Almost everywhere convergence of Bochner-Riesz means for the Hermite operators
Peng Chen, Xuan Thinh Duong, Danqing He, Sanghyuk Lee, Lixin Yan
aa r X i v : . [ m a t h . C A ] J u l ALMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZMEANS FOR THE HERMITE OPERATORS
PENG CHEN, XUAN THINH DUONG, DANQING HE, SANGHYUK LEE, AND LIXIN YAN
Abstract.
Let H = − ∆ + | x | be the Hermite operator in R n . In this paperwe study almost everywhere convergence of the Bochner-Riesz means associ-ated with H which is defined by S λR ( H ) f ( x ) = ∞ P k =0 (cid:0) − k + nR (cid:1) λ + P k f ( x ) . Here P k f is the k -th Hermite spectral projection operator. For 2 ≤ p < ∞ , weprove that lim R →∞ S λR ( H ) f = f a.e.for all f ∈ L p ( R n ) provided that λ > λ ( p ) / λ ( p ) = max (cid:8) n (cid:0) / − /p (cid:1) − / , (cid:9) . Conversely, we also show the convergence generally fails if λ < λ ( p ) / f ∈ L p ( R n ) for 2 n/ ( n − ≤ p such that theconvergence fails. This is in surprising contrast with a.e. convergence of theclassical Bochner-Riesz means for the Laplacian. For n ≥ p ≥ S λR ( H )is as small as only the half of the critical index for a.e. convergence of theclassical Bochner-Riesz means. When n = 1, we show a.e. convergence holdsfor f ∈ L p ( R ) with p ≥ λ >
0. Compared with the classical resultdue to Askey and Wainger who showed the optimal L p convergence for S λR ( H )on R we only need smaller summability index for a.e. convergence. Introduction
Convergence of Bochner-Riesz means of Fourier transform in the L p spaces is oneof the most fundamental problems in classical harmonic analysis. For λ ≥ R >
0, the classical Bochner-Riesz means for the Laplacian on R n are defined by S λR f ( x ) = Z R n e πix · ξ (cid:18) − | ξ | R (cid:19) λ + b f ( ξ ) dξ, ∀ ξ ∈ R n . (1.1)Here t + = max { , t } for t ∈ R and b f denotes the Fourier transform of f . The L p convergence of S λR f → f as R → ∞ is equivalent to the L p boundedness of theoperator S λ := S λ , and the longstanding open problem known as the Bochner-Rieszconjecture is that, for 1 ≤ p ≤ ∞ and p = 2, S λ is bounded on L p ( R n ) if and onlyif λ > λ ( p ) := max n n (cid:12)(cid:12)(cid:12) − p (cid:12)(cid:12)(cid:12) − , o . (1.2) Date : July 29, 2020.2000
Mathematics Subject Classification.
Key words and phrases.
Almost everywhere convergence, Bochner-Riesz means, the Hermiteoperators, trace lemma.
It was shown by Herz [27] that the condition (1.2) on λ is necessary for L p bound-edness of S λ . Carleson and Sj¨olin [10] proved the conjecture when n = 2. After-ward, substantial progress has been made in higher dimensions, for example see[46, 34, 5, 25, 45] and references therein. However, the conjecture still remainsopen for n ≥
3. Concerning pointwise convergence, Carbery, Rubio de Franciaand Vega [8] showed a.e. convergence with the sharp summability exponent for all f ∈ L p ( R n ), lim R →∞ S λR f = f a . e . provided p ≥ λ > λ ( p ) . When n = 2 the result was previously obtainedby Carbery [7] who proved the sharp L p estimates for the maximal Bochner-Rieszmeans. Also, see [14] for earlier partial result based on the maximal Bochner-Rieszestimate in higher dimensions. Regarding the most recent result for the maximalBochner-Riesz estimate, we refer the reader to [35].It is remarkable that the result by Carbery et al. [8] settled the a.e. convergenceproblem up to the sharp index λ ( p ) for 2 ≤ p ≤ ∞ . There are also results at thecritical exponent, i.e., λ = λ ( p ) (for example, see [1, 36]). It should be mentionedthat almost everywhere convergence of S λR f with f ∈ L p , 1 < p < , exhibitsdifferent nature and few results are known in this direction except when dimension n = 2 ([37, 43, 44]). Bochner-Riesz means for the Hermite operator.
In this paper we are con-cerned with almost everywhere convergence of Bochner-Riesz means for the Hermiteoperator H on R n , which is defined by H = − ∆ + | x | = − n X i =1 ∂ ∂x i + | x | , x = ( x , · · · , x n ) . (1.3)The operator H is non-negative and selfadjoint with respect to the Lebesgue mea-sure on R n . For each non-negative integer k , the Hermite polynomials H k ( t ) on R are defined by H k ( t ) = ( − k e t d k dt k (cid:0) e − t (cid:1) , and the Hermite functions h k ( t ) :=(2 k k ! √ π ) − / H k ( t ) e − t / , k = 0 , , , . . . form an orthonormal basis of L ( R ). Forany multiindex µ ∈ N n , the n -dimensional Hermite functions are given by tensorproduct of the one dimensional Hermite functions:Φ µ ( x ) = n Y i =1 h µ i ( x i ) , µ = ( µ , · · · , µ n ) . (1.4)Then the functions Φ µ are eigenfunctions for the Hermite operator with eigenvalue(2 | µ | + n ) and { Φ µ } µ ∈ N n form a complete orthonormal system in L ( R n ). Thus,for every f ∈ L ( R n ) we have the Hermite expansion f ( x ) = X µ h f, Φ µ i Φ µ ( x ) = ∞ X k =0 P k f ( x ) , (1.5)where P k denotes the Hermite projection operator given by P k f ( x ) = X | µ | = k h f, Φ µ i Φ µ ( x ) . (1.6) LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 3
For
R > H of order λ ≥ S λR ( H ) f ( x ) = ∞ X k =0 (cid:18) − k + nR (cid:19) λ + P k f ( x ) . (1.7)The assumption λ ≥ S λR ( H ) to be defined for all R > Concerning the L p convergence of S λR ( H ) f , uniform L p boundedness of S λR ( H )has been studied by a number of authors. In one dimension, it is known [2, 47]that if λ > / S λR ( H ) is uniformly bounded on L p for 1 ≤ p ≤ ∞ and, for1 / > λ ≥ ≤ p ≤ ∞ , S λR ( H ) is uniformly bounded on L p ( R ) if and onlyif λ > (2 / | /p − / | − /
6. In higher dimensions ( n ≥
2) the L p boundednessof S λR ( H ) is not so well understood yet. When λ > ( n − / S λR ( H ) on L p for 1 ≤ p ≤ ∞ . In particular, S λR ( H )converges to f in L ( R n ) if and only if λ > ( n − /
2. For 0 ≤ λ ≤ ( n − / ≤ p ≤ ∞ , p = 2, it still seems natural to conjecture that S λR ( H ) is uniformlybounded on L p ( R n ) if and only if λ > λ ( p ) (see [51, p.259]). Thangavelu alsoshowed k S λR ( H ) f k p ≤ C k f k p if and only if λ > λ ( p ) under the assumption that f is radial, thus the condition λ > λ ( p ) is necessary for L p boundedness of S λR ( H ).The necessity of the condition λ > λ ( p ) for L p boundedness can also be shown bythe transplantation result in [32] which deduces the L p boundedness of S λR from thatof S λR ( H ). Karadzhov [30] verified the conjecture in the range 1 ≤ p ≤ n/ ( n + 2) . The boundedness for p ∈ [2 n/ ( n − , ∞ ] follows from duality. However, it remainsopen to see if the conjecture is true in the range 2 n/ ( n + 2) < p ≤ n/ ( n + 1). Almost everywhere convergence.
Concerning a.e. convergence of S λR ( H ) f , itis known [47, 48] (see also [50, Chapter 3]) that S λR ( H ) f converges to f a.e. forevery f ∈ L p ( R n ) , ≤ p < ∞ , whenever λ > (3 n − /
6. Recently, Chen, Lee,Sikora and Yan [12] studied L p boundedness of the maximal Bochner-Riesz meansfor the Hermite operator H on R n for n ≥
2, that is to say, S λ ∗ ( H ) f ( x ) := sup R> | S λR ( H ) f ( x ) | , and they showed that the operator S λ ∗ ( H ) is bounded on L p ( R n ) whenever p ≥ nn − λ > λ ( p ) . (1.8)As a consequence, we have(1.9) lim R →∞ S λR ( H ) f = f a.e.for f ∈ L p ( R n ) and p , λ satisfying (1.8). For more results regarding the Hermiteexpansion (1.5), the estimate for the Hermite spectral projection, and the Bochner-Riesz means for the Hermite operator, we refer the reader to [31, 48, 49, 51, 33, 20,13, 26] and references therein.The following is the main result of this paper which establishes a.e convergence ofthe operator S λR ( H ) up to the sharp summability index for p ≥ Note that S λR ( H ) f can not be defined with R = 2 k + n if λ < PENG CHEN, XUAN THINH DUONG, DANQING HE, SANGHYUK LEE, AND LIXIN YAN
Theorem 1.1.
