Bochner-Riesz means for the Hermite and special Hermite expansions

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Abstract

We consider the Bochner-Riesz means for the Hermite and special Hermite expansions. Developing further Thangavelu's approach [31], we study their Lp boundedness with the sharp summability index in a local setting. In two dimensions we establish the boundedness on the optimal range of p and extend the previously known range in higher dimensions. Furthermore, we prove a new lower bound on the Lp summability index for the Hermite Bochner-Riesz means in Rd, d2. This invalidates the conventional conjecture which was expected to be true.

Introduction

Let H denote the Hermite operatorΔ+|x|2=i=1di2+xi2,x=(x1,,xd),d1, which is non-negative and selfadjoint with respect to the Lebesgue measure on Rd. The spectrum of the operator H is given by the set 2N0+d. Here N0 denotes the set of nonnegative integers. For each kN0, the Hermite polynomial Hk(t) on R is given by Rodrigues' formula Hk(t)=(1)ket2(d/dt)k(et2), and the L2 normalized Hermite functions hk(t):=(2kk!π)1/2Hk(t)et2/2, kN0 form an orthonormal basis of L2(R). In higher dimensions the d-dimensional Hermite functions are given by the tensor products of hk:Φα(x)=i=1dhαi(xi),α=(α1,,αd)N0d. The Hermite operator and functions, respectively, represent the Hamiltonian and quantum states of the particle for the quantum harmonic oscillator. The functions Φα 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 Hd (see for example [32]).

The set {Φα}αN0d forms a complete orthonormal system in L2(Rd) and the functions Φα are eigenfunctions for the Hermite operator with eigenvalue 2|α|+d where |α|=i=1dαi. Thus, for every fL2(Rd) we have the Hermite expansionf=αN0df,ΦαΦα=λ2N0+dΠλHf, where ΠλH denotes the Hermite spectral projection given byΠλHf=2|α|+d=λf,ΦαΦα.

The Hermite expansion is convergent in L2(Rd) space, but when d2 the expansion λNΠλHf does not converge to f as N in Lp(Rd) unless p=2. This can be shown making use of the transplantation theorem due to Kenig-Stanton-Tomas [15] and Fefferman's counterexample for Lp boundedness of the ball multiplier [6] (also see [30, Theorem 3.1.2]). Thus we are naturally led to consider the Bochner-Riesz mean:Sλδ(H)f:=(1Hλ)+δf:=λ2N0+d(1λλ)+δΠλHf. The summability exponent δ mitigates the influence of new summands ΠλHf which enter into the summation as λ increases. So, the operator Sλδ(H) has more favorable behavior in perspective of Lp summability as δ becomes larger. The classical Bochner-Riesz problem is to determine the optimal summability order δ for which Sλδ(Δ)f converges to f in Lp for a given p[1,]. When d=2, 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 Lp convergence of the Hermite Bochner-Riesz means, that is to say, the problem of determining the optimal δ for which Sλδ(H)f converges to f in Lp. By the uniform boundedness principle, this problem is equivalent to that of characterizing the optimal δ for which the estimateSλδ(H)pC holds with a uniform constant C where Tp:=supfp1Tfp.

When d=1, the problem is almost completely settled except some endpoint cases. Askey and Wainger [1] proved that (1.2) holds with δ=0 if and only if 4/3<p<4. When p4/3 or p4, combining this with the result due to Thangavelu [29], one can show that Sλδ(H) is uniformly bounded on Lp(Rd) if δ>max{23|1p12|16,0}. On the other hand, Thangavelu [29] showed that (1.2) fails to hold if δ<max{23|1p12|16,0}. However, it looks that the estimate (1.2) (or its weaker variants) with δ=23|1p12|16 still remains open when p<4/3 or p>4.

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 Sλδ(Δ) are uniformly bounded on Lp (see Proposition 4.1 and its proof). Thus, by the well known necessary condition for Lp boundedness of Sλδ(Δ) (see for example [9], [6]) we haveδ>δ(d,p):=max{d|1p12|12,0},p2 if the uniform bound (1.2) holds. This naturally leads to the following conjecture.

Conjecture 1.1

Let d2 and p[1,]{2}. The uniform estimate (1.2) holds if and only if (1.3) holds.

Karadzhov [14] verified Conjecture 1.1 for max(p,p)2d/(d2). His result was based on the optimal L2Lp spectral projection estimateΠλH2pCλ12+d2(121p),2d/(d2)p. The estimate plays the role of L2Lp (adjoint) restriction estimate for the sphere in Stein's argument [6] which deduces the sharp Lp bound on Sλδ(Δ) from the L2 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 L2Lp spectral estimate is no longer viable when one tries to prove Lp boundedness with δ satisfying (1.3) when max(p,p)(2,2d/(d2)).

