Sharp interpolation inequalities for discrete operators and applications

We consider interpolation inequalities for imbeddings of the $l^2$-sequence spaces over $d$-dimensional lattices into the $l^\infty_0$ spaces written as interpolation inequality between the $l^2$-norm of a sequence and its difference. A general method is developed for finding sharp constants, extremal elements and correction terms in this type of inequalities. Applications to Carlson's inequalities and spectral theory of discrete operators are given.


Introduction
In this paper we study imbeddings of the sequence space l 2 (Z d ) into l ∞ 0 (Z d ) written in terms of a interpolation inequality involving the l 2norms both of the sequence u ∈ l 2 (Z d ), and the sequence of differences ∇u, where for u ∈ l 2 (Z) and n ∈ Z Du(n) = u(n + 1) − u(n), and for u ∈ l 2 (Z d ) and n ∈ Z d ∇u(n) = {D 1 u(n), . . . , D d u(n)}, Before we describe the content of the paper in greater detail we give a simple but important example [16], namely, let us prove the one-dimensional inequality sup n u(n) 2 ≤ u Du . (1.1) The proof repeats that in the continuous case. For an arbitrary γ ∈ Z we have 2u 2 (γ) = Below we consider separately interpolation inequalities of the form in dimension d = 1, 2 and d ≥ 3. By notational definition K d (θ) is the sharp constant in this inequality. This inequality clearly holds for θ = 1 (with K d (1) = 1), and if it holds for a θ = θ * ∈ [0, 1), then it holds for θ ∈ [θ * , 1], when the 'weight' of the stronger norm u is getting larger (see (1.11)). For d = 1 we show that (1.2) holds for 1/2 ≤ θ ≤ 1 and find explicitly the corresponding sharp constant: In the limiting case θ = 1/2 we have K 1 (1/2) = 1, and we supplement inequality (1.1) (which is, in fact, sharp) with a refined inequality which for any d ∈ (0, 4) has a unique extremal sequence u * with Du * 2 / u * 2 = d.
In the 2D case (1.2) holds for 0 < θ ≤ 1 and the sharp constant is given by (1.5) where K is the complete elliptic integral of the first kind, see (3.8).
The constant K 2 (θ) logarithmically tends to ∞ as θ → 0 + , and for θ = 0 we have the following limiting logarithmic inequality of Brezis-Galluet type: where the constants in front of logarithms and 2π are sharp. The inequality saturates for u = δ, otherwise the inequality is strict. Finally, in dimension three and higher the inequality holds for the limiting exponent θ = 0: where the sharp constant is given by (1. 8) In the three dimensional case the constant K 3 can be evaluated in closed form since it is expressed in terms of the so-called third Watson's triple integral: where (see [3] and the references therein) (1.10) It is natural to compare interpolation inequalities for differences and inequalities for derivatives in the continuous case. While in the continuous case the L ∞ -norm is the strongest (at least locally), in the discrete case the l 1 -norm is the strongest. Obviously, u l ∞ ≤ u l p for p ≥ 1, and therefore u l p ≤ u l q for q ≤ p: u p l p ≤ u p−q l ∞ u q l q ≤ u p−q l p u q l q . Also, unlike the continuous case, the difference operator is bounded : (1.11) Roughly speaking, the situation (at least in the one-dimensional case) is as follows. The discrete inequality (1.2) for d = 1 holds for θ ∈ [1/2, 1], while the corresponding continuous inequality , f ∈ H 1 (Q) holds only for θ = 1/2 in case when Q = R, and for θ ∈ [0, 1/2] for periodic function with zero mean, Q = T 1 . Hence, it makes sense to compare the constants at a unique common point θ * = 1/2 where both constants are equal to 1. For n-order derivatives and differences, n > 1, the constants in the discrete inequalities are strictly greater than those in the continuous case, the corresponding θ * = 1 − 1/(2n).
