They also showed that in some cases also the H.I. R2 (i + a)u 2 u 2 dx C R x dx, u 2 2 C 0 (R 2 \ {0}) hold for some C > 0. This is in great constrast to the ordinary H.I. (with a = 0) which does not hold in R 2. Examples of magnetic fields which gives rise to such an inequality are those of so-called Aharonov-Bohm type (see [32]). In this case it was proved, in the same article, that the optimal constant C takes the form C = min Φ n 2 n Z where Φ = B(x)dx, B = curl a 2π denotes the total flux of the magnetic field B = curl a.

10 0 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS Generalizations of the above results may be found, for example, in [2] or [3].

11 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS Article - A geometrical version of Hardy s inequality for W,p () In a very influential paper [9] H.Brezis and M.Marcus showed that the following improved Hardy inequality holds: Theorem[Brezis,Marcus] Let be convex, then (0.6) u 2 dx 4 u 2 dx + λ() δ 2 (x) u 2 dx, u W,2 0 (), where λ() 4 diam 2 (). They also formulated the question whether the constant λ() could be replaced by a constant of the form D() = γ 2 n or not, where γ is some universal constant. This was proven to be true by M. and T.Hoffmann-Ostenhof and A.Laptev in the following theorem (see [26]): Theorem[M. and T.Hoffmann-Ostenhof,Laptev] Let be convex, then (0.7) where u 2 dx 4 u 2 δ 2 (x) dx + µ n 2 n u 2 dx, u W,2 0 (), µ n = n(n 2)/n S n 2/n 4 They also obtained some refinements involvning logarithmic weights. In my article I generalize this result to the L p case. More precisely, I prove the following inequality : Theorem[Tidblom] Let be convex, then ( ) p p (0.8) u p a(p, n) dx dx + p δ p (x) p n dx, u W,p 0 () The volume dependent constants in front of the remainder terms in (0.7),(0.8) are superior to the inverse diameter constant in (0.6) if the domain in question has a large diameter but small volume, for example if the domain is long and thin. I also show how to obtain similar results for non convex domains. In that case, however, the constant in front of the main term is not likely to be optimal. The method I use is a variant of the elegant, elementary and geometrical method used in [26]. The constant in front of the remainder terms in (0.7),(0.8) is probably far from being optimal. No one has so far being able to find the optimal value. Barbatis, Filippas and Tertikas showed that if is a ball and p = 2, n = 3 then the best constant C n = inf u W,2 0 () B u 2 dx 4 B u 2 dx B u 2 δ 2 dx

12 2 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS is equal to the first eigenvalue, µ of the Dirichlet Laplacian for the unit disc in R 2. For arbitrary n 2 they obtained the estimate (n )(n 3) C n µ Recently, Filippas, Tertikas and Mazya generalized and improved my result by showing that Theorem[Filippas,Tertikas,Maz ya] Suppose R n is a convex domain with δ in <. For < p < n and p q < np, let C() be the best constant in the inequality n p ( ) p ( ) p u p p dx dx + C() u q q dx p δp for C() of the form a(p, n) C() = δ p where δ in = sup δ(x) = the in-radius. in x Then there exist positive constants c i = c i (p, q, n), i =, 2 independent of, such that np n p q c (p, q, n)δin np n p q C() c 2 (p, q, n)δin. This constant is of a stronger type than mine that involved the volume of, since we always have an estimate B n (0, ) δ n in. There are even a lot of examples of domains with infinte volume, but finite in-radius. A simple example of such a domain is an infinite cylinder. In those cases the inverse volume constant become zero whereas the inverse in-radius constant is nonzero. A natural open question left for the future is to see if one can find an even better type of geometric depending constant than the one involvning the inradius.

