Abstract
We consider interpolation inequalities for imbeddings of the -sequence spaces over -dimensional lattices into the spaces written as interpolation inequality between the -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.
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1 Introduction
In this paper we study imbeddings of the sequence space into written in terms of a interpolation inequality involving the -norms both of the sequence , and the sequence of differences , where for and
and for and
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
The proof repeats that in the continuous case. For an arbitrary we have
Below we consider separately interpolation inequalities of the form
in dimension and . By notational definition is the sharp constant in this inequality. This inequality clearly holds for (with ), and if it holds for a , then it holds for , when the ‘weight’ of the stronger norm is getting larger [see (1.11)].
For we show that (1.2) holds for and find explicitly the corresponding sharp constant:
In the limiting case we have , and we supplement inequality (1.1) (which is, in fact, sharp) with a refined inequality
which for any has a unique extremal sequence with .
In the 2D case (1.2) holds for and the sharp constant is given by
where is the complete elliptic integral of the first kind, see (3.8). The constant logarithmically tends to as , and for we have the following limiting logarithmic inequality of Brezis–Gallouet type:
where the constants in front of logarithms and are sharp. The inequality saturates for , otherwise the inequality is strict.
Finally, in dimension three and higher the inequality holds for the limiting exponent :
where the sharp constant is given by
In the three dimensional case the constant 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)
It is natural to compare interpolation inequalities for differences and inequalities for derivatives in the continuous case. While in the continuous case the -norm is the strongest (at least locally), in the discrete case the -norm is the strongest. Obviously, for , and therefore for :
Also, unlike the continuous case, the difference operator is bounded:
Roughly speaking, the situation (at least in the one-dimensional case) is as follows. The discrete inequality (1.2) for holds for , while the corresponding continuous inequality
holds only for in case when , and for for periodic function with zero mean, . Hence, it makes sense to compare the constants at a unique common point where both constants are equal to . For -order derivatives and differences, , the constants in the discrete inequalities are strictly greater than those in the continuous case, the corresponding .
For example, the second-order inequality on the line 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 , of the extremal , where
In the discrete inequality the unique extremal sequence is ,
see (5.7) for the explicit formula for .
In two dimensions in the continuous case the imbedding holds only with a logarithmic correction term involving higher Sobolev norms (and ), which is the well-known Brezis–Gallouet inequality. On the contrary, in the discrete case inequality (1.2) holds for and also requires a logarithmic correction for , see (1.6).
In higher dimensional case the imbedding fails at all, while inequality (1.2) holds for all .
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 Carlson-type inequality. For example, in the 1D case, setting for a function
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
saturating for each at
Developing further this approach we prove a Sobolev -type discrete inequality for a non-limiting exponent
Our explicit estimate for the constant is non-sharp, moreover, it blows up as however, it is sharp in the limit .
Finally, we apply the results on discrete inequalities to the estimates of negative eigenvalues of discrete Schrödinger operators
acting in . Here and . Each discrete interpolation inequality for the imbedding into 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 .
We finally point out that in the continuous case the classical Lieb–Thirring inequality for the negative trace of operator (1.13) in is as follows (see [6, 13, 14])
2 1D case
Since as , without loss of generality we can assume that .
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 for which (2.1) holds at all. Here
Since , we have
and we could have further reduced our treatment to the case when . 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’ Poincaré inequality:
The adjoint to is the operator:
and
To find the sharp constant in (2.1) we consider a more general problem: find , where is the solution of the following maximization problem:
where .
Its solution is found in terms of the Green’s function of the corresponding second-order self-adjoint positive operator, see [1, 18]. The spectrum of the operator is the closed interval , and we set
Then is positive definite
Let be the delta-sequence: , for , and let be the Green’s function of operator (2.5), that is, the solution of the equation:
Then we have by the Cauchy–Schwartz inequality
Furthermore, this inequality is sharp and turns into equality if and only if .
We find in Lemma 2.2 explicit formulas for and . Nevertheless, we now independently prove the following two symmetry properties of and , especially since their counterparts will be useful in the two-dimensional case below.