For example, the second-order inequality on the line R and the corresponding discrete inequality are as follows Both constants are sharp, the second one is strictly greater than the first. Up to a constant factor (and shift of the origin) the family of extremal functions in the first inequality is produced by scaling In the discrete inequality the unique extremal sequence is see (5.7) for the explicit formula for u * (n). In two dimensions in the continuous case the imbedding H 1 ⊂ L ∞ holds only with a logarithmic correction term involving higher Sobolev norms (and θ = 0), which is the well-known Brezis-Gallouet inequality. On the contrary, in the 2D discrete case inequality (1.2) holds for θ ∈ (0, 1] and also requires a logarithmic correction for θ = 0, see (1.6).
Next, we consider applications of discrete interpolation inequalities. Using the discrete Fourier transform and Parseval's identities we show that each discrete interpolation inequality is equivalent to an integral Carslon-type inequality. For example, in the 1D case, setting for a function g ∈ L 2 (0, 2π) we obtain that inequality (1.1) is equivalent to the sharp inequality , with no extremal functions, while the refined inequality (1.4) is equivalent to the inequality Developing further this approach we prove a Sobolev l q -type discrete inequality for a non-limiting exponent (1.12) Our explicit estimate for the constant C(q, d) is non-sharp, moreover, it blows up as q → 2d/(d − 2) however, it is sharp in the limit q → ∞. Finally, we apply the results on discrete inequalities to the estimates of negative eigenvalues of discrete Schrödinder operators acting in l 2 (Z d ). Here −∆ := D * D and V (n) ≥ 0. Each discrete interpolation inequality for the imbedding into l ∞ (Z d ) produces by the method of [7] a collective inequality for families of orthonormal sequences, which, in turn, is equivalent to a Lieb-Thirring estimate for the negative trace. For example, we deduce from (1.7) the estimate which holds for d ≥ 3.
We finally point out that in the continuous case the classical Lieb-Thirring inequality for the negative trace of operator (1.13) is as follows (see [14], [13], [6])
We consider a more general problem of finding sharp constants, existence of extremals and possibly correction terms in the inequalities of the type including, to begin with, the problem of finding those θ for which (2.1) holds at all. Here Since |a − b| ≥ ||a| − |b||, we have and we could have further reduced our treatment to the case when u(n) ≥ 0. However, we shall be dealing below with a more general problem (2.4) which has both sing-definite and non-sign-definite extremals. We have the following 'reverse' Poincare inequality: The adjoint to D is the operator: To find the sharp constant K 1 (θ) in (2.1) we consider a more general problem: find V(d), where V(d) is the solution of the following maximization problem: where 0 < d < 4. Its solution is found in terms of the Green's function of the corresponding second-order self-adjoint positive operator, see [18], [1]. The spectrum of the operator −∆ = D * D is the closed interval [0, 4], and we set (2.5) Let δ be the delta-sequence: δ(0) = 1, δ(n) = 0 for n = 0, and let G λ = {G λ (n)} ∞ n=−∞ ∈ l 2 (Z) be the Green's function of operator (2.5), that is, the solution of the equation: (2.6) Then we have by the Cauchy-Schwartz inequality (2.7) Furthermore, this inequality is sharp and turns into equality if and only if u = const · G λ . We find in Lemma 2.2 explicit formulas for V(d) and G λ (n). Nevertheless, we now independently prove the following two symmetry properties of V(d) and G λ (n), especially since their counterparts will be useful in the two-dimensional case below. (2.8) For λ > 0 and n ∈ Z Proof. For u ∈ l 2 (Z) we define the orthogonal operator T T u = u ⋆ := {(−1) |n| u(n)} ∞ n=−∞ . Then clearly u 2 = u ⋆ 2 and, in addition, (2.10) Therefore if for a fixed d and u = u d we have then for u * = T u it holds which gives that V(4 − d) ≥ V(d). However, the strict inequality here is impossible, since otherwise by repeating this procedure we would have found that V(d) > V(d). This proves (2.8).
