Agmon Kolmogorov Inequalities on l 2 (Z d )

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1 Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN E-ISSN Publishe by Canaian Center of Science an Eucation Agmon Kolmogorov Inequalities on l (Z ) Arman Sahovic Mathematics Department, Imperial College Lonon, Lonon, UK Corresponence: Arman Sahovic, Mathematics Department, Imperial College Lonon, 80 Queen s Gate, Lonon W 3QS, UK. arman@imperial.ac.uk Receive: December 3, 03 Accepte: February 3, 04 Online Publishe: April 9, 04 oi:0.5539/jmr.v6np38 URL: Abstract Lanau Kolmogorov inequalities have been extensively stuie on both continuous an iscrete omains for an entire century. However, the research is limite to the stuy of functions an sequences on R an Z, with no equivalent inequalities in higher-imensional spaces. The aim of this paper is to obtain a new class of iscrete Lanau Kolmogorov type inequalities of arbitrary imension: ϕ l (Z ) μ p, D ϕ p/, l (Z ) ϕ p/ l (Z ) where the constant μ p, is explicitly specifie. In fact, this also generalises the iscrete Agmon inequality to higher imension, which in the corresponing continuous case is not possible. Keywors: Lieb Thirring inequalities, Lanau Kolmogorov inequalities, iscrete inequalities, iscrete spaces, functional inequalities, sequence spaces. Introuction In 9, Hary, Littlewoo an Pólya prove the following inequalities for a function f L (R): f L (, ) f / f /, () L (, ) L (, ) f L (0, ) f / f /, () L (0, ) L (0, ) with the constants an being sharp. These results sparke interest in inequalities involving functions, their erivatives an integrals for a century to come. Specifically, in 93, Lanau prove the following inequality: For Ω R, an f L (Ω): f L (Ω) f / L (Ω) f / L (Ω), with the constant being sharp. This result in turn was motivation for A. Kolmogorov, where in 939 he foun sharp constants for the more general case, using a simple, but very effective inuctive argument to exten the case to higher orer erivatives: f (k) L (Ω) C(k, n) f (n) k/n L (Ω) f k/n L (Ω), where, for k, n N with k < n, he etermine the best constants C(k, n) R for Ω=R. Since then, there has been a great eal of work on what are nowaays known as the Lanau Kolmogorov inequalities, which are in their most general form: f (k) L p K(k, n, p, q, r) f (n) α L q f β L, r with the minimal constant K = K(k, n, p, q, r). The real numbers p, q, r ; k, n N with (0 k < n) an α, β R take on values for which the constant K is finite (Gabushin, 967). However, literature on iscrete equivalents of those inequalities remaine very limite for a long time. In 979, E. T. Copson was one of the first to fin equivalent results for sequences, series an ifference operators. Inee, he foun the iscrete equivalent to () an (). For a square summable sequence, {a(n)} n Z l (Z) an a ifference operator (Da)(n) := a(n + ) a(n), we have: Da l (, ) a / l (, ) D a /, (3) l (, ) 38

