Bochner-Riesz means for the Hermite and special Hermite expansions
Introduction
Let denote the Hermite operator which is non-negative and selfadjoint with respect to the Lebesgue measure on . The spectrum of the operator is given by the set . Here denotes the set of nonnegative integers. For each , the Hermite polynomial on is given by Rodrigues' formula , and the normalized Hermite functions , form an orthonormal basis of . In higher dimensions the d-dimensional Hermite functions are given by the tensor products of : The Hermite operator and functions, respectively, represent the Hamiltonian and quantum states of the particle for the quantum harmonic oscillator. The functions can also be interpreted as basis functions for the bosonic Fock space via the Bargmann transform. For a detailed discussion regarding the matters, we refer the reader to [7]. The Hermite operator also appears in the representation theory of the Heisenberg group (see for example [32]).
The set forms a complete orthonormal system in and the functions are eigenfunctions for the Hermite operator with eigenvalue where . Thus, for every we have the Hermite expansion where denotes the Hermite spectral projection given by
The Hermite expansion is convergent in space, but when the expansion does not converge to f as in unless . This can be shown making use of the transplantation theorem due to Kenig-Stanton-Tomas [15] and Fefferman's counterexample for boundedness of the ball multiplier [6] (also see [30, Theorem 3.1.2]). Thus we are naturally led to consider the Bochner-Riesz mean: The summability exponent δ mitigates the influence of new summands which enter into the summation as λ increases. So, the operator has more favorable behavior in perspective of summability as δ becomes larger. The classical Bochner-Riesz problem is to determine the optimal summability order δ for which converges to f in for a given . When , the problem was settled by Carleson-Sjölin [4]. In higher dimensions progress has been made, however the problem is still left open. See [24], [28], [18] and also see [8], [33] for most recent results and references therein.
In this paper we are concerned with convergence of the Hermite Bochner-Riesz means, that is to say, the problem of determining the optimal δ for which converges to f in . By the uniform boundedness principle, this problem is equivalent to that of characterizing the optimal δ for which the estimate holds with a uniform constant C where .
When , the problem is almost completely settled except some endpoint cases. Askey and Wainger [1] proved that (1.2) holds with if and only if . When or , combining this with the result due to Thangavelu [29], one can show that is uniformly bounded on if . On the other hand, Thangavelu [29] showed that (1.2) fails to hold if . However, it looks that the estimate (1.2) (or its weaker variants) with still remains open when or .
In higher dimensions, unlike one dimension, only partial results are known. By the abovementioned transplantation theorem [15], the bound (1.2) implies that the classical Bochner-Riesz means are uniformly bounded on (see Proposition 4.1 and its proof). Thus, by the well known necessary condition for boundedness of (see for example [9], [6]) we have if the uniform bound (1.2) holds. This naturally leads to the following conjecture.
Conjecture 1.1 Let and . The uniform estimate (1.2) holds if and only if (1.3) holds.
Karadzhov [14] verified Conjecture 1.1 for . His result was based on the optimal – spectral projection estimate The estimate plays the role of – (adjoint) restriction estimate for the sphere in Stein's argument [6] which deduces the sharp bound on from the restriction estimate. However, as shown by Koch and Tataru [17], the range of p where the above estimate is valid can not be extended any further. We refer the reader to [17], [12] and references therein for more about the Hermite spectral projection operator. This means the approach in [14] relying on the – spectral estimate is no longer viable when one tries to prove boundedness with δ satisfying (1.3) when .
Local estimate for Meanwhile, the kernel of is expressed on a critical region as an Airy type integral and such phenomenon does not occur in the case of the classical Bochner-Riesz operator . Taking this into account, Thangavelu [31] speculated that Conjecture 1.1 may fails1 when . Instead of the global estimate (1.2) he considered a local variant of (1.2). To be specific, let us consider the estimate with a constant C independent of λ where are measurable subsets of . It was shown by Thangavelu [31] that (1.4) holds with a compact set E and for when (1.3) holds. The result is clearly sharp in that the estimates fail if because of the aforementioned transplantation result [15]. In analogy to Karadzhov's approach, a form of local – estimate for was utilized. As was shown in [12], the local spectral projection estimate does not extend for , so one can not expect any progress using – estimate for .