It remains to show that for λ > 0 G λ (n) > 0 for all n. Since A(λ) is positive definite, it follows that G λ (0) = (A(λ)G λ , G λ ) > 0. We use the maximum principle and suppose that for some n = 1, G λ (n) < 0. Since G λ (n) → 0 as n → ∞ and G λ (0) > 0, it follows that G λ attains a global strictly negative minimum at some point n > 1 (the case n < −1 is similar). Then the sum of the first three terms in (2.20) is non-positive and the fourth term is strictly negative, which contradicts δ(n) = 0. This proves that G λ (n) ≥ 0 for all n. Finally, to prove strict positivity, we suppose that G λ (n) = 0 for some n > 1. Then we see from (2.20) that G λ (n − 1) + G λ (n + 1) = 0, and what has already been proved gives G λ (n−1) = G λ (n+1) = 0. Repeating this we reach n = 1 giving that G λ (0) = 0, which is a contradiction.
To denote the three norms of G λ we set (2.11) Lemma 2.1. The functions f , g and h satisfy Proof. Let λ > 0. Then A(λ) = D * D + λ. Taking the scalar product of (2.6) with G λ we have Differentiating this formula with respect to λ we obtain where we used that G λ +A(λ)G ′ λ = 0, which, in turn, follows from (2.6). The case λ < −4 is treated similarly taking into account that now satisfies the functional equation Proof. It follows from (2.9) and (2.11) that .
and we obtain from (2.12) Next, we find explicit formulas for f , g and h.
We finally point out that the equality (2.9) can now be also verified by a direct calculation: We can now give the solution to the problem (2.4).
Proof. It follows from (2.7) that for any u ∈ l 2 (Z) and, furthermore, for with u * λ 2 = 1 the above inequality turns into equality. Next, using (2.17) we find the formula for the function d The inverse function λ(d) is given by (2.26) and with this λ(d) we have Therefore u * λ(d) is the extremal sequence in (2.4) and its solution is Remark 2.1. It is worth pointing out that in accordance with Proposition 2.1 and Corollary 2.1 we directly see here that Corollary 2.2. For any u ∈ l 2 (Z) inequality (1.1) holds, the constant 1 is sharp and no extremals exist. The following refined inequality holds: Proof. Inequality (2.27) follows from (2.25) by homogeneity.
Proof. The proof is similar to the proof of Theorem 2.5 in [18] where the classical Sobolev spaces were considered. For convenience we include some details. We first observe that inequality (2.1) cannot hold for θ < 1/2, since otherwise we would have found that The case θ = 1/2 was treated above and we assume in what follows that θ > 1/2. We set (2.29) Then, using (2.7), we have We have taken into account in the last equality that .
Hence, the supremum in the above formula is a (unique) maximum on λ ∈ R + of the function In view of (2.12) this gives Hence (2.29) is satisfied for u * = G λ * the two inequalities in (2.30) become equalities, and u * is the unique extremal. The graph of the function K 1 (θ) is shown in Fig.1 on the right. Here K 1 (1/2) = 1 corresponds to (1.1), and K 1 (1) = 1 corresponds to the trivial inequality u(0) 2 ≤ u 2 with extremal u = δ.
Remark 2.2. In this theorem we do not use the formula (2.25) for V(d). However, if we do, then finding K 1 (θ) for θ ∈ [1/2, 1] becomes very easy. In fact, by the definition of V(d) and homogeneity, The corresponding d * = (4θ − 2)/θ ≤ 2 and λ(d * ) > 0, see (2.26). This also explains why the region of negative λ does not play a role in Theorem 2.2.

2D case
In this section we consider the two-dimensional inequalities and address the same problems as in the previous section. We set . As in the 1D case we shall be dealing with the following extremal problem: The resolvent set of −∆ + λ is (−∞, −8) ∪ (0, ∞) and as before we consider the positive self-adjoint operator operator Our main goal is to find the Green's function of it: more precisely, G λ (0, 0).
Proof. The proof is completely analogous to that of Proposition 2.1 and Corollary 2.1, where the functions f (λ), g(λ) and h(λ) have the same meaning as in (2.11) and satisfy (2.12). The operator T is as follows T u(n, m) = (−1) |n+m| u(n, m).
where K(k) is the complete elliptic integral of the first kind: . . (3.9) Therefore for λ > 0, using (2.23) where the last integral was calculated by transforming general elliptic integrals to the standard form (see formula 3.147.7 in [8]).
