Sharp resolvent estimates
outside of the uniform boundedness range
In this paper we are concerned with resolvent estimates for the Laplacian in Euclidean spaces. Uniform resolvent estimates for were shown by Kenig, Ruiz and Sogge  who established rather a complete description of the Lebesgue spaces allowing such uniform resolvent estimate. The estimates have a variety of applications, particularly, to proving uniform Sobolev and Carleman estimates. Recently, interest in such estimates was renewed in connection to the Carleman estimate related to inverse problems. Especially, on a compact Riemannian manifold , an interesting new phenomenon was discovered by Dos Santos Ferreira, Kenig and Salo . Precisely, the estimate
holds with independent of when for . Later it was shown by Bourgain, Shao, Sogge and Yao  that the region of can not be extended to a wider range in general, and under the same assumption on the range of , the range of was further extended by Shao and Yao . However, even in the Euclidean spaces, the problem of obtaining sharp – bounds depending on has not been considered in general framework which admits all possible . In this paper, we present a complete picture of sharp – resolvent estimates, which may depend on . The resolvent estimates in Euclidean space seem to be expected to behave in a simpler way compared with those on manifolds. However, it turns out that, for some , the estimates exhibit unexpected behavior which is similar to those on compact Riemannian manifolds. We also obtain the non-uniform sharp resolvent estimates for the fractional Laplacians and a new result for the Bochner–Riesz operators of negative index.
Key words and phrases:resolvent estimate
2010 Mathematics Subject Classification:35B45, 42B15
1. Introduction and main results
In this paper we are concerned with the resolvent estimate for the Laplacian which is of the form
When the estimate is simply the classical Hardy–Littlewood–Sobolev inequality. If the left hand side can not be defined even as a distribution without additional assumption. Throughout this article we assume . The inequality (1.1) and its variants (especially, with independent of ) have applications to various related problems. Among them are uniform Sobolev estimates, unique continuation properties [31, 30], limiting absorption principles , absolute continuity of the spectrum of periodic Schrödinger operators  and eigenvalue bounds for Schrödinger operators with complex potentials [17, 18]. As just mentioned, (1.1) has been usually considered with independent of but the sharp bounds which are allowed to be dependent on are not studied in a general framework. The primary purpose of this paper is to provide complete characterization of the sharp – bounds for the resolvent operators up to a multiplicative constant.
Uniform resolvent estimate
In their celebrated work  Kenig, Ruiz and Sogge showed that, for certain pairs of , the constant in (1.1) can be chosen uniformly in . More precisely, for , it was shown that there is a uniform constant such that (1.1) holds if and only if and , or equivalently lies on the open line segment whose endpoints are
(See Figure 2.) They used these estimates to show uniform Sobolev estimates for second order elliptic differential operators on the same range of (see [31, Theorem 2.2]). When 111The midpoint of the line segment in Figure 2. the same estimate was also obtained by Kato and Yajima [32, pp. 493–494] by a different approach.
The result in  gives complete characterization of the range of which admits the uniform resolvent estimate. However, it is not difficult to see that if in (1.1) is allowed to be dependent on , there is a larger set of for which the estimate (1.1) holds. To be precise, for let us set
where denotes the space of Schwartz functions on .
Let , and . Then if and only if which is given by
In view of Proposition 1.1 it is natural to ask what is the sharp value of which depends on . For some such estimate (modulo a constant multiplication) can be deduced by interpolation between estimates in [31, 25] and the easy bound
which directly follows from the Fourier transform and Parseval’s identity. Some of related results can be found in . Moreover, these estimates turn out to be sharp (see (1.5) Proposition 1.3 below). But, the sharp bound for with general can not be deduced from interpolation between previously known estimates. For the purpose we need to make use of theory of oscillatory integral operators of Carleson–Sjölin type under the additional elliptic condition ([11, 27, 46, 36, 23], also see Section 2.1 below).
Boundedness of the associated multiplier operators
To obtain the sharp resolvent estimates, it is convenient to consider bounds for the associated multiplier operators. Clearly,
Here and denote the Fourier and inverse Fourier transforms on , respectively. Since the multiplier becomes singular as approaches to the set it is reasonable to expect that the bound gets worse as . Thanks to homogeneity and scaling, we have that
Thus we may assume that , to get the sharp bounds for . Indeed, when , it was shown in  that there is a uniform constant , independent of , such that
if lies in either the open line segment of which endpoints are and (see (1.2) and Figure 2), or the line of duality restricted to (see [31, Lemma 2.2(b) and Theorem 2.3]). Later, Gutiérrez ([25, Theorem 6]) extended (1.6) to the optimal range of . More precisely, she proved that the uniform bound (1.6) is true if lies in the set
This region is the closed trapezoid from which the closed line segments joining and are removed (see Figure 2). She also established the – (restricted weak type) analogues of (1.6) when is either or , where
Failure of (1.6) for has been actually known before in the studies of the Bochner–Riesz operators of negative orders (see Section 2.6). In fact, the necessity of the conditions and follow since (1.6) combined with (1.4) implies – boundedness of the restriction-extension operator on the sphere (see Theorem 2.14 and [31, pp. 341–342]), which is a constant multiple of the Bochner–Riesz operator (2.49) of order . The other two conditions and can be obtained by the Knapp type example (see Böjeson ) and a simple argument involved with the Littlewood–Paley projection (see Proof of Proposition 1.1), respectively.
