A quantified Tauberian theorem and local decay of -semigroups
We prove a quantified Tauberian theorem for functions under a new kind of Tauberian condition. In this condition we assume in particular that the Laplace transform of the considered function extends to a domain to the left of the imaginary axis, given in terms of an increasing function and is bounded at infinity within this domain in terms of a different increasing function . Our result generalizes [4, Theorem 4.1]. We also prove that the obtained decay rates are optimal for a very large class of functions and . Finally we explain in detail how our main result improves known decay rates for the local energy of waves in odd-dimensional exterior domains.
In the last decade there has been much activity in the field of quantified Tauberian theorems for functions of a real variable [20, 2, 10, 3, 6, 21, 5, 9, 4]. See also [23, 24] and references therein for quantified Tauberian theorems on sequences and  for Dirichlet series. We refer to  and [1, Chapter 4] for a general overview on Tauberian theory.
Let be a Banach space and be a locally integrable function. For some continuous and increasing function let us define
The above mentioned articles impose essentially the Tauberian condition that the function has a bounded derivative (in the weak sense), the Laplace transform extends across the imaginary axis to and it satisfies a growth condition, also expressed in terms of in at infinity. The decay rate (the rate of convergence to zero) is then determined in terms of . For example, a polynomially growing gives a polynomial decay rate and an exponentially growing gives a logarithmic decay rate. In general could also have a finite number of singularities on the imaginary axis , but we are not interested in this situation in the present article.
The pioneering works [20, 2] focus on polynomial decay for orbits of -semigroups. A generalization for functions (as formulated above) and to arbitrary decay rates was given in  for the first time. There the authors also improved the decay rates from [20, 2]. In  it was shown that the results of  are optimal in the case of polynomial decay. We want to emphasize at this point that the main result of  for the special case of a truncated orbit of a unitary group of operators (i.e. for some bounded operators ) were already obtained in the earlier article  with the same rate of decay. Actually the authors only formulated a theorem on polynomial decay but in the retrospective it is not difficult to generalize their proof to arbitrary decay rates.
A major contribution to the field of Tauberian theorems is the recent article . The authors extended the known Tauberian theorems to -rates of decay. On the basis of a technique already applied in  the authors showed the optimality of their results in the case of polynomial decay. Another important observation, made in , concerns the above mentioned growth condition. In  it was assumed that the norm of is bounded by in . This condition was weakened in  in case of polynomial decay, and later in  assuming merely that can be bounded by a polynomial in .
The aim of the present article is to further generalize the growth condition on in . That is, we introduce a second continuous and increasing function and assume that the norm of is bounded by in . The decay rate is then given in terms of and .
Let denote the right-continuous right-inverse of given by for all . Let
We are now ready to state our first main result, a generalization of [4, Theorem 4.1].
Let be a Banach space, , and be a locally integrable function such that its -th weak derivative is in for some . Assume that there exist continuous and increasing functions satisfying
such that the Laplace transform of extends analytically to and
Then there exists a constant such that
Note that a function with is polynomially bounded. In fact, holds for all . In particular the Laplace transform of is well-defined in the interior of as an absolutely convergent integral.
One can drop condition (i) on but then one has to replace by the function given by .
We are not able to prove the theorem for in condition (ii). In Section 2.3 the reader can find a short discussion on a slightly weaker constraint on .
If we replace in (1) by for some and set we recover [4, Theorem 4.1]. Our theorem applies perfectly to local energy decay of waves in odd-dimensional exterior domains. Here is typically a spatially truncated orbit of a solution to the wave equation and one is often confronted with the situation that is constant and is asymptotically larger than any polynomial. In this situation no known Tauberian result applies directly. One might guess that one can apply the Phragmén-Lindelöf principle to get a better estimate on on a smaller domain to the left of the imaginary axis. Indeed this works, and as shown in  one can apply known Tauberian theorems after this procedure. However in Section 5 we discuss the application to local decay of waves in exterior domains in detail and show that this procedure yields a weaker estimate than a direct application of Theorem 1.1.
Let be a Banach space, , and be a locally integrable function such that for some . Let and be as in Theorem 1.1. Assume that the Fourier transform of is of class and its derivatives satisfy for all
Then there exists a constant such that
Note that the Fourier transform of is well-defined in the sense of tempered distributions since is polynomially bounded (compare with Remark 1.2).
A theorem of this type (for , and ) was formulated for the first time in . A main contribution of the authors was also to provide a new and easier to understand technique - on the basis of Ingham’s original proof of the unquantified version  - for proving Tauberian theorems. For example in  and  one main difficulty is to choose contours for integration in the complex plane in a clever way. In  the authors avoid this technicality by considering the derivatives of the Fourier transform of instead of the Laplace transform.