2 Da l (0, ) a / l (0, ) D a / l (0, ), (4) with the constants an yet again being sharp. Z. Ditzian (98/83) then extene those results to establish best constants for a variety of Banach spaces, aing equivalent results for continuous shift operators f (x + h) f (x); x R, f Ł (R). Comparing inequalities such as () an (), with (3) an (4) respectively, it was suspecte that sharp constants were ientical for equivalent iscrete an continuous Lanau Kolmogorov inequalities for p = q = r. Inee, in the cases p =,,, this was true for the whole an semi-axis. However, the general case has since been shown to be false, as for example emonstrate in 988 by M. K. Kwong an A. Zettl, where they prove that for many values of p, the iscrete constants are strictly greater than the continuous ones. Another important special case of the Lanau Kolmogorov inequalities is the Agmon inequality, proven by Agmon (00). Viewe as an interpolation inequality between L (R) an L (R), he states the following: f L (R) f / L (R) f / L (R). Thus, throughout this paper we shall call, for a omain Ω, a function f L (Ω), a sequence ϕ l (Ω), α, β being Q-value functions of the integers k, n with k n an constants C(Ω, k, n), D(Ω, k, n) R: f (k) L (Ω) C(Ω, k, n) f α(k,n) L (Ω) f (n) β(k,n) L (Ω), (5) D k ϕ l (Ω) D(Ω, k, n) ϕ α(k,n) l (Ω) Dn ϕ β(k,n) l (Ω). (6) Agmon Kolmogorov inequalities, where (6), for Ω = Z will be the central concern of this paper. Specifically we only require the case where k = 0 an n =, whereas the other inequalities, i.e. those concerne with higher orer, have been iscusse in Sahovic (03). These have a variety of applications in spectral theory for example. They can be use to obtain the Generalise Sobolev inequality an thus afterwars to obtain a variety of Lieb Thirring class inequalities. These in turn have vast applications in the theory of quantum mechanics. Alternatively, Agmon Kolmogorov inequalities on higher imensional iscrete spaces are a bit of a novelty in the fiel iscrete inequalities, as usually only one-imensional inequalities are stuie. The inuction metho contains intrinsic ieas translateable to other classes of inequalities.. Agmon Kolmogorov Inequalities Over Z We introuce our notations for the -imensional inner prouct space of square summable sequences. For a vector of integers ζ = (ζ,...,ζ ) Z, we say {ϕ(ζ)} ζ Z l (Z ), if an only if the following norm is finite: ϕ l (Z ) = ( ζ Z ϕ(ζ) p ) /. Then, for ϕ, φ l (Z ),welet<.,. > be the inner prouct on l (Z ): ϕ, φ = ζ Z ϕ(ζ)φ(ζ). We then let D,...,D be the partial ifference operators efine by: (D i ϕ)(ζ) =ϕ(ζ,...,ζ i +,...,ζ ) ϕ(ζ,...,ζ ). The iscrete graient D shall thus take the following form: D ϕ(ζ,ζ,...,ζ )=(D ϕ(ζ), D ϕ(ζ),...,d ϕ(ζ)). Thus, combining this efinition with that of our norm above, we obtain: Further, we require the following notation: D ϕ l (Z ) = D ϕ l (Z ) D ϕ l (Z ). 39

3 Definition. For a sequence ϕ(ζ) l (Z ) with ζ = (ζ,..., ζ ) Z, for 0 k we efine: [ϕ] k =... ϕ(ζ). ζ Z ζ k Z Remark We ientify that [ϕ] 0 = ϕ(ζ) an if we apply this operator for k =, i.e. sum across all coorinates, we obtain the l (Z )-norm: [ϕ] = ϕ l (Z ). We are intereste in a higher-imensional version of the iscrete Agmon inequality (Sahovic, 00), which estimates the sup-norm of a sequence φ l (Z) as follows: φ l (Z) φ l (Z) Dφ l (Z). Thus we commence by lifting this estimate to encompass more variables: Lemma. (Agmon Cauchy Inequality) For the operator D k+, acting on a sequence ϕ(ζ) l (Z ), we have: sup [ϕ] k [D k+ ϕ] / k+ [ϕ]/ k+. ζ k+ Z Proof. Using the iscrete Agmon inequality on the (k + ) th coorinate, we fin: ϕ(ζ,...,ζ ) ( D k+ ϕ(ζ,...,ζ k, l,ζ k+,...,ζ ) ) / ( ϕ(ζ,...,ζ k, l,ζ k+,...,ζ ) ) /. l Z l Z Now we sum with respect to the other coorinates:... ϕ(ζ,...,ζ )... [( D k+ ϕ(ζ,...,ζ k, l,ζ k+,...,ζ ) ) / ζ Z ζ k Z ζ Z ζ k Z l Z ( ϕ(ζ,...,ζ k, l,ζ k+,...,ζ ) ) / ], l Z an use the Cauchy Schwartz inequality on the k th coorinate:... ϕ(ζ,...,ζ )... [( D k+ ϕ(ζ,...,ζ k, l,ζ k+,...,ζ, ) ) / ζ Z ζ k Z ζ Z ζ k Z ζ k Z l Z / We repeat this process to finally obtain: ( ϕ(ζ,...,ζ k, l,ζ k+,...,ζ ) ) / ]. ζ k Z l Z... ϕ(ζ,...,ζ ) (... D k+ ϕ(ζ,...,ζ k, l,ζ k+,...,ζ ) ) / ζ Z ζ k Z ζ Z ζ k Z l Z (... ϕ(ζ,...,ζ k, l,ζ k+,...,ζ ) ) /. ζ Z ζ k Z l Z We estimate the l (Z )-norm of a partial ifference operator with the l (Z )-norm of the sequence itself: Lemma.3 For a sequence ϕ l (Z ) an for i {,...,}, we have: D i ϕ l (Z ) ϕ l (Z ). Proof. We show the argument for D an note that ue to symmetry the other cases follow immeiately. D ϕ l (Z ) = ϕ(ζ +,...,ζ ) (ζ,...,ζ ) ζ Z ( ϕ(ζ +,...,ζ ) + ϕ(ζ,...,ζ ) ) ζ Z ζ Z = 4 ϕ(ζ,...,ζ ) = 4 ϕ l (Z ). ζ Z 40