Proposition 2.1
For
For and
Proof
For we define the orthogonal operator
Then clearly and, in addition,
Therefore if for a fixed and we have
then for it holds
which gives that . However, the strict inequality here is impossible, since otherwise by repeating this procedure we would have found that . This proves (2.8).
Turning to (2.9) we note that and we see from (2.10) that
and, consequently,
Therefore, if for , solves
then
Since , using definition (2.5) we obtain
which gives
and proves the equality in (2.9).
It remains to show that for for all . Since is positive definite, it follows that . We use the maximum principle and suppose that for some , . Since as and , it follows that attains a global strictly negative minimum at some point (the case 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 . This proves that for all . Finally, to prove strict positivity, we suppose that for some . Then we see from (2.20) that , and what has already been proved gives . Repeating this we reach giving that , which is a contradiction.
To denote the three norms of we set
Lemma 2.1
The functions and satisfy
Proof
Let . Then . Taking the scalar product of (2.6) with we have
Differentiating this formula with respect to we obtain
where we used that , which, in turn, follows from (2.6). The case is treated similarly taking into account that now .
Corollary 2.1
The function defined as follows
satisfies the functional equation
Proof
It follows from (2.9) and (2.11) that
Hence, and we obtain from (2.12)
Next, we find explicit formulas for , and .
Lemma 2.2
The Green’s function belongs to , and both for and
Furthermore the elements can be found explicitly: for
for
Proof
In view of (2.12), for the proof of (2.17) it suffices to find only . We consider two cases: and . For the sequence solves (2.6), which takes the form , or component-wise
We multiply each equation by and sum the results from to . Setting
we obtain
or
which gives .
In the case when Eq. (2.6) becomes and we merely have to change the sign of and we obtain:
and
Since , it follows that .
Using the integral
we finally obtain both for , and
Finally, to obtain the explicit formula (2.18) (which will not be used below) we observe that the Eq. (2.20) for positive (and negative) is a homogeneous linear recurrence relation with constant coefficients. The characteristic equation is
with roots
For the general -solution of (2.20) is for and for . Since we already know that , it follows that . Substituting into (2.20) with we obtain , which gives
and proves (2.18). The proof of (2.19) in the case is totally similar, we only have to use the second root with .
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).
Theorem 2.1
For any the solution of the maximization problem (2.4) is given by
The supremum in (2.4) is the maximum that is attained at a unique sequence , where
for ; for , .
Proof
It follows from (2.7) that for any
and, furthermore, for
with the above inequality turns into equality.
Next, using (2.17) we find the formula for the function defined in (2.15)
The inverse function is given by (2.26) and with this we have
Therefore 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 and . For the inverse function we have the functional equation .
Corollary 2.2
For any inequality (1.1) holds the constant is sharp and no extremals exist. The following refined inequality holds:
For any the inequality saturates for where see (2.26) with . For .
Proof
Inequality (2.27) follows from (2.25) by homogeneity.
Since , we obtain inequality (1.1), and since as the constant is sharp. In view of the refined inequality (2.27) there can be no extremals in the original inequality (1.1).
We now consider (2.1) for .
Theorem 2.2
Inequality (2.1) holds only for . The sharp constant is
For each there exists a unique extremal sequence.
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 , since otherwise we would have found that , , a contradiction with (2.25): as .
The case was treated above and we assume in what follows that . We set
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 of the function
attained at , which gives (2.28). To see that the constant is sharp we use that , and . In view of (2.12) this gives
Hence (2.29) is satisfied for the two inequalities in (2.30) become equalities, and is the unique extremal.
The graph of the function is shown in Fig. 1 on the right. Here corresponds to (1.1), and corresponds to the trivial inequality with extremal .
Remark 2.2
In this theorem we do not use the formula (2.25) for . However, if we do, then finding for becomes very easy. In fact, by the definition of and homogeneity, is the smallest constant for which for all . Therefore
The corresponding and , see (2.26). This also explains why the region of negative does not play a role in Theorem 2.2.
3 2D case
In this section we consider the two-dimensional inequalities
and address the same problems as in the previous section.
We set
Then for . As in the 1D case we shall be dealing with the following extremal problem:
where .
The resolvent set of is 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, .