To prove Theorem 1.5 we adapt the proof of [9, Theorem 2.1(b)]. That is - for - we decompose into two terms with the help of some suitably chosen and scaled convolution kernel with . Then we estimate the -norm of and in terms of and , solely assuming respectively the bounds on all derivatives . Finally we optimize the sum of these two estimates by choosing for a sufficiently small .
We improve the techniques of  in the following way: We estimate from above by a Poisson integral which makes it possible to apply a fundamental result on Carleson measures. We note that this technique was already applied in . Compared to the proof in  we get a better estimate on by choosing a better convolution kernel . Also the Fourier transform of our convolution kernel is a -function which simplifies the prove slightly. Our choice of is based on the Denjoy-Carleman theorem on quasi-analytic functions.
The paper is organized as follows. In Section 2 and 3 we prove Theorem 1.5 and 1.1, respectively. In Section 4 we prove the optimality of Theorem 1.1 for a very large class of possible choices of and . This is even new in the case where . To prove the optimality we make a similar construction as in . As a side product this construction also shows that there actually exist functions satisfying the hypotheses of Theorem 1.1 for increasing faster than any polynomial in , but do not satisfy (1) if one replaces by a polynomial in (see Remark 4.7). This proves that Theorem 1.1 is a proper generalization of [4, Theorem 4.1]. A short discussion on the optimal choice of in (2) is included in Subsection 4.1. In Subsection 5.1 we explain how to get local decay rates for -semigroups from our results. Finally in Subsection 5.2 we apply this to local energy decay of waves in odd-dimensional exterior domains.
We denote and . By we denote the natural numbers including . For we define to be the natural numbers greater or equal to . By we denote a strictly positive constant which may change implicitly their value from line to line. Every statement in this article which includes remains true if one replaces by a larger constant. Other strictly positive constants, having the names are not allowed to change their values - except it is explicitly stated. Analogously are strictly positive constants which might be replaced by smaller constants without invalidating any statement in our article. We say that a function decays rapidly if for any there exists a constant such that .
2. Proof of Theorem 1.5
Without loss of generality we may assume that . If this was not satisfied we could replace by for some function with and arbitrary. This neither changes the asymptotics of at infinity nor does it change the growth of and its derivatives at infinity considerably. To see this note that the Fourier transform of satisfies
Now let us extend by zero on the negative numbers. By our additional assumptions we see that the extended function is -times continuously differentiable on the whole real line and .
Let with and be a function to be fixed later in the proof. Let
be its inverse Fourier transform. Note that is a Schwartz function with . For let and . Let us decompose
Here by we denote the -times convolution of with itself. We also define (delta-function). Note that .
2.1. Estimation of
Let us define the Poisson kernel by
Recall that by Young’s inequality the Poisson kernel acts as a continuous operator on via convolution.
Let and . Let be a locally integrable function such that . Let be as above. Then there exists a constant (only depending on and ) such that
holds for all and .
It is clear from the proof that in the statement of the lemma one can replace by any positive and integrable kernel bounded from below by for some . We then define . Unfortunately this is not consistent with the definition of , but for the Carleson measure argument below it is more convenient to define as above.
Let us define two antiderivatives of
Furthermore we define the following auxiliary function
We observe that the derivative of is plus a factor times the delta function at zero. This observation is the reason why we split the integral from the following calculation at .
First we consider the case .
We need to explain why the partial integration executed from line two to three produces no boundary terms at and . At zero there are no boundary terms since vanishes at and the two limits exist. Recall that is polynomially bounded. Moreover the function decays rapidly at infinity. Thus there are no boundary terms at plus or minus infinity. Finally the last equality together with the fact that decays rapidly implies
Now we consider the case . Let us define recursively for . Clearly . We prove now via induction on . Observe that for any the function decays rapidly. For the inductive hypothesis is precisely (7). Assume that the hypothesis is valid for some . Then by (7) for replaced by
From here we can finish the proof as in the case . ∎
Since the -norm of the Poisson kernel is (for any ) we see from Young’s inequality that for any and it holds that . If and if we set we deduce from Lemma 2.1 that
holds for all . If we compare this with (4) we see that this already yields the desired estimate on in the case . If we need a slightly more involved argument based on a property of Carleson-measures.
Therefore let and let be a Borel measure on the upper half-plane . Now we ask for which measures an inequality
holds for all with a constant not depending on ? Note that the inequality is a special case of (9) for with being the one-dimensional Hausdorff measure of the line . Actually for one can characterize the class of all measures for which (9) holds for all . These measures are called Carleson measures (see [13, Theorem I.5.6.]). Let be a bounded continuous function with bounded variation. Then the one-dimensional Hausdorff measure of
is a Carleson measure. Now let for and for . If we set to be the Carleson measure corresponding to this particular choice of then we deduce that for
From this together with Lemma 2.1 we deduce
Let and define . (i) Then for we have
(ii) For we have
In both cases does not depend on .
2.2. Estimation of
The following Lemma is only necessary if .
There exists a such that for all .