Let ≤ p < ∞ and λ ≥ . Then, for any f ∈ L p ( R n ) we have (1.9) whenever λ > λ ( p ) / . In particular, for n = 1 , (1.9) holds for all f ∈ L p ( R ) whenever λ > . Conversely, if (1.9) holds for all f ∈ L p ( R n ) with n ≥ and n/ ( n − < p < ∞ , we have λ ≥ λ ( p ) / . Except the endpoint cases Theorem 1.1 almost completely settles the a.e. con-vergence problem of S λR ( H ) f as R → ∞ with f ∈ L p ( R n ), p ≥
2. As is alreadymentioned, S λR ( H ) converges in L p only if λ > λ ( p ). Surprisingly, we only need thehalf of the critical summability index λ ( p ) in order to guarantee a.e. convergenceof S λR ( H ) f . Unlike the classical Bochner-Riesz means the critical indices for L p convergence and a.e. convergence for Hermite operators do not match.Let us now recall from [9, pp.320-321] (also [36]) how the sharpness of the result in[8] can be justified for the classical Bochner-Riesz means S λR . In order to considera.e. convergence of S λR f with f ∈ L p , S λR f should be defined at least as a tempereddistribution for f ∈ L p . If so, by duality S λ is defined from Schwartz class S to L p .This implies the convolution kernel K λ of S λ is in L p ′ , so it follows that λ > λ ( p )because K λ ∈ L p ′ if and only if λ > λ ( p ). However, this kind of argument doesnot work for the Bochner-Riesz means for the Hermite operator since S λR ( H ) f iswell defined for any f ∈ L p . To show the necessity part of Theorem 1.1 we makeuse of the Nikishin-Maurey theorem by which a.e. convergence implies a weightedinequality for the maximal operator S λ ∗ ( H ). We show such a maximal estimate cannot be true if λ < λ ( p ) /
2. See Proposition 4.1 below.The sufficiency part of Theorem 1.1 relies on the maximal estimate which is a typicaldevice in the study of almost everywhere convergence. In order to show (1.9)we consider the corresponding maximal operator S λ ∗ ( H ) and prove the followingweighted estimate, from which we deduce a.e. convergence of S λR ( H ) f via thestandard argument. Theorem 1.2.
Let ≤ α < n . The operator S λ ∗ ( H ) is bounded on L ( R n , (1 + | x | ) − α ) if λ > max n α − , o . Conversely, if S λ ∗ ( H ) is bounded on L ( R n , (1+ | x | ) − α ) , then λ ≥ max (cid:8) ( α − / , (cid:9) . Once we have Theorem 1.2 it is easy to deduce the sufficiency part of Theo-rem 1.1. Indeed, via a standard approximation argument (see, for example, [41]and [49, Theorem 2]) Theorem 1.2 establishes a.e. convergence of S λR ( H ) f for all f ∈ L ( R n , (1 + | x | ) − α ) provided that λ > max (cid:8) ( α − / , (cid:9) . Now, for given p ≥ λ > λ ( p ) / α such that α > n (1 − /p ) and λ > max (cid:8) ( α − / , (cid:9) . Our choice of α ensures that f ∈ L ( R n , (1 + | x | ) − α ) if f ∈ L p as it follows by H¨older’s inequality. Therefore, this yields a.e. convergenceof S λR ( H ) f for f ∈ L p ( R n ) if λ > λ ( p ) / L estimate in the study of pointwise convergence for Bochner-Riesz means goes back to Carbery et al. [8]. It turned out that the same strategyis also efficient for similar problems in different settings. For example, see [1, 36]for a.e. convergence of the classical Bochner-Riesz means at the critical index λ ( p ) LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 5 with p > n/ ( n −
1) and see [23, 29] for a.e. convergence for the Bochner-Rieszmeans associated with the sub-Laplacian on the Heisenberg group.
Square function estimate on weighted L -space. The proof of the sufficiencypart of Theorem 1.2 relies on a weighted L -estimate for the square function S δ which is defined by(1.10) S δ f ( x ) = (cid:18)Z ∞ (cid:12)(cid:12)(cid:12) φ (cid:16) δ − (cid:16) − Ht (cid:17)(cid:17) f ( x ) (cid:12)(cid:12)(cid:12) dtt (cid:19) / , < δ < / , where φ is a fixed C ∞ function supported in [2 − , − ] with | φ | ≤ The followingis our main estimate.
Proposition 1.3.
Let < δ ≤ / , < ǫ ≤ / , and let ≤ α < n . Then, thereexists a constant C > , independent of δ and f , such that (1.11) Z R n | S δ f ( x ) | (1 + | x | ) − α dx ≤ CδA ǫα,n ( δ ) Z R n | f ( x ) | (1 + | x | ) − α dx, where A ǫα,n ( δ ) := ( δ − ǫ , ≤ α ≤ , if n = 1 ,δ − α , < α < n, if n ≥ . (1.12)A similar estimate with the homogeneous weight | x | − α was obtained by Carbery etal. [8] for the square function associated to the Laplacian ∆: S δ f ( x ) := (cid:16) Z ∞ (cid:12)(cid:12)(cid:12) φ (cid:16) δ − (cid:16) t − ∆ (cid:17)(cid:17) f ( x ) (cid:12)(cid:12)(cid:12) dtt (cid:17) / . Though we make use of the weighted L estimate as in [8] there are notable differ-ences which are due to special properties of the Hermite operator and they even-tually lead to improvement of the summability indices. Let P k be the Littlewood-Paley projection operator which is given by d P k f ( ξ ) = φ (2 − k | ξ | ) b f ( ξ ) for φ ∈ C ∞ c (2 − , S δ ( P k f ) can be reduced to the equivalent estimate for S δ ( P f ). This tells thatcontributions from different dyadic frequency pieces are basically identical. How-ever, this is not the case for S δ f . As for the Hermite case estimate (1.11), the highand low frequency parts exhibit considerably different natures. Unlike the classicalBochner-Riesz operator, we need to handle them separately. Basic estimates.
As is to be seen in Section 3 below, the proof of Proposition 1.3mainly depends on the following two lemmas.
Lemma 1.4.
Let α ≥ . Then, the estimate k (1 + | x | ) α f k ≤ C k (1 + H ) α f k (1.13) holds for any f ∈ S ( R n ) . Here, S ( R n ) stands for the class of Schwartz functionsin R n . Here, for any bounded function M the operator M ( H ) is defined by M ( H ) = ∞ P k =0 M (2 k + n ) P k . PENG CHEN, XUAN THINH DUONG, DANQING HE, SANGHYUK LEE, AND LIXIN YAN
Clearly, this can not be true if H is replaced by − ∆. It should be noted that theestimate (1.13) becomes more efficient when we deal with the low frequency partof the function. The second is a type of trace lemma (Lemma 1.5) for the Hermiteoperator. In fact, we obtain Lemma 1.5.