Local Lp estimate for Sλδ(H)  Meanwhile, the kernel of Sλδ(H) 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 Sλδ(Δ). Taking this into account, Thangavelu [31] speculated that Conjecture 1.1 may fails1 when max(p,p)(2d/(d2),2). Instead of the global estimate (1.2) he considered a local variant of (1.2). To be specific, let us consider the estimateχESλδ(H)χFpC with a constant C independent of λ where E,F are measurable subsets of Rd. It was shown by Thangavelu [31] that (1.4) holds with a compact set E and F=Rd for 2(d+1)d1p when (1.3) holds. The result is clearly sharp in that the estimates fail if δ<δ(d,p) because of the aforementioned transplantation result [15]. In analogy to Karadzhov's approach, a form of local L2L2(d+1)/(d1) estimate for ΠλH was utilized. As was shown in [12], the local spectral projection estimate does not extend for p<2(d+1)/(d1), so one can not expect any progress using L2Lp estimate for ΠλH.

We shall show that Conjecture 1.1 is generally not true when max(p,p)(2d/(d2),2). 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 δ>δ(d,p) 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 Lp convergence of Sλδ(H) in a local setting to make progress on the current state regarding Lp boundedness of the Hermite Bochner-Riesz means.

As far as the authors are aware, concerning on Lp 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 Lp bound on Sλδ(Δ). More precisely, if the estimate (1.4) holds with E,F=B(0,ϵ) for any ϵ>0, from the transplantation theorem ([15]) we see that the Bochner-Riesz operator Sλδ(Δ) is uniformly bounded on Lp(Rd).

To state our first result, we introduce some notations. Let us setp0(d)={23d+23d2if d0(mod 2),23d+13d3if d1(mod 2), andD(x,y):=1+x,y2|x|2|y|2,(x,y)Rd×Rd. The following is our first result.

Theorem 1.2

Let d2. Suppose that E,FRd are compact sets such that E×FD(c0):={(x,y)Rd×Rd:|x|,|y|1c0,D(x,y)>c02} for some 0<c0<1. Then there is a constant C independent of λ such thatχEλSλδ(H)χFλpC, provided that p>p0(d) and δ>δ(d,p) where Eλ, Fλ denote the dilated sets λE, λF, respectively.

When d=2, Theorem 1.2 establishes the estimate (1.5) on the optimal range p, that is to say, (1.5) holds if and only if δ>δ(d,p), p2. The quantity D(x,y) plays important role in determining the natures of the kernels of Πλ and Sλδ(H). 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 PH (see (2.6) below). When D(x,y)>0, the phase function has non-degenerate critical points. However, if D(x,y)=0, the stationary points are not non-degenerate, so the integrals behave as they were Airy functions.

From Theorem 1.2, we have the following Lp convergence result.

Corollary 1.3

Suppose that p>p0(d) and δ>δ(d,p). Then for any compact set KRd and compactly supported fLp(Rd), we havelimλK|Sλδ(H)f(x)f(x)|pdx=0.

Now we consider the twisted Laplacian, which is closely related to the Hermite operator. The twisted Laplacian L on CdR2d which is defined byL=j=1d((xj12iyj)2+(yj+12ixj)2),x,yRd has the same discrete spectrum 2N0+d as H. The associated eigenfunctions are the special Hermite functions Φα,β which are given by the Fourier-Wigner transform of the Hermite functions. Indeed, for any multi-index α,βN0d,Φα,β(z):=(2π)d2Rdeix,ξΦα(ξ12y)Φβ(ξ+12y)dξ,z=x+iy. Then it follows that LΦα,β=(2|β|+d)Φα,β, thus Φα,β is an eigenfunction of L with the eigenvalue 2|β|+d and the eigenspaces are infinite dimensional. Additionally, Φα,β satisfies (Δz+14|z|2)Φα,β=(|α|+|β|+d)Φα,β, which means Φα,β is an eigenfunction of the Hermite operator Δz+14|z|2. This is the reason that Φα,β is called the special Hermite function.