Remark 3.1. The equality in (3.5) also follows from (3.9) by changing the variables (x, y) → (x ′ + π, y ′ + π) and using the fact that the integrand is even.
We also see from the first formula that for small positive θ the leading term in the second factor in (3.14) is while the first factor tends to 1. This proves (3.13). For example, In the limiting case θ = 0 inequality (3.1) holds with a logarithmic correction term of Brezis-Galouet type [4], [1].
The solution of the extremal problem (3.2) is given in terms of the functions f (λ), g(λ) and h(λ): (3.16) where E(k) is the complete elliptic integral of the second kind: , (3.17) where λ(d) is the inverse function of the function d(λ): Their graphs are shown in Fig. 3. Finally, V(4) = 1 and u * = δ.
Proof. We act as in Theorem 2.1, the essential difference being that we now do not have a formula for the inverse function λ(d), by means of which we construct the extremal element for each d. Although d(λ) is given explicitly, the monotonicity of it required for the existence of the inverse function is a rather general fact and can be verified as in [18,Theorem 2.1], where the continuous case was considered.

Figure 3. Graphs of d(λ) and λ(d).
We now find an explicit majorant V 0 (d) for the implicitly defined solution V(d). In view of the symmetry (3.4) it suffices to study the case d → 0 only and then, by replacing d → d(8−d)/8 we get the symmetric expansions valid for both singularities. We have the following expansions Truncating the first expansion and solving d = (5 ln 2 − ln λ − 1)λ, we have is the −1th branch of the Lambert function. Using the known asymptotic expansions for the Lambert function, we get the following expression for λ(d) .
Using the expansions at λ = ∞ in (3.19) and (3.21) we find that where the constants in front of logarithms and 2π are sharp. The inequality saturates for u = δ, otherwise the inequality is strict.

3D case
In the three-dimensional case the following result holds which is somewhat similar to the classical Sobolev inequality for the limiting exponent.

(4.4)
The constant is sharp and there exists a unique extremal element, which does not lie in l 2 (Z 3 ), but rather in l ∞ 0 (Z 3 ), but whose gradient does belong to l 2 (Z 3 ). Furthermore, as we already mentioned in §1, we have the closed form formula for K 3 (0) (see [3]) Proof. We have to find the fundamental solution G λ (k, l, m) of the equation Similarly to the 1D and 2D cases we find that the function As before we have the inequality which saturates for u = const · G λ . For λ > 0 as in the 1D and 2D cases we have g λ ∈ L 2 (T 3 ), and, hence, G λ ∈ l 2 (Z 3 ) for λ > 0. In particular, using (3.10) we find (4.7) However, unlike the previous two cases, now g λ is integrable for all λ ≥ 0 including λ = 0: g λ ∈ L 1 (T 3 ) for λ ≥ 0. Therefore the Green's function G 0 is well defined and belongs to l ∞ 0 (Z 3 ). We point out, however, that since g 0 / ∈ L 2 (T 3 ), it follows that G 0 / ∈ l 2 (Z 3 ). For λ = 0, the integrand has only a logarithmic singularity at x = 0 and we obtain We now see that f (λ) := G λ (0, 0, 0) is continuous on λ ∈ [0, ∞) and is of the order 1/λ at infinity. This gives that for θ ∈ (0, 1) the function λ θ f (λ) vanishes both at the origin and at infinity. Hence, it attains its maximum at a (generically) unique point λ * (θ), and the claim of the theorem concerning the case θ ∈ (0, 1) follows in exactly the same way as in Theorem 2.2.
Setting λ = 0 in (4.6) we obtain (4.3) with (4.4). It remains to verify that ∇G 0 ∈ l 2 (Z 3 ). To see this we use notation (2.11) and Lemma 2.1. We obtain Since the integral on right-hand side is bounded for λ = 0 we have ∇G 0 2 < ∞. Finally, G λ has strictly positive elements for λ ≥ 0, since we have as before the maximum principle. In the case when λ = 0 we use, in addition, the fact that G 0 ∈ l ∞ 0 . The proof is complete.