We shall show that Conjecture 1.1 is generally not true when . In fact, on a certain range of p we obtain a new lower bound on the summability index δ (see Proposition 4.1) for the uniform bound (1.2). This invalidates Conjecture 1.1. Thus, in order to prove boundedness for one has to consider a weaker alternative as was done [31]. It would be interesting to determine whether (1.2) holds up to the new lower bound but for the present the problem seems to be beyond reach. Instead, we first look into convergence of in a local setting to make progress on the current state regarding boundedness of the Hermite Bochner-Riesz means.
As far as the authors are aware, concerning on boundedness of the Hermite Bochner-Riesz means no further progress has been made beyond Thangavelu's result ([31]) until now. In this paper, we extend the range of p for which (1.4) holds under a suitable condition on E and F. Even if (1.4) is a weaker variant of the global estimate, the local estimate is still strong enough to imply the sharp bound on . More precisely, if the estimate (1.4) holds with for any , from the transplantation theorem ([15]) we see that the Bochner-Riesz operator is uniformly bounded on .
To state our first result, we introduce some notations. Let us set and The following is our first result.
Theorem 1.2 Let . Suppose that are compact sets such that for some . Then there is a constant C independent of λ such that provided that and where , denote the dilated sets , , respectively.
When , Theorem 1.2 establishes the estimate (1.5) on the optimal range p, that is to say, (1.5) holds if and only if , . The quantity plays important role in determining the natures of the kernels of and . More precisely, as to be seen in Section 2, the kernels can be expressed as a sum of one-dimensional oscillatory integrals which are associated with the phase function (see (2.6) below). When , the phase function has non-degenerate critical points. However, if , the stationary points are not non-degenerate, so the integrals behave as they were Airy functions.
From Theorem 1.2, we have the following convergence result.
Corollary 1.3 Suppose that and . Then for any compact set and compactly supported , we have
Now we consider the twisted Laplacian, which is closely related to the Hermite operator. The twisted Laplacian on which is defined by has the same discrete spectrum as . The associated eigenfunctions are the special Hermite functions which are given by the Fourier-Wigner transform of the Hermite functions. Indeed, for any multi-index , Then it follows that , thus is an eigenfunction of with the eigenvalue and the eigenspaces are infinite dimensional. Additionally, satisfies , which means is an eigenfunction of the Hermite operator . This is the reason that is called the special Hermite function.
For , by we denote the projection to the eigenspace of with the eigenvalue λ, i.e., Since is an orthonormal basis of , so one can expand f into the series of special Hermite functions. In fact, we have As seen before in the case of Hermite expansion, by the transplantation theorem in [15] and Fefferman's counterexample [6], this series fails to converge in unless . So, we consider the Bochner-Riesz mean for the special Hermite expansion which is defined by By the uniform boundedness principle, the convergence of for all is equivalent to the uniform estimate The problem has been studied by many authors. By the transplantation theorem in [15] we see the estimate (1.7) implies boundedness of the classical Bochner-Riesz operator in . Thus (1.7) holds only if , . It seems to be plausible to conjecture that the uniform estimate (1.7) holds if . In [30], Thangavelu verified the conjecture for . Later, the range was extended to by Ratnakumar, Rawat, and Thangavelu [20]. Further progress was made by Thangavelu [31] who showed the local estimate holds provided that and . The corresponding global estimate was later established by Stempak and Zienkiewicz [27], i.e., they showed that the estimate (1.7) holds for and . The common key ingredient of the previous results is the – projection estimate of the form with C independent of λ, which was combined with Stein's argument [6]. The projection estimate (1.8) was shown by Stempak and Zienkiewicz [27] for . Later, Koch and Ricci [16] proved that (1.8) holds if and only if (also, see [13] for - estimates for ). So, further improvement is no longer possible via the estimate (1.8) when .
Local convergence of Currently, no result with the sharp summability exponent is known when . Following the approach in [31], we consider a local variant of (1.7) and prove new estimate with the sharp summability exponent outside the aforementioned range of p. The following is our result regarding the Bochner-Riesz means for the special Hermite expansion.
Theorem 1.4 Let , , , and . Suppose that are compact sets satisfying for all , . Then there exists a constant C independent of λ such that where , denote the dilated sets , , respectively.