For λ2N0+d, by ΠλL we denote the projection to the eigenspace of L with the eigenvalue λ, i.e.,ΠλLf:=β:2|β|+d=λαN0df,Φα,βΦα,β,fL2(Cd). Since {Φα,β}α,β is an orthonormal basis of L2(Cd), so one can expand f into the series of special Hermite functions. In fact, we havef=λ2N0+dΠλLf,fL2(Cd). 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 Lp unless p=2. So, we consider the Bochner-Riesz mean for the special Hermite expansion which is defined bySλδ(L)f=μ2N0+d(1μλ)+δΠλLf. By the uniform boundedness principle, the Lp convergence of Sλδ(L)f for all fLp is equivalent to the uniform estimateSλδ(L)fpCfp. The problem has been studied by many authors. By the transplantation theorem in [15] we see the estimate (1.7) implies Lp boundedness of the classical Bochner-Riesz operator in R2d. Thus (1.7) holds only if δ>δ(2d,p), p2. It seems to be plausible to conjecture that the uniform estimate (1.7) holds if δ>δ(2d,p). In [30], Thangavelu verified the conjecture for 2dd1p. Later, the range was extended to 2(3d+1)3d2<p by Ratnakumar, Rawat, and Thangavelu [20]. Further progress was made by Thangavelu [31] who showed the local estimate χESλδ(L)pC holds provided that δ>δ(2d,p) and 2(2d+1)2d1<p. The corresponding global estimate was later established by Stempak and Zienkiewicz [27], i.e., they showed that the estimate (1.7) holds for 2(2d+1)2d1<p and δ>δ(2d,p). The common key ingredient of the previous results is the L2Lp projection estimate of the formΠλL2pCλd12dp 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 2(2d+1)2d1<p. Later, Koch and Ricci [16] proved that (1.8) holds if and only if 2(2d+1)2d1p (also, see [13] for Lp-Lq estimates for ΠλL). So, further improvement is no longer possible via the estimate (1.8) when 2(2d+1)2d1>p.

Local Lp convergence of Sλδ(L)  Currently, no result with the sharp summability exponent δ(2d,p) is known when 2(2d+1)2d1>p. 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 d1, 0<c0<2, p>p0(2d), and δ>δ(2d,p). Suppose that E,FR2d are compact sets satisfying |zz|2c0 for all zE, zF. Then there exists a constant C independent of λ such thatχEλSλδ(L)χFλpC, where Eλ, Fλ denote the dilated sets λE, λF, respectively.

Corollary 1.5

Let p,δ be as in Theorem 1.4. Then for any compact set KR2d and compactly supported fLp(Cd), we havelimλK|Sλδ(L)f(x)f(x)|pdx=0.

When d=1, from the result we have a complete characterization of (p,δ) for which the local convergence (1.10) holds. The assumption that |zz|2c0 for all zE, zF can be regarded as a counterpart of the assumption on D(x,y) in Theorem 1.2. If |zz|2+c0 for all zE, zF, the kernel of the operator χEλSλδ(L)χFλ rapidly decays (see (3.11)). Thus the assumption can be relaxed so that ||zz|2|c0 for all zE, zF.

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 R2×R 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., L2Lp 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 Sλδ(H), Sλδ(L) have been unknown due to complexity associated to the special functions. This is why all the previous results were obtained relying on the L2Lp 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 ΠλH and ΠλL. By making use of the Schrödinger propagators eitH, eitL (see (2.1), (3.1)), of which kernel representation is well known, we obtain explicit expressions for the kernels of the operators Sλδ(H), Sλδ(L). 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 p<2(d+1)/(d1) (or p<2(2d+1)/(2d1)). 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, λ2N0+d and we identify Cd with R2d.

  • For given A,B>0, we write BA if there is a constant C>0 such that BCA. Here, if C has to be taken to be small enough, we use the notation BA to mean that A is sufficiently larger than B. Furthermore, AB denotes that AB and BA.

  • Bd(x,r)={yRd:|yx|<r}.

  • For an operator T we denote by T(x,y) (or T(z,z)) the kernel of T.

  • x:=(x1,,xd), x:=(x1,,xd), so xy=(xiyj)1i,jd. In particular, if a(x)=(a1(x),,ad(x)) is a Cd-valued differentiable function on Rd, then we have xa(x)=(xjai(x))1i,jd.

  • By Id we denote the d×d identity matrix. If the value of d is clear from the context, we simply denote Id by I.

  • For 1id, ei denotes the i-th standard basis in Rd.

  • For SRd and a constant a>0, we denote aS={ax:xS}.

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:f×g(z)=R2df(zw)g(w)ei2z,Swdw, where S is a skew-symmetric 2d×2d matrix given byS=(0IdId0). By the argument using the Weyl transform, it can be shown that ΠλLf=f×ςk, where k=λd2 and ςk(z)=(2π)dLkd1(12|z|2)e

Lower bound on the summability index of Sλδ(H)

In this section we obtain a new lower bound on the summability index δ for uniform boundedness of Sλδ(H) on Lp.

Proposition 4.1

Let d1 and 2<p. The uniform bound Sλδ(H)pC holds only if δ>δ(d,p) andδγ(d,p):=13p+d3(121p).

In particular, when d=1 the uniform bound (1.2) holds only if δ0 for 2p4 and δ13p+13(121p) for p4. This coincides with Thangavelu's result [29, Theorem 2.1]. In higher dimensions, i.e., d2, it is necessary for (1.2) that δ>δ(d,p) if 2(2d1)2d3p; δγ(d,p)>δ(d,p) if 2(d+1)dp2(

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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