For α > , there exists a constant C > such that the estimate k χ [ k,k +1) ( H ) k L ( R n ) → L ( R n , (1+ | x | ) − α ) ≤ Ck − (1.14) holds for every k ∈ N . In our proof of (1.11) this inequality (1.14) takes the place of the classical tracelemma which was the main tool in [8]. The trace lemma tells that a function in theSobolev space ˙ W α, ( R n ) can be restricted to S n − as an L function. By takingFourier transform and Plancherel’s theorem, this can be equivalently formulated asfollows: Z R n (cid:12)(cid:12) χ [1 − ǫ, ǫ ] ( √− ∆) f ( x ) (cid:12)(cid:12) dx ≤ Cǫ Z R n | f ( x ) | | x | α dx. In contrast with the case of the Laplacian where the trace inequality should take ascaling-invariant form, that is to say, the weight should be homogeneous, we havethe inhomogeneous weight (1 + | x | ) − α in both of the estimates (1.13) and (1.14).As to be seen later, this is related to the fact that the spectrum of the Hermiteoperator H is bounded away from the origin.We show Proposition 1.3 by making use of both of the estimates (1.13) and (1.14).The proof of Proposition 1.3 divides into two parts depending on size of frequencyin the spectral decomposition (1.6). For the high frequency part ( k & δ − in (1.6))the key tool is the estimate (1.14), which we combine with spatial localizationargument based on the finite speed of propagation of the wave operator cos( t √ H ).The estimate (1.14) can be compared with the restriction-type estimate due toKaradzhov [30]: k χ [ k,k +1) ( H ) k → p ≤ Ck n ( − p ) − , ∀ k ≥ . (1.15)The bound in (1.14) is much smaller than that in (1.15) when k is large. So, theestimate (1.14) becomes more efficient in the high frequency regime. In fact, theestimate (1.15) was used to show the sharp L p –bounds on S δ for 2 n/ ( n − ≤ p ≤ ∞ , n ≥
2, [12, Proposition 5.6]. In the low frequency part ( k . δ − in (1.6)),inspired by [12, Lemma 5.7], we directly obtain the estimate using the estimate(1.13). The estimate (1.13) does not seem to be so efficient since the bound getsworse as the frequency increases, but it is remarkable that this bound is goodenough to yield the sharp result in Theorem 1.2 via balancing the estimates for lowand high frequencies (see Remark 3.2). Organization of the paper.
The rest of the paper is organized as follows. InSection 2 we prove Lemma 1.4, Lemma 1.5 and the Littlewood-Paley inequalityfor the Hermite operator, which provide basic estimates required for the proof ofProposition 1.3. We give the proof of the sufficiency part of Theorem 1.2 in Section3 by establishing the square function estimate in Proposition 1.3. In Section 4 weshow the sharpness of the summability indices, hence we complete the proofs ofTheorem 1.1 and Theorem 1.2.
LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 7 Some weighted estimates for the Hermite operator
In this section, we prove Lemma 1.4, Lemma 1.5 and the Littlewood-Paley in-equality for the Hermite operator in R n , which are to be used in the proof of thesufficiency part of Theorem 1.2 in Section 3.2.1. Proof of Lemma 1.4.
In order to show Lemma 1.4, we use the followingLemmas 2.1, 2.2 and 2.3.
Lemma 2.1.
For all φ ∈ S ( R n ) , we have k φ k ≤ k Hφ k and k H k φ k ≤ k H k + m φ k for any k, m ∈ N .Proof. This follows from the fact that the first eigenvalue of H is bigger than orequal to 1. (cid:3) Lemma 2.2.
Let n = 1 . Then, for all φ ∈ S ( R ) , k x φ k + (cid:13)(cid:13)(cid:13) d dx φ (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) x ddx φ (cid:13)(cid:13)(cid:13) ≤ k Hφ k . Proof.
Since k Hφ k = h ( − d dx + x ) φ, ( − d dx + x ) φ i , a simple calculation shows k Hφ k = (cid:13)(cid:13)(cid:13) d dx φ (cid:13)(cid:13)(cid:13) + 2Re D ddx φ, xφ E + 2 (cid:13)(cid:13)(cid:13) x ddx φ (cid:13)(cid:13)(cid:13) + k x φ k . We now observe 2Re h ddx φ, xφ i = (cid:10) ddx φ, xφ (cid:11) + h xφ, ddx φ i = − h φ, φ i . This andthe above give k x φ k + (cid:13)(cid:13)(cid:13) d dx φ (cid:13)(cid:13)(cid:13) + 2 (cid:13)(cid:13)(cid:13) x ddx φ (cid:13)(cid:13)(cid:13) = k Hφ k + 2 k φ k ≤ k Hφ k as desired. For the last inequality we use Lemma 2.1. (cid:3) Lemma 2.3.
Let n = 1 . Then, for φ ∈ S ( R ) and for k ∈ N , we have k x k φ k ≤ C k k H k φ k and k Hx k − φ k ≤ D k k H k φ k . (2.1) Proof.
We begin with noting that, if k = 1, the first estimate in (2.1) holds with C = √ D = 1. We now proceed to prove(2.1) for k ≥ k − C k − and D k − . A computation gives(2.2) H ( x k − φ ) = − [(2 k − k − − k − k − x k − φ − k − x ddx (cid:0) x k − φ (cid:1) + x k − Hφ.
By (2.2), Lemma 2.2, and our induction assumption we see that k H ( x k − φ ) k ≤ (2 k − k − k x k − φ k + 2(2 k − k x ddx ( x k − φ ) k + k x k − Hφ k ≤ (2 k − k − C k − k H k − φ k + 2(2 k − C k H ( x k − φ ) k + C k − k H k φ k ≤ (2 k − k − C k − k H k − φ k + 2(2 k − C D k − k H k − φ k + C k − k H k φ k . PENG CHEN, XUAN THINH DUONG, DANQING HE, SANGHYUK LEE, AND LIXIN YAN
Hence, we get the estimate k H ( x k − φ ) k ≤ D k k H k φ k with D k = (2 k − k − C k − + 4(2 k − C D k − + C k − . On other hand, wealso have k x k φ k ≤ C k H ( x k − φ ) k ≤ C k k H k φ k with C k = C D k , k ≥
2. So, we readily get the estimates in (2.1). This completesthe proof. (cid:3)
Now we are ready to prove Lemma 1.4.
Proof of Lemma 1.4.
Define H i = − ∂ ∂x i + x i , i = 1 , , . . . , n . By Lemma 2.3 k| x i | k f k ≤ C k k H ki f k for all positive natural numbers k ∈ N . Hence, we have k (1 + | x | ) k f k ≤ C k (cid:0) k f k + n X i =1 k| x i | k f k (cid:1) ≤ C k (cid:0) k f k + n X i =1 k H ki f k (cid:1) . Since all H i are non-negative selfadjoint operators and commute strongly (thatis, their spectral resolutions commute), the operators Q ni =1 H ℓ i i are non-negativeselfadjoint for all ℓ i ∈ Z + . Hence1 + n X i =1 H ki ≤ (1 + n X i =1 H i ) k = (1 + H ) k for all k ∈ N . Combining this with the above inequality we get k (1 + | x | ) k f k ≤ C k D(cid:16) n X i =1 H i (cid:17) k f, f E = C k k (1 + H ) k f k . This proves estimates (1.13) for all α ∈ N . Now, by virtue of L¨owner-Heinz in-equality (see, e.g., [15, Section I.5]) we can extend this estimate to all α ∈ [0 , ∞ ).This completes the proof of Lemma 1.4. (cid:3) The proof of Lemma 1.5: Trace lemma for the Hermite operator.
Our proof of the estimate (1.14) is inspired by the argument in [4, Theorem 3.3]where the authors obtained local smoothing estimate for the Hermite Schr¨odingerpropagator.
Proof.