Proposition 3.1
For
For and
Finally the function satisfies
Proof
The proof is completely analogous to that of Proposition 2.1 and Corollary 2.1, where the functions , and have the same meaning as in (2.11) and satisfy (2.12). The operator is as follows
Lemma 3.1
For the Green’s function and
where is the complete elliptic integral of the first kind:
Proof
Setting
and acting as in Lemma 2.2 we find that
and
Therefore for , using (2.23)
where the last integral was calculated by transforming general elliptic integrals to the standard form (see formula in [8]).
Since is even, we see from (3.5) that formula (3.7) works both for and .
Remark 3.1
The equality in (3.5) also follows from (3.9) by changing the variables and using the fact that the integrand is even.
Theorem 3.1
The inequality
holds for . For the sharp constant is
and for each there exists a unique extremal sequence
Finally with and
The graph of the function is shown in Fig. 2.
Proof
Similarly to Theorem 2.2, we have
where, of course, is given by (3.7). We have the following asymptotic expansions
Hence, for we see that both at and , and the supremum in (3.14) is the maximum, which proves (3.11) and (3.12).
We also see from the first formula that for small positive the leading term in the second factor in (3.14) is
while the first factor tends to . This proves (3.13). For example,
In the limiting case inequality (3.1) holds with a logarithmic correction term of Brezis–Gallouet type [1, 4].
The solution of the extremal problem (3.2) is given in terms of the functions , and :
where is the complete elliptic integral of the second kind:
and where we used .
Theorem 3.2
The solution of problem (3.2) is
where is the inverse function of the function :
and where . Here is defined on , satisfies (3.6) and monotonically increases from to and then from to . The inverse function is defined on and satisfies
Their graphs are shown in Fig. 3. Finally and .
Proof
We act as in Theorem 2.1, the essential difference being that we now do not have a formula for the inverse function , by means of which we construct the extremal element for each . Although 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.
We now find an explicit majorant for the implicitly defined solution . In view of the symmetry (3.4) it suffices to study the case only and then, by replacing we get the symmetric expansions valid for both singularities. We have the following expansions
Truncating the first expansion and solving , we have
where is the th branch of the Lambert function. Using the known asymptotic expansions for the Lambert function, we get the following expression for
Using
and substituting (3.20) into the first expansion we get
where . This justifies our choice of the approximation to :
The constant instead of (and the numerator ) are chosen so that for we have .
The asymptotic expansion of at shows that for , where is sufficiently small.
Using the expansions at in (3.19) and (3.21) we find that
Since
it follows that for for a small . Corresponding to is the finite interval on which computer calculations show that the inequality still holds. This gives that
for all and hence, by symmetry, for .
Thus, we have proved the following inequality.
Theorem 3.3
For
where the constants in front of logarithms and are sharp. The inequality saturates for otherwise the inequality is strict.
4 3D case
In the three-dimensional case the following result holds which is somewhat similar to the classical Sobolev inequality for the limiting exponent.
Theorem 4.1
Let . Then for any
where for and its sharp value for is given by
and there exists a unique extremal element which belongs to .
In the limiting case inequality (4.1) still holds:
where
The constant is sharp and there exists a unique extremal element, which does not lie in , but rather in , but whose gradient does belong to . Furthermore as we already mentioned in Sect. 1, we have the closed form formula for see [3])
Proof
We have to find the fundamental solution of the equation
Similarly to the 1D and 2D cases we find that the function
satisfies
As before we have the inequality
which saturates for .
For as in the 1D and 2D cases we have , and, hence, for . In particular, using (3.10) we find
However, unlike the previous two cases, now is integrable for all including : for . Therefore the Green’s function is well defined and belongs to . We point out, however, that since , it follows that .
For , the integrand has only a logarithmic singularity at and we obtain
We now see that is continuous on and is of the order at infinity. This gives that for the function vanishes both at the origin and at infinity. Hence, it attains its maximum at a (generically) unique point , and the claim of the theorem concerning the case follows in exactly the same way as in Theorem 2.2.
Setting in (4.6) we obtain (4.3) with (4.4). It remains to verify that . To see this we use notation (2.11) and Lemma 2.1. We obtain
Since the integral on right-hand side is bounded for we have . Finally, has strictly positive elements for , since we have as before the maximum principle. In the case when we use, in addition, the fact that . The proof is complete.