4 This implies that we can obtain an estimate for any mixe ifference operator as follows: D...D k ϕ l (Z ) D...D l D l+...d k ϕ l (Z ). Therefore, by eliminating l ifference operators, our inequality will contain the constant l. We arrive at our main result, the Agmon Kolmogorov inequalities on l (Z ). Theorem.4 For a sequence ϕ l (Z ), an p {,..., }: ϕ l (Z ) μ p, D ϕ p/ l (Z ) ϕ p/ l (Z ), where μ p, = ( κ / p, ), p/ an κ p, is a constant to be etermine in the following section. Proof. We use Lemma. an Lemma.3 repeately: ϕ l (Z ) [D ϕ] / [ϕ] / [D D ϕ] /4 [D ϕ] /4 [D ϕ] /4 [ϕ] /4 [D...D ϕ] / [ϕ] / = D...D ϕ / l (Z ϕ / ) l (Z ) ϕ l (Z ) D...D ϕ l (Z ) ϕ l (Z ). We have generate an estimate by norms, with exactly norms originating from the term [D ϕ] /. All those will thus involve the operator D, or more formally: Ξ =, where we let Ξ = { D a...d ak D ϕ l (Z ) a i a j i j ; {a,...,a k } {,...,}}. We note that we coul also employ estimates by D i ϕ l (Z ) for any i {,..., }, but our inequality will not change ue to our symmetrising argument. Similarly, we have norms originating from the term [ϕ] /, whose estimates will not involve the operator D. Hence Ξ =, where we let Ξ = { D a...d ak ϕ l (Z ) a i a j i j ; {a,...,a k } {,...,}}. We will now apply Lemma.3 repeately, to reuce the orer of the operator insie the norms to either 0 or. We recognise that we have to estimate all ξ Ξ by ξ = D ϕ l (Z ) or alternatively by ϕ l (Z ). Hence, we choose a p {0,..., } to estimate p elements in Ξ by D ϕ l (Z ), leaving p elements in Ξ to be estimate by ϕ l (Z ). However, for all elements ξ Ξ, we have to provie an estimate by ξ = ϕ l (Z ) only. This means we have p elements in Ξ = Ξ Ξ to be estimate by ϕ l (Z ): ϕ l (Z ) κ p, D ϕ p l (Z ) ϕ p l (Z ), where κ p, remains a constant of the form z with z Q, which we leave to be ientifie in the next section. We thus obtain the following estimate: We now exploit the symmetry of the argument: ϕ + /p l (Z ) κ/p p, D ϕ l (Z ) ϕ (+ p)/p l (Z ). ϕ + /p l (Z ) κ /p p, ( D ϕ l (Z ) D ϕ l (Z ) ) ϕ (+ p)/p l (Z ) = κ /p p, Dϕ l (Z ) ϕ (+ p)/p l (Z ), 4

5 an finally rearrange: ϕ l (Z ) ( κ / p, ) p/ D ϕ p/ l (Z ) ϕ p/ l (Z ). 3. The Constant κ p, It remains to ientify the constant κ p,, we thus give: Theorem 3. We have, for arbitrary imension an p {,..., }: κ p, = p. We will break the proof own into several steps. The metho for fining κ p, will rely largely on the following observation: Let τ(ξ) be the orer of the operator containe in any given ξ Ξ. Then we let Ω i = {ξ τ(ξ)=i}, be the set of all terms in the estimate whose operator has a given orer i. InΞ we have i, an in Ξ,0 i. Lemma 3. For the size of Ω i, we have for : For Ξ : Ω i =( ), i, i an Ξ : Ω i =( ), 0 i. i Proof. We follow by inuction an prove the case of Ξ, noting that the argument for Ξ is symmetrically ientical. We have alreay seen that the formula is correct for =, an now we assume it is true for = l, i.e. for 0 i l : Ω i =( l ), i an thus we have the following list: Ξ ξ ξ ξ Ω 0 Ω Ω... Ω l Z l : D l...d D ( l 0 ) (l ) (l )... (l l ) Now each term of a given orer τ will, by the Agmon Cauchy inequality (Lemma.), generate a term of orer τ an one of orer τ +. Thus we have: Ξ ξ ξ ξ Ω 0 Ω Ω... Ω l Z l+ : D l+...d D ( l 0 ) (l 0 ) + (l ) (l ) + (l )... (l Now we apply the stanar combinatorial ientity a C b + a C b+ = a+ C b+ an consier a C 0 = a C a =, which immeiately implies: Ξ ξ ξ ξ Ω 0 Ω Ω... Ω l+ Z l+ : D l+...d D ( l 0 ) (l ) (l )... (l l ) l ) an hence for = l +, we have: Ω i =( l i ), completing our inuctive step. As iscusse previously, if we consier to estimate a given ξ Ξ using Lemma.3, we will, for example, obtain D...D k ϕ l (Z ) D...D l D l+...d k ϕ l (Z ). We can see that we generate a factor of for every partial ifference operator we eliminate, an thus have, for ξ Ξ an ξ Ξ with orer τ( ξ) an τ( ξ) respectively: ξ τ( ξ) D ϕ l (Z ), an ξ τ( ξ) ϕ l (Z ). 4