To show (1.14), it is sufficient to show Z [ − M,M ] n | χ [ k,k +1) ( H ) f ( x ) | dx ≤ CM k − k f k (2.3)for every M ≥
1. Indeed, the estimate (1.14) immediately follows by decomposing R n into dyadic shells and applying (2.3) to each of them because α > LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 9
Let us prove (2.3). For every f ∈ S ( R n ), we may write its Hermite expansion f ( x ) = P µ h f, Φ µ i Φ µ ( x ) as in (1.5). Considering this spectral decomposition,clearly we may decompose(2.4) f = n X i =1 f i such that f , . . . , f n are orthogonal to each other and, for 1 ≤ i ≤ n , µ i ≥ | µ | /n whenever h f i , Φ µ i 6 = 0 (see for example, [4]). Recalling that the Hermite functionsΦ µ are eigenfunctions for the Hermite operator H , it is clear that χ [ k,k +1) ( H ) f i ( x ) = X | µ | + n = k h f i , Φ µ i Φ µ ( x ) . Note that µ i ∼ | µ | if h f i , Φ µ i 6 = 0, so in order to show (2.3) it is enough to showthat(2.5) Z [ − M,M ] n | χ [ k,k +1) ( H ) f i ( x ) | dx ≤ CM X | µ | + n = k µ − i |h f i , Φ µ i| for each i = 1 , . . . , n and M >
0. By symmetry we have only to show (2.5) with i = 1. For the purpose we do not need the particular structure of f , so let us set g := f for a simpler notation.Let us write g ( x ) = P µ c ( µ )Φ µ ( x ) with c ( µ ) = h g, Φ µ i . Hence, we have | χ [ k,k +1) ( H ) g ( x ) | = X | µ | + n = k X | ν | + n = k c ( µ ) c ( ν ) n Y i =1 h µ i ( x i ) h ν i ( x i ) . Using this, by Fubini’s theorem it follows that Z [ − M,M ] n | χ [ k,k +1) ( H ) g ( x ) | dx ≤ X | µ | + n = k X | ν | + n = k c ( µ ) c ( ν ) Z M − M h µ ( x ) h ν ( x ) dx n Y i =2 h h µ i , h ν i i . Since h µ i are orthogonal to each other, we have µ i = ν i for i = 2 , . . . , n whenever h h µ i , h ν i i 6 = 0 and we also have µ = ν since 2 | µ | + n = k = 2 | ν | + n . Thus, Z [ − M,M ] n | χ [ k,k +1) ( H ) g ( x ) | dx ≤ X | µ | + n = k | c ( µ ) | Z M − M h µ ( x ) dx . Therefore, to complete the proof it suffices to show that Z M − M h µ ( t ) dt ≤ CM µ − / . If µ ≤ M , the estimate is trivial because k h µ k = 1. Hence, we may assume µ > M . By the property of the Hermite functions (see [50, Lemma 1.5.1]) thereexists a constant C > | h µ ( t ) | ≤ Cµ − / provided that t ∈ [ − M, M ]and µ > M . Thus, we get the desired estimate, which completes the proof ofLemma 1.5. (cid:3) An extension of the estimate (1.14) . We modify the estimate (1.14) intoa form which is suitable for our purpose. For any function F with support in [0 , ≤ q < ∞ , we define k F k N ,q := N N X ℓ =1 sup λ ∈ [ ℓ − N , ℓN ) | F ( λ ) | q /q , N ∈ N . (2.6)For q = ∞ , we put k F k N , ∞ = k F k ∞ (see [13, 17, 20]). Then we have the followingresult which is a generalization of Lemma 1.5. Lemma 2.4.
For α > we have Z R n | F ( √ H ) f ( x ) | (1 + | x | ) − α dx ≤ CN k δ N F k N , Z R n | f ( x ) | dx for any function F with support in [ N/ , N ] and N ∈ N , where δ N F ( λ ) is definedby F ( N λ ) .Proof. Since the operator F ( √ H ) is selfadjoint, it is sufficient to show the dualestimate(2.7) Z R n | F ( √ H ) f ( x ) | dx ≤ CN k δ N F k N , Z R n | f ( x ) | (1 + | x | ) α dx. By orthogonality Z R n | F ( √ H ) f ( x ) | dx ≤ N X ℓ = N / (cid:13)(cid:13)(cid:13) χ (cid:2) ℓ − N , ℓN (cid:1) ( √ H ) F ( √ H ) f (cid:13)(cid:13)(cid:13) because F is supported in [ N/ , N ]. Note that k χ [ ( ℓ − N , ℓN ) ( √ H ) f k ≤ k χ (cid:2) ( ℓ − N , ( ℓ − N +2 (cid:1) ( H ) f k . Hence, it follows that Z R n | F ( √ H ) f ( x ) | dx ≤ N X ℓ = N / sup λ ∈ (cid:2) ℓ − N , ℓN (cid:1) | F ( λ ) | (cid:13)(cid:13)(cid:13) χ (cid:2) ( ℓ − N , ( ℓ − N +2 (cid:1) ( H ) f (cid:13)(cid:13)(cid:13) . Since ℓ − N ∼ N , applying (1.14) we obtain Z R n | F ( √ H ) f ( x ) | dx ≤ CN N X ℓ = N / sup λ ∈ (cid:2) ℓ − N , ℓN (cid:1) | F ( λ ) | Z R n | f ( x ) | (1 + | x | ) α dx. Thus, the estimate (2.7) follows from (2.6). (cid:3)
In Lemma 1.5, the estimate (1.14) is established for α > α to 0 < α ≤
1. We recall that [ · , · ] θ stands for the complex interpolation bracket(for example, see [3, 6]). LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 11
Lemma 2.5.
Let ( A i , B i ) , i = 1 , and ( A, B ) be interpolation pairs. Suppose T isa bilinear operator defined on ⊕ i =1 ( A i ∩ B i ) with values in A ∩ B such that k T ( x , x ) k A ≤ M Y i =1 k x i k A i , k T ( x , x ) k B ≤ M Y i =1 k x i k B i . Then, for θ ∈ [0 , , we have k T ( x , x ) k [ A,B ] θ ≤ M − θ M θ Y i =1 k x i k [ A i ,B i ] θ . Thus T can be extended continuously from ⊕ i =1 [ A i , B i ] θ into [ A, B ] θ for any θ ∈ [0 , . For the proof of Lemma 2.5, we refer the reader to [6, p.118]. Making use of Lemma2.5, we obtain the following result.
Lemma 2.6.
Let < α ≤ , N ∈ N , and let F be a function supported in [ N/ , N ] .Then, for any ε > , we have Z R n | F ( √ H ) f ( x ) | (1 + | x | ) − α dx ≤ C ε N α ε k δ N F k N ,q Z R n | f ( x ) | dx (2.8) with q = 2 α − (1 + ε ) for some constant C ε > independent of f and F .Proof. Clearly, (2.8) holds for N = 1. Now we fix N ∈ N and N ≥
2, and let A bea collection of functions supported in [ N/ , N ] which is given by A := n G ∈ L loc : G ( x ) = a χ [ N , [ N / N ) ( x )+ N X i =[ N / a i χ [ i − N , iN ) ( x ) , a , a i ∈ C o . Then, for any Borel set Q , we define a normalized counting measure ν on R bysetting ν ( Q ) := 1 N − [ N / − (cid:18) n i : 2 i − N ∈ Q, i = [ N /
4] + 2 , . . . , N o + C NQ (cid:19) , where N ≥ C NQ = ( N/ ∈ Q ;0 if N/ Q. We also define an L q norm on A by k G k L q ( dν ) := (cid:16) Z NN/ | G ( x ) | q dν ( x ) (cid:17) /q . Hence, k G k L q ( dν ) ∼ k δ N G k N ,q , and the space A equipped with this norm becomesa Banach space which is denoted by A q . It also follows (for example, see [3]) that[ A , A ∞ ] s = A q , q = 1 − s , s ∈ [0 , . Let us denote by B α the space B α := (cid:8) f ∈ L loc ( R n ) : Z R n | f ( x ) | (1 + | x | ) α dx < ∞ (cid:9) equipped with the norm k f k B α := k f k L ( R n , (1+ | x | ) − α ) . Thus we have [ B ν , B ] θ = B s , s = (1 − θ ) ν. We consider the bilinear operator T given by T ( G, f ) := G ( √ H ) f. By Lemma 2.4 and duality, for any ε > k T ( G, f ) k L ( R n ) ≤ CN / k G k A k f k B ε . Since R R n | G ( √ H ) f ( x ) | dx ≤ k δ N G k ∞ R R n | f ( x ) | dx, we also have k T ( G, f ) k L ( R n ) ≤ k G k A ∞ k f k B . Now, taking A = A , A = B ε , B = A ∞ , B = B , and A = B = L ( R n ), weapply Lemma 2.5 to get k T ( G, f ) k L ( R n ) ≤ CN (1 − θ ) / k G k A / (1 − θ ) k f k B (1 − θ )(1+ ε ) for θ ∈ [0 , − θ )(1 + ε ) = α . Then, equivalently, for G ∈ A we have(2.9) Z R n | G ( √ H ) f ( x ) | dx ≤ C ε N α ε k δ N G k N , ε ) /α Z R n | f ( x ) | (1 + | x | ) α dx. We now extend the estimate (2.9) to general functions supported in [ N/ , N ]. Fora function F supported in [ N/ , N ], set a = sup λ ∈ [ N , [ N / N ) | F ( λ ) | and a i =sup λ ∈ [ i − N , iN ) | F ( λ ) | , i = [ N /
4] + 2 , . . . , N and define G ∈ A by G ( x ) = a χ [ N , [ N / N ) ( x ) + N X i =[ N / a i χ [ i − N , iN ) ( x ) . Then, we clearly have k δ N F k N ,q = k δ N G k N ,q and | F ( x ) | ≤ | G ( x ) | for x ∈ R .Since h| F | ( √ H ) f, f i = P k ∈ N + n | F | ( √ k ) P n +2 | µ | = k |h f, Φ µ i| , we have Z R n | F ( √ H ) f ( x ) | dx ≤ X k ∈ N + n | G | ( √ k ) X n +2 | µ | = k |h f, Φ µ i| = Z R n | G ( √ H ) f ( x ) | dx. Since k δ N F k N ,q = k δ N G k N ,q , using (2.9) we get Z R n | F ( √ H ) f ( x ) | dx ≤ C ε N α ε k δ N F k N , ε ) /α Z R n | f ( x ) | (1 + | x | ) α dx. By duality the desired estimate follows. This completes the proof of Lemma 2.6. (cid:3)
Littlewood-Paley inequality for the Hermite operator.