The graph of is shown in Fig. 4.
Remark 4.1
Higher dimensional cases are treated similarly, in particular, for and
In Sect. 6 we give an independent elementary proof of this inequality.
5 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
Accordingly, the operator is
Here
As before, we have to find the Green’s function solving . Furthermore, for finding it suffices to solve this equation for . Setting
and arguing as in Lemma 2.2 we get from (5.2)
so that
Now a word for word repetition of the argument in Theorem 2.2 gives that
Therefore we see from (5.4) that if and only if
For example, for supremum is the maximum attained at , giving
We only mention that in the general case
however, the corresponding substitution produces a long (but explicit) formula for , and instead we present in Fig. 5 the graph of the sharp constant , where and .
Finally, it is possible to find explicitly. In fact, the free recurrence relation has the characteristic equation
or , which decomposes into two quadratic equations
with four roots , where , , , where
Since for , it follows that any symmetric -solution of (5.2) is of the form , and since, in addition is real, we have
Setting and we obtain a linear system for
where is given in (5.4) and
Solving this system we find :
and, consequently, the formula for with :
where is given in (5.6).
Thus, we obtain the following result.
Theorem 5.1
Inequality (5.1) holds for . In particular in the limiting case
In the general case
where is given in (5.5) and in (5.4) see also (5.7) For the unique extremal is . For and
Remark 5.1
It is not difficult to find the function , that is, the solution of the maximization problem
where . For this purpose we also need the expression for the Green’s function in the region , which is as follows
Using (5.4), (5.10) we can write down a parametric representation of as in Theorem 3.2, but instead we merely show its graph in Fig. 7.
This time we do not have the maximum principle, and the Green’s function is not positive for all , but is rather oscillating with exponentially decaying amplitude, see Fig. 6. Nor do we have the symmetry in Fig. 7 that we have seen in the first-order inequalities in the one- and two-dimensional cases, see (2.8) and (3.4). The maximum is attained at corresponding to . The component of the resolvent set corresponds to and corresponds to .
It is worth to compare the results so obtained in the discrete case with the corresponding interpolation inequalities for Sobolev spaces in the continuous case. It is well known that the interpolation inequality on the whole line
where , holds only for . The sharp constant was found in [17]:
Thus, for first-order inequalities both in the discrete and continuous cases the constants are equal to , while for the second-order inequalities we see from (5.8) and (5.12) that
The next theorem states that for higher order inequalities the constants in the discrete case are always strictly greater than those in the continuous case.
Theorem 5.2
Let and let . The inequality
where
holds for and
For all supremum is the maximum. If , then for the constants in the continuous and discrete inequalities satisfy
Proof
Following the scheme developed above we look for the solution of the equation
and as in (5.3) find that
which proves (5.15) (whenever the supremum is finite). Using and changing the variable , where we have
Clearly , and we have to study as . The integral converges uniformly for , since the denominator is greater then for and is greater then for observing that uniformly for . Therefore
which proves, in the first place, that the right-hand side in (5.15) is finite if and only if and, secondly, that non-strict inequality (5.16) holds. Finally, for we have strict inequality since
For we have , and
is strictly decreasing not only at but for all , the fact that we have already seen in (2.24).
Remark 5.2
Inequality (5.13) holds for , that is, when the weight of the stronger norm, which is the -norm, increases. Accordingly, inequality (5.11) for periodic functions with mean value zero holds for in the complementary interval , when the weight of the stronger norm, which is the -norm of the -th derivative, increases:
where
A general method for finding sharp constants in interpolation inequalities of ---type was developed in [1, 11, 18], which was also used in the discrete case in the present paper. For example, for
where is the Green’s function of the equation
and
For the limiting the constant is the same as on : . The graph of on the interval is shown in Fig. 1 on the left. Observe that .
6 Applications
6.1 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 is sharp and cannot be attained at a non identically zero sequence . 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 , , by setting for a sequence
Accordingly, inequality (6.3) for is equivalent (as was first probably observed in [15]) to (6.2) by setting and further restricting (and ) to even functions. Furthermore, the unique (up to scaling) extremal function in (6.3) on the whole axis produces the extremal function in (6.2).