6 We note here that κ p, will not epen on which l (Z )-norms in Ξ are chosen to be estimate by ξ = ϕ l (Z ). The reason for this is transparent when consiering that the sum of all the orers i= τ( ξ i ) is a constant an nees to be reuce to the constant p τ( ξ )=p, generating a unique κ p,. Lemma 3.3 The min p κ p, will be attaine at p = an takes on the following explicit form: κ, = i( i ). i=0 Proof. Our minimum constant for Ξ in fact occurs if we choose all ξ Ξ to be estimate by D ϕ l (Z ), i.e. choose p =, the maximum p possible. Our minimum constant, enote by ρ, for all terms in Ξ will thus be: ρ = τ( ξ k ). k= Instea of examining each iniviual element ξ, we consier that all ξ of equal orer i generate the same constant, namely i. Thus we collect all ξ of the same orer, an obtain: ρ = (i ) Ωi = i= i= (i )( i ). Then we nee to estimate all ξ Ξ, an we procee as for Ξ. All ξ nee to be estimate by ϕ l (Z ), each generating the constant i, forming the equivalent pattern as that of Ξ. We thus obtain, for the minimal constant ρ : ρ = i Ωi = i( i ). i=0 i=0 We now see that ρ = ρ, an: κ, = ρ ρ = i( i ). i=0 We are now finally in a position to prove Theorem 3.: Proof. (Proof of Theorem 3.) We are left to analyse the constant s epenence on our choice of p. First we note that in aition to the constant generate above, we will have chosen p terms to be further reuce to ϕ l (Z ), each generating a power of. Hence we aitionally nee to multiply κ, by p. Thus our final constant will be: κ p, = p i( i ) = p+ i=0 i( i ). i=0 Then we can simplify this further by consiering the binomial formula ( + X) n = n k=0 (n k )Xk.Weifferentiate with respect to X an set X = : n n n = k ( n k ). Thus we arrive at: κ p, = p. References Agmon, S. (00). Lectures on elliptic bounary value problems. Provience, RI: AMS Chelsea Publishing. Revision of the 965 original by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Copson, E. T. (979). Two series inequalities. Proc. Roy. Soc. Einburgh Sect. A, 83(-), k=0

7 Ditzian, Z. (98/83). Discrete an shift Kolmogorov type inequalities. Proc. Roy. Soc. Einburgh Sect. A, 93(3-4), Gabushin, V. N. (967). Inequalities for the norms of a function an its erivatives in the metrics lp. Matem. Zametki,, Hary, G. H., Littlewoo, J. E., & Pólya, G. (9). Inequalities (n e.). Cambrige: University Press. Kolmogorov, A. N. (939). On inequalities between the upper bouns of the successive erivatives of an arbitrary function on an infinite interval. Kwong, K. M., & Zettl, A. (988). Lanau s inequality for the ifference operator. Proceeings of the American Mathematical Society, 04, Lanau, E. (93). Ungleichungen für zweimal ifferenzierbare funktionen. Proc. Lonon Math. Soc., 3, Sahovic, A. (00). New constants in iscrete Lieb Thirring inequalities for Jacobi matrices. J. Math. Sci. (N. Y.), 66(3), Problems in mathematical analysis. No. 45. Sahovic, A. (03). Spectral Bouns for Polyiagonal Jacobi Matrix Operators. arxiv:3.90 [math.fa]. Copyrights Copyright for this article is retaine by the author(s), with first publication rights grante to the journal. This is an open-access article istribute uner the terms an conitions of the Creative Commons Attribution license ( 44