We now recall afew standard results in the theory of spectral multipliers of non-negative selfadjointoperators (see for example, [19, 20]). By the Feynman-Kac formula we have theGaussian upper bound on the semigroup kernels p t ( x, y ) associated to e − tH :(2.10) 0 ≤ p t ( x, y ) ≤ (4 πt ) − n/ exp (cid:18) − | x − y | t (cid:19) for all t >
0, and x, y ∈ R n . LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 13
Proposition 2.7.
Fix a non-zero C ∞ bump function ϕ on R such that supp ϕ ⊆ (1 , . Let ϕ k ( t ) = ϕ (2 − k t ) , k ∈ Z , for t > . Then, for any − n < α < n , (2.11) (cid:13)(cid:13)(cid:13)(cid:16) ∞ X k = −∞ (cid:12)(cid:12) ϕ k ( √ H ) f (cid:12)(cid:12) (cid:17) / (cid:13)(cid:13)(cid:13) L ( R n , (1+ | x | ) α ) ≤ C p k f k L ( R n , (1+ | x | ) α ) . This can be proved by following the standard argument, for example, see [40, Chap-ter IV]. We include a brief proof for convenience of the reader.
Proof.
Let us denote by { r k } k ∈ Z the Rademacher functions. Set F ( t, λ ) := ∞ X k = −∞ r k ( t ) ϕ k ( λ ) . Let η be a nontrivial cutoff function such that η ∈ C ∞ c ( R + ). A straightforwardcomputation shows that sup R> k ηF ( t, Rλ ) k C β ≤ C β uniformly in t ∈ [0 ,
1] forevery integer β > n/ | x | α belongs to the A class if and only if − n < α < n (see [24, Example 7.1.7]). Thus,1 + | x | α ∈ A and so is (1 + | x | ) α for − n < α < n . Then we may apply [19, Theorem3.1] to get (cid:13)(cid:13) F ( t, H ) f (cid:13)(cid:13) L ( R n , (1+ | x | ) α ) ≤ C (cid:13)(cid:13) f (cid:13)(cid:13) L ( R n , (1+ | x | ) α ) with C > t ∈ [0 , . Since P ∞ k = −∞ | ϕ k ( H ) f | ∼ = R (cid:12)(cid:12) F ( t, H ) f (cid:12)(cid:12) dt bythe property of the Rademacher functions, taking integration in t on both sides ofthe above inequality yields (2.11). This proves Proposition 2.7. (cid:3) Proof of sufficiency part of Theorem 1.2
In this section we prove the sufficiency part of Theorem 1.2, that is to say, theoperator S λ ∗ ( H ) is bounded on L ( R n , (1 + | x | ) − α ) whenever 0 ≤ α < n and λ > max { α − , } . To do so, as mentioned before, we make use of the square functionto control the maximal operator.3.1.
Reduction to square function estimate.
We begin with recalling the wellknown identity (cid:18) − | m | R (cid:19) λ = C λ, ρ R − λ Z R | m | ( R − t ) λ − ρ − t ρ +1 (cid:18) − | m | t (cid:19) ρ dt, where λ > λ > ρ , and C λ, ρ = 2Γ( λ + 1) / (Γ( ρ + 1)Γ( λ − ρ )). Using the argumentin [41, pp.278–279], we have S λ ∗ ( H ) f ( x ) ≤ C ′ λ, ρ sup
2. Via dyadic decomposition, we write x ρ + = P k ∈ Z − kρ φ ρ (2 k x ) for some φ ρ ∈ C ∞ c (2 − , − ). Thus(1 − | ξ | ) ρ + =: φ ρ ( ξ ) + ∞ X k =1 − kρ φ ρk ( ξ ) , where φ ρk = φ (2 k (1 − | ξ | )) , k ≥
1. We also note that supp φ ρ ⊆ { ξ : | ξ | ≤ × − } and supp φ ρk ⊆ { ξ : 1 − − − k ≤ | ξ | ≤ − − k − } . Using (3.1), for λ > ρ + 1 / S λ ∗ ( H )( f ) ≤ C (cid:16) sup Proof of sufficiency part of Theorem 1.2. Since λ > 0, choosing η > λ − η > 0, we set ρ = λ − − η . Withour choice of ρ we can use (3.2). It is easy to obtain estimate for the first term inthe right hand side of (3.2). Since (1 + | x | ) − α is an A weight, by virtue of (2.10)a standard argument (see for example [12, Lemma 3.1]) yields (cid:13)(cid:13)(cid:13) sup 0. Hence, this gives the desired boundedness of S λ ∗ ( H ) on L ( R , (1 + | x | ) − α ) for n = 1 and 0 ≤ α < n ≥ 2, we assume α > α is later to be extended by interpolation. Similarly as before, we use (1.11) with δ = 2 − k to get (cid:13)(cid:13)(cid:13) ∞ X k =1 − kρ (cid:16) Z ∞ (cid:12)(cid:12)(cid:12) φ ρk (cid:0) t − √ H (cid:1) f (cid:12)(cid:12)(cid:12) dtt (cid:17) / (cid:13)(cid:13)(cid:13) L ( R n , (1+ | x | ) − α ) ≤ C ∞ X k =0 − k ( λ − α − − η ) k f k L ( R n , (1+ | x | ) − α ) . Thus, taking small enough η we see that S λ ∗ ( H ) is bounded on L ( R n , (1 + | x | ) − α )for n ≥ < α < n provided that λ > α − . On the other hand, we notethat S λ ∗ ( H ) is bounded on L ( R n ) for any λ > 0, see for example, [12, Corollary LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 15 S λ ∗ ( H )is bounded on L ( R n , (1 + | x | ) − α ) for any 0 < α ≤ λ > 0. This provesthe sufficient part of Theorem 1.2. (cid:3) To complete the proof of the sufficiency part of Theorem 1.2, it remains to proveProposition 1.3.3.2. Weighted inequality for the square function. In this subsection, we es-tablish Proposition 1.3. For the purpose we decompose S δ into high and lowfrequency parts. Let us set S lδ f ( x ) =: (cid:16) Z δ − / / (cid:12)(cid:12)(cid:12) φ (cid:16) δ − (cid:16) − Ht (cid:17)(cid:17) f ( x ) (cid:12)(cid:12)(cid:12) dtt (cid:17) / , S hδ f ( x ) =: (cid:16) Z ∞ δ − / (cid:12)(cid:12)(cid:12) φ (cid:16) δ − (cid:16) − Ht (cid:17)(cid:17) f ( x ) (cid:12)(cid:12)(cid:12) dtt (cid:17) / . Since the first eigenvalue of the Hermite operator is larger than or equal to 1, φ (cid:0) δ − (cid:0) − Ht (cid:1)(cid:1) = 0 if t ≤ φ ⊂ (2 − , − ) and δ ≤ / 2. Thus, it isclear that S δ f ( x ) ≤ S lδ f ( x ) + S hδ f ( x ) . (3.3)In order to prove Proposition 1.3 it is sufficient to show the following. Lemma 3.1. Let A ǫα,n ( δ ) be given by (1.12) . Then, for all < δ ≤ / and < ǫ ≤ / , we have the following estimates: Z R n | S lδ f ( x ) | (1 + | x | ) − α dx ≤ CδA ǫα,n ( δ ) Z R n | f ( x ) | (1 + | x | ) − α dx, (3.4) Z R n | S hδ f ( x ) | (1 + | x | ) − α dx ≤ CδA ǫα,n ( δ ) Z R n | f ( x ) | (1 + | x | ) − α dx. (3.5)Both of the proofs of the estimates (3.4) and (3.5) rely on the generalized