Let . Since is essentially the right-inverse of we have
The inverse of the function is asymptotically equal to for large . Hence there exists a such that . Thus
At this point in the proof we fix a having one additional property. We assume that the derivatives of satisfy for some
Note that (11) can not be satisfied by any if we would replace by since then would be analytic and hence can not have compact support and at the same time. The Denjoy-Carleman111A special version of the Denjoy-Carleman theorem (sufficient for our considerations) reads as follows: Let be the set of -functions on supported on such that (11) holds for a sequence such that is increasing. Then contains a non-zero function if and only if . theorem (see e.g. [15, Theorem 1.3.8] or ) gives a description of those sequences which allow for compactly supported non-zero functions satisfying the inequality in (11). In particular, the Denjoy-Carleman theorem implies that our choice of is admissible for the existence of such a . Conversely it implies that there is no which satisfies (11) with .
Now we proceed with the estimation of . Therefore we have to estimate for . First let us consider . Let . Integration by parts -times yields
To verify the following calculations recall (3) and (11). We estimate the integral very roughly from above: length of interval of integration times supremum of the integrand within this interval. We also use Stirling’s formula implying for example that for appropriate constants .
The second inequality is valid for sufficiently large . Now let us set and . The constant will be chosen later. Then the condition (ii) on implies
The constant in the last inequality does not depend on . Moreover
If we choose sufficiently small Lemma 2.4 implies that
Clearly (12) remains valid if one replaces by and by its -th power . It is not difficult to check that also satisfies (11) if one replaces by . Therefore (14) remains true after replacing by . This together with Lemma 2.3 proves Theorem 1.5.
2.3. A remark on condition (ii) for
Our proof breaks down if we allow to be zero in condition (ii) in Theorem 1.1 (and 1.5). This is essentially due to the fact that by the Denjoy-Carleman theorem a function satisfying (11) for is necessarily quasi-analytic. This means that for a single but all automatically implies . However, one can weaken (ii) slightly by choosing for some given and
This allows to replace (ii) by the condition
Again choosing is forbidden for any .
3. Proof of Theorem 1.1
Lemma 3.1 below implies that Theorem 1.1 and Theorem 1.5 are equivalent. To prepare the formulation of this lemma we introduce some notation. Let , , , be continuous and increasing functions. For measurable and polynomially bounded and extended by zero on the negative real numbers we consider two distinct conditions. The first one is
This condition implicitly states that the Laplace transform of can be extended to . Let be the Fourier-transform of . The second condition is
This condition implicitly states that the Fourier transform is a -function.
The following lemma relates these conditions to each other under a mild condition on .
Let be a measurable and polynomially bounded function with for some and . We extend by zero on the negative real numbers and denote by its Fourier transform. (a) If satisfies (16) then satisfies (15) with
The constant depends only on , the constant depends only on .
Before proving this lemma we finish the proof of Theorem 1.1. Since satisfies (15) for and , Lemma 3.1 implies that (16) is true for and given as in part (b) of the lemma. In the following we assume large enough to satisfy . Note that condition (i) in Theorem 1.1 implies the existence of a (small) constant such that (for large )
This immediately yields for large
Therefore for some implies that . The proof of Theorem 1.1 is complete.
Proof of Lemma 3.1.
Let us begin with the easier part (a). Hadamard’s formula shows that (16) implies that is analytic in . Let and let . Then
Let us now prove part (b). Let us fix , let and let be the positively oriented circle of radius around in the complex plane. Note that is indeed included in the closure of the union of and . Let and be the intersection of with and , respectively. By Cauchy’s formula we have
Let us first estimate :
Let us now estimate :
It is an easy exercise to show that the integral of over can be estimated from above by a constant times . Therefore by Hölder’s inequality we get for large
A similar (and easier) estimate is true for the other summands . This together with (17) yields the claim. ∎
4. Optimality of Theorem 1.1
In this section we show that under the assumptions of Theorem 1.1 and for one can - up to improvement of the constant - not get a faster decay rate than the one already given by the theorem. To show this we use almost the same method as in . There the authors showed the optimality in the very particular case that and for .
Let and let be continuous and increasing functions satisfying for some increasing function
Then there exists a real number , not depending on and a locally integrable function with such that
If instead of (ii) we have the stronger assumption that there exists a such that
Note that condition (i) is only a very mild restriction. In fact, a typical situation where (i) is violated is that is a constant and grows at most polynomially. But then Theorem 1.1 implies exponential decay for . This in turn implies, that the integral which defines is absolutely convergent in a small strip to the left of the imaginary axis. In particular extends analytically to this strip and is bounded there. So our results are trivially optimal in that case.
Before we prove the Theorem we need a similar lemma as in . Given a compactly supported measure on we use the following notation for and
To simplify the notation we extend and symmetrically to the negative real axis.
Let and be as in Theorem 4.1. There exists a and , only depending on and , such that for all and there exists and a compactly supported Borel measure on such that