13 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS 3 Article 2 - A Hardy inequality in the Half-space Consider an unbounded convex domain,, in R n. One might wonder what type of remainder terms one can add to the optimal Hardy inequality : ( ) p p u p dx p δ dx, p C 0 () in this case. The remainder terms obtained in [20],[26] and [44] of the form C() dx where C() = a(p, n) p n resp. C() = b(p, n) δ p in are usually not very good here, since the first one is always equal zero and the other one might be zero too. Let us consider one of the simplest examples of an unbounded domain, namely the upper half-space in R n, R n +. Every one dimensional linear Hardy inequality on the positive real axis (0.9) 0 ( ) p p u p dt p 0 t p dt + 0 v(t) dt immediately give rise to a corresponding inequality in R n + simply by changing t to x n and integrating over all the other variables. Hovever, it is known that you cannot have an inequality like (0.9) where v(t) is nonnegative a.e. Therefore, you have to do something else if you want a positive remainder term. The first improved optimal Hardy inequality in the litterature, is precisely about this situation. The proof is due to V.G.Mazya and appears in his book Sobolev spaces : Theorem[Maz ya] (0.0) u 2 dx 4 R n + R n + He begins by proving the inequality u 2 dx + x 2 n 6 R n + u 2 dx, u C x n (x 2 n + x 2 n) 0 (R n +). 2 Theorem[Maz ya] (0.) x n p u p dx Rn dx, u C R (2p) n p (x 2 n + x 2 n) 0 (R n ) 2 and then he substitutes u(x) = x n /2 v(x) in (0.), which leads to inequality (0.0) after some integrations by parts. By using elementary methods, based on a vector field approach as in [6], I will show how to improve the constant /6 in (0.0) to /8.

14 4 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS I will also prove an old conjecture by Mazya which says that there should exist positive constants C(p, τ) > 0 such that ( ) p p u p dx dx + C(p, τ) R n p + R n δp + R n x p τ dx. + n (x 2 n + x 2 τ/2 n) The same method can be used prove other inequalities for unbounded domains as well.

15 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS 5 Article 3 - Geometric many particle Hardy inequalities In this article we study Hardy inequalities related to Schrödinger operators describing many particle systems. In R 3N we show that N C(N) x i x j 2 in the quadratic form sense, where the x k are points in R 3 and C(N) = 4( + K(N)), where K(N) will be defined below. Also, the asymptotical behaviour of K(N) is investigated. We prove that K(N) N 2, 2 and hence that the optimal choice of C(N) satisfies C(N) 2N. Furthermore, we prove that K(N) lim inf N N > δ > 0. In particular we get that K(N) = O(N) when N. The problem of finding the sharp value of K(N) is directly related to the problem of finding the optimal configuration of points which maximizes the quotient Q N (x,..., x N ) = i j N i j,i k,j k 2 N i j where r ij = x i x j and R ijk is the circumradius of the triangle obtained by letting it s corners to be the points x i, x j, x k. In fact K(N) is defined as the supremum of this quotient. For three points the optimal configuration turns out to be the equilateral traingle and for four points the tetrahedron is optimal. For larger values of N, the problem is much harder. This problem of finding the right configuration of points gives a nice connection between the theory of Hardy inequalities and Geometrical combinatorics. r 2 ij R 2 ijk,

16 6 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS Article 4 - Various results in the theory of Hardy Inequalities Here we give more examples of H.I. that can be proved with the same type of techniques used in my previous articles. I will, among other things, prove that, in the upper half-space in R n the following inequality holds true : n u 2 u 2 dx 4 x 2 n k dx, u C x2 0 (R n +) n R n + k=0 R n + where the constant /4 in front of each term is optimal. In the domain = {x,..., x n } I will prove the inequality u 2 dx ( ) u 2 dx + (n ) 2 u 2 dx, 4 x 2 x 2 n x x 2 n where (n ) 2 also is optimal. This will be used to prove that u 2 dx ( ) u 2 dx 4 x 2 x 2 n ( n ) + c(n) u 2 dx, x k (x x 2 n) 2 where (n ) 2 c(n) = 2 n(2n + n 2). This c(n) is however most likely not optimal. k= In later sections I will make some comments of how to generalize and improve some known inequalities. As an example, I will generalize my results in article to include higher order derivatives and vector valued functions (in the one dimensional case). I will also calculate an explict expression for the largest possible angle, α, such that u 2 dx u α 4 α 2 δ(x) dx, u 2 C 0 (), where α is a conical sector of angle α in R 2 and δ(x) = dist(x, α ). This has been done numerically by E.B. Davies in [3] and the result up to three decimals was that α The exact value is ( α = π + 4 arctan 4 Γ ( ) Γ ( 4 ) 2 ) ( = π + 4 arctan Some questions about similar problems will also be disdussed. In the last section, we will see how optimal H.I. with remainder terms can be used to prove existence and uniqueness in certain special cases of the stationary Navier-Stokes equations { ν v + n k= v k v x k = p + f(x) div v = 0, ).

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