tracelemmata, Lemma 2.4 and Lemma 2.6. Though, there are distinct differences in theirproofs. As for (3.4) we additionally use the estimate (1.13) which is efficient for thelow frequency part. Regarding the estimate (3.5) we use the spatial localizationargument which is based on the finite speed of propagation of the Hermite waveoperator cos( t √ H ). Similar strategy has been used to related problems, see forexample [12]. In this regards, our proof of the estimate (3.5) is similar to that in[8]. In high frequency regime the localization strategy becomes more advantageoussince the associated kernels enjoy tighter localization. This allows us to handle theweight (1 + | x | ) − α in an easier way. The choice of δ − in the definitions of S lδ , S hδ is made by optimizing the estimates which result from two different approaches,see Remark 3.2.3.3. Proof of (3.4) : low frequency part. We start with Littlewood-Paley de-composition associated with the operator H . Fix a function ϕ ∈ C ∞ supported in { ≤ | s | ≤ } such that P ∞−∞ ϕ (2 k s ) = 1 on R \{ } . By the spectral theory wehave X k ϕ k ( √ H ) f := X k ϕ (2 − k √ H ) = f . (3.6)for any f ∈ L ( R n ). Using (3.6), we get(3.7) | S lδ f ( x ) | ≤ C X ≤ k ≤ − log √ δ Z k +2 k − (cid:12)(cid:12)(cid:12) φ (cid:18) δ − (cid:18) − Ht (cid:19)(cid:19) ϕ k ( √ H ) f ( x ) (cid:12)(cid:12)(cid:12) dtt for f ∈ L ( R n ) ∩ L ( R n , (1 + | x | ) − α ). To exploit disjointness of the spectralsupport φ (cid:0) δ − (cid:0) − Ht (cid:1)(cid:1) we make additional decomposition in t . For k ∈ Z and i = 0 , , · · · , i = [8 /δ ] + 1 we set I i = (cid:2) k − + i k − δ, k − + ( i + 1)2 k − δ (cid:3) , (3.8)so that [2 k − , k +2 ] ⊆ ∪ i i =0 I i . Define η i adapted to the interval I i by setting η i ( s ) = η (cid:18) i + 2 k − − s k − δ (cid:19) , (3.9)where η ∈ C ∞ c ( − , 1) and P i ∈ Z η ( · − i ) = 1. For simplicity we also set φ δ ( s ) := φ ( δ − (cid:0) − s )) . We observe that, for t ∈ I i , φ δ ( s/t ) η i ′ ( s ) = 0 only if i − iδ − ≤ i ′ ≤ i + iδ + 3 . Hence, for t ∈ I i we have φ δ (cid:0) t − √ H (cid:1) ϕ k ( √ H ) = i +10 X i ′ = i − φ δ (cid:0) t − √ H (cid:1) ϕ k ( √ H ) η i ′ ( √ H ) , and thus Z k +2 k − (cid:12)(cid:12)(cid:12) φ δ (cid:0) t − √ H (cid:1) ϕ k ( √ H ) f (cid:12)(cid:12)(cid:12) dtt ≤ C X i i +10 X i ′ = i − Z I i (cid:12)(cid:12)(cid:12) φ δ (cid:0) t − √ H (cid:1) ϕ k ( √ H ) η i ′ ( √ H ) f (cid:12)(cid:12)(cid:12) dtt . Substituting this into (3.7), we have that | S lδ f ( x ) | ≤ C X ≤ k ≤ − log √ δ X i i +10 X i ′ = i − Z I i (cid:12)(cid:12)(cid:12) φ δ (cid:0) t − √ H (cid:1) η i ′ ( √ H ) ϕ k ( √ H ) f ( x ) (cid:12)(cid:12)(cid:12) dtt . Now we claim that, for 1 ≤ t ≤ δ − / ,(3.10) Z R n | φ δ (cid:0) t − √ H (cid:1) g ( x ) | (1 + | x | ) − α dx ≤ CA ǫα,n ( δ ) Z R n | (1 + H ) − α/ g ( x ) | dx. Before we begin to prove it, we show that this concludes the proof of estimate (3.4).Combining (3.10) with the preceding inequality, we see that R R n | S lδ f ( x ) | (1 + | x | ) − α dx is bounded by CA ǫα,n ( δ ) X ≤ k ≤ − log √ δ X i i +10 X i ′ = i − Z I i Z R n (cid:12)(cid:12)(cid:12) η i ′ ( √ H ) ϕ k ( √ H )(1 + H ) − α/ f ( x ) (cid:12)(cid:12)(cid:12) dx dtt . LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 17 Since the length of interval I i is comparable to 2 k − δ , taking integration in t andusing disjointness of the spectral supports, we get Z R n | S lδ f ( x ) | (1 + | x | ) − α dx ≤ CδA ǫα,n ( δ ) Z R n (cid:12)(cid:12)(cid:12) (1 + H ) − α/ f ( x ) (cid:12)(cid:12)(cid:12) dx. This, being combined with Lemma 1.4, yields the desired estimate (3.4).We now show the estimate (3.10). Let us consider the equivalent estimate(3.11) Z R n | φ δ (cid:0) t − √ H (cid:1) (1 + H ) α/ g ( x ) | (1 + | x | ) − α dx ≤ CA ǫα,n ( δ ) Z R n | g ( x ) | dx. We first show the estimate for the case n ≥ 2. Let N = 8[ t ] + 1. Note thatsupp φ δ (cid:0) · /t (cid:1) ⊂ [ N/ , N ]. By Lemma 2.4, Z R n | φ δ (cid:0) t − √ H (cid:1) (1 + H ) α/ g ( x ) | (1 + | x | ) − α dx ≤ CN (cid:13)(cid:13)(cid:13) φ δ (cid:0) t − N u (cid:1) (1 + N u ) α/ (cid:13)(cid:13)(cid:13) N , Z R n | g ( x ) | dx. We now estimate k φ δ (cid:0) t − N u (cid:1) (1 + N u ) α/ k N , . Note that supp φ δ (cid:0) t − N u (cid:1) ⊂ [ t √ − δN , t √ − δ/ N ] . Since the length of the interval [ t √ − δN , t √ − δ/ N ] ∼ δ and N ∼ t ≤ δ − / , we get (cid:13)(cid:13) φ δ (cid:0) t − N u (cid:1) (1 + N u ) α/ (cid:13)(cid:13) N , ≤ (cid:13)(cid:13) φ δ (cid:0) t − N u (cid:1) (1 + N u ) α/ (cid:13)(cid:13) ∞ (cid:13)(cid:13) χ [ t √ − δN , t √ − δ/ N ] (cid:13)(cid:13) N , ≤ CN α − . (3.12)Thus, noting 1 / ≤ t ≤ δ − / and α > 1, we obtain Z R n | φ δ (cid:0) t − √ H (cid:1) (1 + H ) α/ g ( x ) | (1 + | x | ) − α dx ≤ CN α − Z R n | g ( x ) | dx ≤ Cδ / − α/ Z R n | g ( x ) | dx, which gives (3.10) in the dimensional case n ≥ n = 1. Let 0 ≤ α < N = 8[ t ] + 1. Note thatsupp φ δ (cid:0) · /t (cid:1) ⊂ [ N/ , N ]. By Lemma 2.6, for any ε > ≤ C ε N α ε (cid:13)(cid:13)(cid:13) φ δ (cid:0) t − N u (cid:1) (1 + N u ) α/ (cid:13)(cid:13)(cid:13) N , ε ) α Z R | g ( x ) | dx. As before, in the same manner as in (3.12) we have k φ δ (cid:0) t − N u (cid:1) (1 + N u ) α/ k N , ε ) α ≤ CN α ( ε − ε , so it follows that Z R n | φ δ (cid:0) t − √ H (cid:1) (1 + H ) α/ g ( x ) | (1 + | x | ) − α dx ≤ Cδ − αε ε ) Z R n | g ( x ) | dx because 1 ≤ t ≤ δ − / . This gives (3.10) with n = 1 and the proof of estimate (3.4)is completed. (cid:3) Proof of (3.5) : high frequency part. We now make