In the similar way, discrete inequalities have equivalent integral analogs. Let be the discrete Fourier transform , where
Then for with on the th place
Therefore
and, finally,
Thus, we have proved the following result.
Theorem 6.1
Let . The inequality
established in Theorem 2.2 is equivalent to the inequality
for . Here see (2.28) In the limiting case inequality (2.27) is equivalent to
Proof
The proof follows from Theorem 2.2 and (6.4). We also point out that for inequality (6.6) saturates for
for no extremals exist and maximizing sequence is obtained by letting ; finally for , (6.6) saturates at constants.
For each , and inequality (6.7) saturates at
For , .
Remark 6.1
Corresponding to (5.8) is the integral inequality
which turns into equality for
Remark 6.2
The integral analog of the two dimensional discrete inequality is
where , and is defined in (3.11).
Remark 6.3
In the -dimensional case, , for the exponent the Parseval’s identities (6.5) provide an independent elementary proof of (4.8). In fact, setting we have
which proves (4.8).
This approach can be generalized to the -case for the proof of the discrete Sobolev type inequality in the non-limiting case (1.12). Here in addition to the Parseval’s identity we also use the Hausdorff–Young inequality (see, for instance, [2]):
where and .
In fact, we have and and by the Riesz–Thorin interpolation theorem
where , . We also observe that (6.11) becomes an equality for and .
Setting in (1.12), , and using the auxiliary inequality (6.12), (6.13) below, we obtain
It remains to prove (6.12). By Hölder’s inequality and (6.11) we have
where
Thus we obtain the following result.
Theorem 6.2
Let and . Then
where is defined in (6.13).
Remark 6.4
We do not claim that the constant here is sharp. Moreover, it blows up for , while it can be shown that the inequality still holds. However, the constant is sharp in the opposite limit , see (4.8).
6.2 Spectral inequalities for discrete operators
Interpolation inequalities characterizing imbeddings of Sobolev spaces into the space of bounded continuous functions have important applications in spectral theory. The original fruitful idea in [7] has been generalized in [6] to give best-known estimates for the Lieb–Thirring constants in estimates for the negative trace of Schrödinger operators.
In this section we apply our sharp interpolation inequalities with the method of [7] for estimates of the negative trace of the discrete operators [16].
We write the inequalities obtained above in the unform way
where is as in (5.14) and belongs to a certain subinterval of uniquely defined in the corresponding theorem:
Theorem 6.3
Let be a family of sequences that are orthonormal with respect to the natural scalar product in . We set
Then for as in (6.15) and
Proof
For arbitrary we construct a sequence
Applying (6.14) and using orthonormality we obtain for a fixed
We now set :
or
Summing over and using orthonormality we obtain (6.17).
Corollary 6.1
Setting in Theorem 6.3 we obtain a family of interpolation inequalities for
In particular to mention a few examples with limiting
in dimension
Remark 6.5
The last inequality holding in dimension 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].
We consider the discrete Schrödinger operator
acting on as follows
Theorem 6.4
Let and let as then the negative spectrum of is discrete and satisfies the estimate
Proof
Suppose that there exists negative eigenvalues , with corresponding orthonormal eigenfunctions :
Taking the scalar product with , summing the the results with respect to , and using (6.16), Hölder inequality and (6.17), we obtain
6.3 Examples
. Then and the negative trace of the operator
satisfies
. Then and the negative trace of the operator
satisfies
, , . Then and the negative trace of the operator
satisfies
In particular, in three dimensions
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Acknowledgments
Funding for this research was provided by the grant of the Russian Federation Government to support scientific research under the supervision of leading scientist at Siberian Federal University, no. 14.Y26.31.0006, and from the Russian Foundation for Basic Research (grant no. 12-01-00203).
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Ilyin, A., Laptev, A. & Zelik, S. Sharp interpolation inequalities for discrete operators and applications. Bull. Math. Sci. 5, 19–57 (2015). https://doi.org/10.1007/s13373-014-0060-8
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DOI: https://doi.org/10.1007/s13373-014-0060-8