Corollary 1.5 Let be as in Theorem 1.4. Then for any compact set and compactly supported , we have
When , from the result we have a complete characterization of for which the local convergence (1.10) holds. The assumption that for all , can be regarded as a counterpart of the assumption on in Theorem 1.2. If for all , , the kernel of the operator rapidly decays (see (3.11)). Thus the assumption can be relaxed so that for all , .
Our approach Our results are based on the estimate for the oscillatory integral operator of Carleson-Sjölin type. We refer the reader forward to Section 2.5 for more regarding the oscillatory integral operator. The optimal estimate for the Carleson-Sjölin type operator in has been known since the works of Carleson-Sjölin [4] and Hörmander [10]. Also estimates for oscillatory integral operator beyond the regime of the Stein-Tomas restriction theorem, i.e., – spectral projection estimate, have been known for many years (for example, see [19]). However, such estimates could not be put to use to study Bochner-Riesz means for the Hermite and special Hermite expansions. Unlike the classical Bochner-Riesz operator, the precise forms of the kernels of , have been unknown due to complexity associated to the special functions. This is why all the previous results were obtained relying on the – spectral projection estimates.
In order to show Theorem 1.2, Theorem 1.4, we follow a strategy inspired by the recent works by Jeong and the authors [12], [13] on the spectral projection operators and . By making use of the Schrödinger propagators , (see (2.1), (3.1)), of which kernel representation is well known, we obtain explicit expressions for the kernels of the operators , . Exploiting those expressions with the method of stationary phase, we obtain asymptotic expansions of the kernels. So, they reduce the matter to obtaining the sharp estimate for the associated oscillatory integral operators. We prove that those oscillatory integral operators satisfy the Carleson-Sjölin condition [4], [10], [23]. In two dimensions this allows us to obtain the optimal results. However, in higher dimensions the Carleson-Sjölin condition alone is not enough, as was shown by Bourgain [3], to give the sharp bound for (or ). We make use of the fact that additional ellipticity assumption on the second fundamental form of the phase makes it possible to obtain the sharp estimate for the Carleson-Sjölin type operator on an extended range. This observation was first made by one of the authors [19]. To utilize such improved range, we verify through explicit computation that the phases in the abovementioned asymptotic expansions satisfy the ellipticity condition (see Lemma 2.13, Lemma 3.10). Then, we can use the previously known oscillatory integral estimates in two and three dimensions ([10], [19]) and the more recent result due to Guth, Hickman, and Iliopoulou [8] in higher dimensions.
Notation For the rest of the paper, and we identify with .
- •
For given , we write if there is a constant such that . Here, if C has to be taken to be small enough, we use the notation to mean that A is sufficiently larger than B. Furthermore, denotes that and .
- •
.
- •
For an operator T we denote by (or ) the kernel of T.
- •
, , so . In particular, if is a -valued differentiable function on , then we have .
- •
By we denote the identity matrix. If the value of d is clear from the context, we simply denote by I.
- •
For , denotes the i-th standard basis in .
- •
For and a constant , we denote .
Section snippets
Bochner-Riesz means for the Hermite expansion: proof of Theorem 1.2
In this section we prove Theorem 1.2. We begin by obtaining an explicit expression of the kernel of the Bochner-Riesz means. For the purpose we make use of Mehler's formula for the Schrödinger propagator.
Bochner-Riesz means for the special Hermite expansion: proof of Theorem 1.4
In this section we consider the Bochner-Riesz means for the special Hermite expansion. Basically, we follow the same strategy for the Hermite expansion. We begin with noting that the expression (1.6) can be simplified by making use of the twisted convolution which is defined as follows: where S is a skew-symmetric matrix given by By the argument using the Weyl transform, it can be shown that , where and
Lower bound on the summability index of
In this section we obtain a new lower bound on the summability index δ for uniform boundedness of on .
Proposition 4.1 Let and . The uniform bound holds only if and
In particular, when the uniform bound (1.2) holds only if for and for . This coincides with Thangavelu's result [29, Theorem 2.1]. In higher dimensions, i.e., , it is necessary for (1.2) that if ; if
Acknowledgement
This work was supported by the National Research Foundation of Korea (NRF) grants No. NRF-2021R1A2B5B02001786 (Sanghyuk Lee) and No. NRF-2021R1I1A3A04035040 (Jaehyeon Ryu).
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