use of the finite speedof propagation of the wave operator cos( t √ H ). From (2.10), it is known (see forexample [16]) that the kernel of the operator cos( t √ H ) satisfies(3.13) supp K cos( t √ H ) ⊆ D ( t ) := { ( x, y ) ∈ R n × R n : | x − y | ≤ t } , ∀ t > . For any even function F with b F ∈ L ( R ) we have F ( t − √ H ) = 12 π Z + ∞−∞ b F ( τ ) cos( τ t − √ H ) dτ. Thus from the above we have(3.14) supp K F ( t − √ H ) ⊆ D ( t − r )whenever supp b F ⊆ [ − r, r ]. This will be used in what follows.Fixing an even function ϑ ∈ C ∞ c which is identically one on {| s | ≤ } and supportedon {| s | ≤ } , let us set j = [ − log δ ] − ζ j ( s ) := ϑ (2 − j s ) and ζ j ( s ) := ϑ (2 − j s ) − ϑ (2 − j +1 s ) for j > j . Then, we clearly have1 ≡ X j ≥ j ζ j ( s ) , ∀ s > . Recalling that φ δ ( s ) = φ ( δ − (1 − s )), for j ≥ j we set φ δ,j ( s ) = 12 π Z ∞−∞ ζ j ( u ) c φ δ ( u ) cos( su ) du. (3.15)By a routine computation it can be verified that | φ δ,j ( s ) | ≤ ( C N ( j − j ) N , | s | ∈ [1 − δ, δ ] ,C N j − j (1 + 2 j | s − | ) − N , otherwise , (3.16)for any N and all j ≥ j (see also [14, page 18]). By the Fourier inversion formula,we have φ (cid:0) δ − (cid:0) − s (cid:1)(cid:1) = X j ≥ j φ δ,j ( s ) , s > . (3.17)By the finite speed propagation property (3.14), we particularly have(3.18) supp K φ δ,j ( √ H/t ) ⊆ D ( t − j +1 ) = (cid:8) ( x, y ) ∈ R n × R n : | x − y | ≤ j +1 /t (cid:9) . Now from (3.6), it follows that(3.19) | S hδ f ( x ) | ≤ X k ≥ − log √ δ Z k +2 k − (cid:12)(cid:12)(cid:12) φ (cid:0) δ − (cid:0) − t − H (cid:1)(cid:1) ϕ k ( √ H ) f ( x ) (cid:12)(cid:12)(cid:12) dtt . For k ≥ − log √ δ and j ≥ j , let us set E k,j ( t ) := D(cid:12)(cid:12)(cid:12) φ δ,j (cid:0) t − √ H (cid:1) ϕ k ( √ H ) f ( x ) (cid:12)(cid:12)(cid:12) , (1 + | x | ) − α E . Using the above inequality (3.19), (3.17), and Minkowski’s inequality, we have(3.20) Z R n | S hδ f ( x ) | (1 + | x | ) − α dx ≤ C X k ≥ − log √ δ (cid:16) X j ≥ j (cid:16) Z k +2 k − E k,j ( t ) dtt (cid:17) / (cid:17) . LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 19 In order to make use of the localization property (3.18) of the kernel, we need todecompose R n into disjoint cubes of side length 2 j − k +2 . For a given k ∈ Z , j ≥ j ,and m = ( m , · · · , m n ) ∈ Z n , let us set Q m = h j − k +2 (cid:0) m − (cid:1) , j − k +2 (cid:0) m + 12 (cid:1)(cid:17) ×· · ·× h j − k +2 (cid:0) m n − (cid:1) , j − k +2 (cid:0) m n + 12 (cid:1)(cid:17) , which are disjoint dyadic cubes centered at 2 j − k +2 m with side length 2 j − k +2 .Clearly, R n = ∪ m ∈ Z n Q m . For each m , we define g Q m by setting g Q m := [ m ′ ∈ Z n : dist ( Q m ′ ,Q m ) ≤√ n j − k +3 Q m ′ , and denote M := (cid:8) m ∈ Z n : Q ∩ g Q m = ∅ (cid:9) . For t ∈ [2 k − , k +2 ] it follows by (3.18) that χ Q m φ δ,j (cid:0) t − √ H (cid:1) χ Q m ′ = 0 if g Q m ∩ Q m ′ = ∅ for every j, k . Hence, it is clear that φ δ,j (cid:0) t − √ H (cid:1) ϕ k ( √ H ) f = X m , m ′ : dist ( Q m ,Q m ′ ) 4) such that θ ( s ) = 1 for s ∈ ( − , ψ ,δ ( s ) := θ ( δ − (1 − s )) , ψ ℓ,δ ( s ) := θ (2 − ℓ δ − (1 − s )) − θ (2 − ℓ +1 δ − (1 − s ))for all ℓ ≥ P ∞ ℓ =0 ψ ℓ,δ ( s ) and φ δ,j ( s ) = P ∞ ℓ =0 (cid:0) ψ ℓ,δ φ δ,j (cid:1) ( s ) for all s > . We put it into (3.21) to write (cid:16) Z k +2 k − E k,j ( t ) dtt (cid:17) ≤ ∞ X ℓ =0 (cid:16) X m Z k +2 k − E k,j, ℓ m ( t ) dtt (cid:17) / , (3.23)where E k,j, ℓ m ( t ) = D(cid:12)(cid:12) χ Q m (cid:0) ψ ℓ,δ φ δ,j (cid:1)(cid:0) t − √ H (cid:1) χ g Q m ϕ k ( √ H ) f ( x ) (cid:12)(cid:12) , (1 + | x | ) − α E . Recalling (3.8) and (3.9), we observe that, for every t ∈ I i , it is possible that ψ ℓ,δ ( s/t ) η i ′ ( s ) = 0 only when i − ℓ +6 ≤ i ′ ≤ i + 2 ℓ +6 . Hence,( ψ ℓ,δ φ δ,j )( t − √ H ) = i +2 ℓ +6 X i ′ = i − ℓ +6 ( ψ ℓ,δ φ δ,j )( t − √ H ) η i ′ ( √ H ) , t ∈ I i . From this and Cauchy-Schwarz’s inequality we have E k,j, ℓ m ( t ) ≤ C ℓ i +2 ℓ +6 X i ′ = i − ℓ +6 E k,j, ℓ m ,i ′ ( t )for t ∈ I i where E k,j, ℓ m ,i ′ ( t ) := D(cid:12)(cid:12) χ Q m (cid:0) ψ ℓ,δ φ δ,j (cid:1)(cid:0) t − √ H (cid:1) η i ′ ( √ H ) (cid:2) χ g Q m ϕ k ( √ H ) f (cid:3) ( x ) (cid:12)(cid:12) , (1 + | x | ) − α E . Combining this with (3.23), we get(3.24) (cid:16) Z k +2 k − E k,j ( t ) dtt (cid:17) ≤ C ∞ X ℓ =0 ℓ/ (cid:16) X m X i Z I i i +2 ℓ +6 X i ′ = i − ℓ +6 E k,j, ℓ m ,i ′ ( t ) dtt (cid:17) / . To continue, we distinguish two cases: j > k ; and j ≤ k . In the latter case theassociated cubes have side length ≤ | x | ) α behaves like aconstant on each cube Q m , so the desired estimate is easier to obtain. The firstcase is more involved and we need to distinguish several cases which we separatelyhandle.3.4.1. Case j > k . From the above inequality (3.24) we now have (cid:16) Z k +2 k − E k,j ( t ) dtt (cid:17) ≤ I ( j, k ) + I ( j, k ) + I ( j, k ) , (3.25)where I ( j, k ) := [ − log δ ] − X ℓ =0 ℓ/ (cid:16) X m ∈ M X i Z I i i +2 ℓ +6 X i ′ = i − ℓ +6 E k,j, ℓ m ,i ′ ( t ) dtt (cid:17) / , (3.26) I ( j, k ) := ∞ X ℓ =[ − log δ ] − ℓ/ (cid:16) X m ∈ M X i Z I i i +2 ℓ +6 X i ′ = i − ℓ +6 E k,j, ℓ m ,i ′ ( t ) dtt (cid:17) / , (3.27) I ( j, k ) := ∞ X ℓ =0 ℓ/ (cid:16) X m M X i Z I i i +2 ℓ +6 X i ′ = i − ℓ +6 E k,j, ℓ m ,i ′ ( t ) dtt (cid:17) / . (3.28)We first consider the estimate for I ( j, k ) which is the major one. The estimates for I ( j, k ), I ( j, k ) are to be obtained similarly but easier. In fact, concerning I ( j, k ),the weight (1 + | x | ) − α behave as if it were a constant, and the bound on I ( j, k ) ismuch