When , as far as the authors are aware, the corresponding results regarding the uniform resolvent estimate (1.6) are not explicitly stated anywhere else before, although the – mapping properties of the closely related Bochner–Riesz operaters of negative order are well known (see e.g., [1, 12] and references therein). However, the method in  can be applied to obtain (1.6) provided that is contained in the pentagon
Conjecture regarding – resolvent estimate with .
Having seen that we have the uniform bound (1.6) on the optimal range , we now proceed to investigate the (non-uniform) sharp bounds with which lie outside of the uniform boundedness range. As becomes clear later, the problem is closely related to sharp – boundedness of the Bochner-Riesz operators of negative orders (see Section 2.6). The non-uniform bounds on the resolvents have been used to study eigenvalues of the Schrödinger operators with complex potentials (for example, see [18, 14]).
In order to state our results we introduce some notations which denote points and regions in the closed unit square . For each we set
Similarly, for every subset of we define by
For , we denote by the convex hull of the points . In particular, if , denotes the closed line segment connecting and in . We also denote by and the open interval and the half-open interval , respectively.
For every and every , define a nonnegative number
The definition of naturally leads to division of into the four regions
and we also see that
In Section 5 we obtain the following lower bounds for .
Let . Suppose that . Then, for ,
where the implicit constant is independent of .
As mentioned in the above, when , . For , it is likely that by adapting Fefferman’s disproof of disk multiplier conjecture  one can show . However, for the other with it seems to be natural to expect that the lower bound in (1.12) is also an upper bound.
For with and , let us set
Let and . There exists an absolute constant , depending only on , and , such that, for ,
Sharp – resolvent estimate with
Our main result is that Conjecture 1 is true for most of cases of . For the statement of the result we introduce additional notations. Let , , and be defined by
If , note that , and , .
It is also possible to obtain similar results regarding the Laplace-Beltrami oprator on compact manifolds (). To prove the sharp resolvent estimates (1.13) we dyadically decompose the multipliers by taking into account the region of where the multiplier gets singular as . Such idea is now classical in the context of the Bochner–Riesz conjecture (e.g. [13, 35]). It is important to obtain the optimal – bounds for each of the operators which are given by the dyadic decomposition. For the purpose we use the Carleson–Sjölin reduction ([11, 46]), and combine this with Theorem 2.2 in Section 2.1 () and bilinear estimate for the extension operator associated to the hypersurfaces of elliptic type (). For more details, see Section 2 (Corollary 2.12).
As mentioned in the above, the restricted weak type estimates with when were shown in . In Section 4 we provide a different proof of those restricted weak type estimates for , together with the weak type estimates when is in the half open line segment (see Figure 4 and Figure 4). This upgrades the endpoint case of uniform Sobolev estimate in  from the restricted weak type to the weak type for when . Also, for satisfying , the uniform resolvent estimate (1.6) follows by duality and interpolation. (For an additional simple argument involving frequency localization and Young’s inequality is necessary to cover the case .)
When it is also possible and much simpler to obtain the sharp resolvent estimates. For we write , where (see [47, p, 203]). Since the kernel is bounded and integrable, Young’s inequality and (1.5) yield
for all such that . Following the argument in Section 5.2 one can easily check that the estimates are sharp.
Resolvent estimates on compact Riemannian manifolds
Let be a -dimensional compact Riemannian manifold without boundary. When Dos Santos Ferreira, Kenig, and Salo proved in  that for any fixed the uniform estimate
holds for all .444Here we choose the branch of , , such that the imaginary part is positive. Note that . In the complex plane this region excludes a neighborhood of the origin and a parabolic region opening to the right. Shortly afterwards, Bourgain, Shao, Sogge and Yao  proved that if is Zoll, then the region cannot be significantly improved by showing that
whenever for all , and as . However, in some cases where the manifold has favorable geometry such as the flat torus or Riemannian manifolds with nonpositive sectional curvature, the range of for (1.15) can be extended (see ). Shao and Yao  proved the off-diagonal – estimate of (1.15) for satisfying , and , but it is not known whether this range of is optimal even for which satisfy . In  Frank and Schimmer observed that the argument in  can be applied to establish – analogue of (1.15) when and . They also obtained the estimate
with independent of by proving an off-diagonal restricted weak type bound for the parametrix constructed in .
Regions of spectral parameters where uniform resolvent estimate is allowed
Since we now have sharp resolvent estimates which depend on the spectral parameter , it is possible, for each given , to describe the region of for which the resolvent estimates are uniform.
The – bound for is uniform in while the uniform estimate (1.15) on compact manifold holds only for (see Figure 5). Thus, we may reasonably expect that the bound behaves better on than on compact manifolds. However, as is to be seen below, it is rather surprising that, for certain , the bound for has a similar behavior with those on compact manifolds and the profile of the -region where is uniformly bounded changes dramatically depending on the values of .
For which satisfy , and we define the region of spectral parameters by
A remarkably interesting phenomena occurs when . To describe this let us divide into the three subsets , , and , given by
Location of the eigenvalues of
The sharp resolvent estimates (Theorem 1.4) can be used to specify the location of eigenvalues of non-self-adjoint Schrödinger operators acting in , . As was shown in [17, 18], if acts in one can use the Birman–Schwinger principle, but this is not the case when acts in , .
Let and let be the constant which appears in (1.13). Fix a positive number (we choose if ). Suppose that, for some ,
If is an eigenvalue of acting in , then must lie in .