The graph of K 3 (θ) is shown in Fig. 4. .
(4.8) In §6 we give an independent elementary proof of this inequality.

Higher order difference operators
The method developed above admits a straight forward generalization to higher order difference operators. We consider the second-order operator in the one dimensional case: where −∆u(n) := D * Du(n) = − u(n + 1) − 2u(n) + u(n − 1) .
As before, we have to find the Green's function G λ solving A(λ)G λ = δ. Furthermore, for finding K 1,2 (θ) it suffices to solve this equation for λ > 0. Setting and arguing as in Lemma 2.2 we get from (5.2) Now a word for word repetition of the argument in Theorem 2.2 gives that Therefore we see from (5.4) that K 1,2 (θ) < ∞ if and only if 3 4 ≤ θ ≤ 1.
Setting n = 0 and n = 1 we obtain a linear system for a(λ) where G λ (0) is given in (5.4) and Solving this system we find a(λ): and, consequently, the formula for G λ (n) with λ > 0: (5.7) where q(λ) is given in (5.6).
Remark 5.1. It is not difficult to find the function V(d), that is, the solution of the maximization problem V(d) := sup u(0) 2 : u ∈ l 2 (Z), u 2 = 1, ∆u 2 = d , (5.9) where d ∈ [0, 16]. For this purpose we also need the expression for the Green's function G λ (0) in the region λ ≤ −16, which is as follows  This time we do not have the maximum principle, and the Green's function G λ (n) is not positive for all n, but is rather oscillating with exponentially decaying amplitude, see Fig. 6. Nor do we have the symmetry V(d) = V(16 − d) in Fig. 7 that we have seen in the firstorder inequalities in the one-and two-dimensional cases, see (2.8) and (3.4). The maximum is attained at d = 6 corresponding to u = δ. The component λ ∈ (0, ∞) of the resolvent set corresponds to d ∈ (0, 6) and λ ∈ (−∞, −16) corresponds to d ∈ (6,16).

Applications
Discrete and integral Carlson inequalities. We now discuss applications of the inequalities for the discrete operators, and our first group of results concerns Carlson inequalities. The original Carlson inequality [5] is as follows: where the constant π is sharp and cannot be attained at a non identically zero sequence {a k } ∞ k=1 . This inequality has attracted a lot of interest and has been a source of generalizations and improvements (see, for example, [10], [12] and the references therein, and also [18] for the most recent strengthening of (6.1)). Inequality (6.1) has an integral analog (with the same sharp constant) As was first observed in [9], inequality (6.1) is equivalent to the inequality for periodic functions f ∈ H 1 per (0, 2π), Accordingly, inequality (6. 3) for f ∈ H 1 (R) is equivalent (as was first probably observed in [15]) to (6.2) by setting g = F f and further restricting g (and f ) to even functions. Furthermore, the unique (up to scaling) extremal function f * (x) = e −|x| in (6.3) on the whole axis produces the extremal function g * (t) = 1/(1 + t 2 ) in (6.2).
where D n is as in (5.14) and θ belongs to a certain subinterval of [0, 1] uniquely defined in the corresponding theorem: Then for θ as in (6.15) and θ < 1 (6.17) Proof. For arbitrary ξ 1 , . . . , ξ N ∈ R we construct a sequence f ∈ l 2 (Z d ) Applying (6.14) and using orthonormality we obtain for a fixed k f (k) 2 ≤ K(θ) We now set ξ j := u (j) (k): Summing over k ∈ Z d and using orthonormality we obtain (6.17).  Remark 6.5. The last inequality holding in dimesion three and higher curiously resembles the celebrated Ladyzhenskaya inequality that is vital for the uniqueness of the weak solutions of the two-dimensional Navier-Stokes system: We now exploit the equivalence between the inequalities for orthonormal families and spectral estimates for the negative trace of the Schrödinger operators [14].