smaller than what we need to show because of rapid decay of the associatedmultipliers. Estimate of the term I ( j, k ) . We claim that, for any N > I ( j, k ) ≤ C N ( j − j ) N (cid:0) δA ǫα,n ( δ ) (cid:1) / (cid:16) Z R n | ϕ k ( √ H ) f ( x ) | (1 + | x | ) − α dx (cid:17) / where A ǫα,n ( δ ) is defined in (1.12).Let us first consider the case n ≥ 2. For (3.29), it suffices to show E k,j, ℓ m ,i ′ ( t ) ≤ C N − ℓN ( j − j ) N k δ Z R n (cid:12)(cid:12) η i ′ ( √ H ) (cid:2) χ g Q m ϕ k ( √ H ) f (cid:3) ( x ) (cid:12)(cid:12) dx (3.30)for any N > t ∈ I i being fixed and i − ℓ +6 ≤ i ′ ≤ i + 2 ℓ +6 . Indeed, sincethe supports of η i are boundedly overlapping, (3.30) gives(3.31) X i Z I i i +2 ℓ +6 X i ′ = i − ℓ +6 E k,j, ℓ m ,i ′ ( t ) dtt . C N − ℓ ( N − ( j − j ) N k δ (cid:13)(cid:13) χ g Q m ϕ k ( √ H ) f (cid:13)(cid:13) . LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 21 Recalling (3.26), we take summation over ℓ and m ∈ M to get I ( j, k ) ≤ C N j α/ ( j − j )( N − α ) / k (1 − α ) / δ (cid:16) ( k − j ) α X m ∈ M (cid:13)(cid:13) χ g Q m ϕ k ( √ H ) f (cid:13)(cid:13) (cid:17) / . Since j > k and m ∈ M , we note that (1 + | x | ) α ≤ C ( j − k ) α if x ∈ Q m . It followsthat(3.32) 2 ( k − j ) α X m ∈ M (cid:13)(cid:13) χ g Q m ϕ k ( √ H ) f (cid:13)(cid:13) ≤ C Z R n | ϕ k ( √ H ) f ( x ) | (1 + | x | ) − α dx. Noting that j = [ − log δ ] − k ≥ [ − log δ ], we obtain I ( j, k ) ≤ C N ( j − j )( N − α ) / δ / − α/ (cid:18)Z R n | ϕ k ( √ H ) f ( x ) | (1 + | x | ) − α dx (cid:19) / , which clearly gives (3.29) since N > ψ ℓ,δ ⊆ (1 − ℓ +2 δ, ℓ +2 δ ), and sosupp ( ψ ℓ,δ φ δ,j ) ( · /t ) ⊂ [ t (1 − ℓ +2 δ ) , t (1 + 2 ℓ +2 δ )]. Thus, setting R = [ t (1 + 2 ℓ +2 δ )],by Lemma 2.4 we get(3.33) E k,j, ℓ m ,i ′ ( t ) ≤ R k ( ψ ℓ,δ φ δ,j ) (cid:0) R · /t (cid:1) k R , Z R n (cid:12)(cid:12)(cid:12) η i ′ ( √ H ) (cid:2) χ g Q m ϕ k ( √ H ) f (cid:3) ( x ) (cid:12)(cid:12)(cid:12) dx for 0 ≤ ℓ ≤ [ − log δ ] − 3. We note that the support of ( ψ ℓ,δ φ δ,j ) (cid:0) R · /t (cid:1) is containedin ⊂ [ R − t (1 − ℓ +2 δ ) , R − t (1 + 2 ℓ +2 δ )] and R δ ≥ 1. Thus, we get k ( ψ ℓ,δ φ δ,j ) (cid:0) (1 + 2 ℓ +2 δ ) · (cid:1) k R , ≤ C k ( ψ ℓ,δ φ δ,j ) (cid:0) (1 + 2 ℓ +2 δ ) · (cid:1) k ∞ (cid:0) ℓ +3 δ (cid:1) / . On the other hand, if ℓ ≥ 1, then ψ ℓ,δ ( s ) = 0 for s ∈ (1 − ℓ δ, ℓ δ ), whichtogether with (3.16) and j = [ − log δ ] − k ψ ℓ,δ φ δ,j (cid:0) (1 + 2 ℓ +2 δ ) · (cid:1) k L ∞ ≤ C N ( j − j ) N − ℓN , ℓ ≥ N < ∞ . Since R ∼ k , combining these two estimates with (3.33) we getthe desired (3.30).Now we prove (3.29) with n = 1. This case can be handled in the same manner asbefore, so we shall be brief. The only difference is that we use Lemma 2.6 insteadof Lemma 2.4. Indeed, by following the same argument in the above and usingLemma 2.6, we get E k,j, ℓ m ,i ′ ( t ) ≤ C N − ℓN ( j − j ) N (cid:0) δ ℓ + k (cid:1) α ǫ Z R n (cid:12)(cid:12) η i ′ ( √ H ) (cid:2) χ g Q m ϕ k ( √ H ) f (cid:3) ( x ) (cid:12)(cid:12) dx for any N > 0. Once we have the above estimate, one can deduce the estimate (3.29)without difficulty. Estimate of the term I ( j, k ). As is clear in the decomposition of E k,j , the term I ( j, k ) is a tail part and we can obtain an estimate which is stronger than we needto have. In fact, we show I ( j, k ) ≤ C N ( j − j ) N δ N (cid:16) Z R n | ϕ k ( √ H ) f ( x ) | (1 + | x | ) − α dx (cid:17) / (3.35) for any N > 0. Indeed, we clearly have E k,j, ℓ m ,i ′ ( t ) ≤ (cid:13)(cid:13)(cid:13) ( ψ ℓ,δ φ δ,j ) (cid:16) t − √ H (cid:17) η i ′ ( √ H ) (cid:2) χ g Q m ϕ k ( √ H ) f (cid:3)(cid:13)(cid:13)(cid:13) ≤ k ( ψ ℓ,δ φ δ,j ) (cid:0) · /t (cid:1) k ∞ Z R n (cid:12)(cid:12)(cid:12) η i ′ ( √ H ) (cid:2) χ g Q m ϕ k ( √ H ) f (cid:3) ( x ) (cid:12)(cid:12)(cid:12) dx. From the definition of ψ ℓ,δ and (3.16) we have k ( ψ ℓ,δ φ δ,j ) (cid:0) · /t (cid:1) k ∞ ≤ C N j − j (2 j + ℓ δ ) − N , ℓ ≥ [ − log δ ] − . Thus, it follows that E k,j, ℓ m ,i ′ ( t ) ≤ C N j − j ) (2 j + ℓ δ ) − N Z R n (cid:12)(cid:12)(cid:12) η i ′ ( √ H ) (cid:2) χ g Q m ϕ k ( √ H ) f (cid:3) ( x ) (cid:12)(cid:12)(cid:12) dx. After putting this in (3.27) we take summation over m ∈ M to obtain E ( j, k ) ≤ C N δ / − N ( j − j ) − Nj ( j − k ) α/ × ∞ X ℓ =[ − log δ ] − − ℓ ( N − (cid:16) ( k − j ) α X m ∈ M (cid:13)(cid:13) χ g Q m ϕ k ( √ H ) f (cid:13)(cid:13) (cid:17) / . As before we may use (3.32) since j > k . Since j = [ − log δ ] − k ≥ [ − log δ ], taking sum over ℓ we obtain (3.35). Estimate of the term I ( j, k ) . We now prove the estimate I ( j, k ) ≤ C N ( j − j ) N δ / (cid:18)Z R n | ϕ k ( √ H ) f ( x ) | (1 + | x | ) − α dx (cid:19) / . (3.36)We begin with making an observation that C − (1 + | j − k +2 m | ) ≤ | x | ≤ C (1 + | j − k +2 m | ) , x ∈ Q m (3.37)provided that m M . Thanks to this observation the estimates for E k,j, ℓ m ,i ′ ( t ) ismuch simpler. By (3.37) it is clear that E k,j, ℓ m ,i ′ ( t ) ≤ (1 + | x m | ) − α (cid:13)(cid:13)(cid:13) ( ψ ℓ,δ φ δ,j ) (cid:16) t − √ H (cid:17)(cid:13)(cid:13)(cid:13) → (cid:13)(cid:13)(cid:13) η i ′ ( √ H ) (cid:2) χ g Q m ϕ k ( √ H ) f (cid:3)(cid:13)(cid:13)(cid:13) . Since k ( ψ ℓ,δ φ δ,j )( t − √ H ) k → ≤ k ( ψ ℓ,δ φ δ,j ) k ∞ , it follows from (3.34) that we have E k,j, ℓ m ,i ′ ( t ) ≤ C N (1 + | x m | ) − α j − j ) N − ℓN (cid:13)(cid:13) η i ′ ( √ H ) (cid:2) χ g Q m ϕ k ( √ H ) f (cid:3)(cid:13)(cid:13) . Using this and disjointness of the spectral supports, successively, we get X m M X i Z I i i +2 ℓ +6 X i ′ = i − ℓ +6 E k,j, ℓ m ,i ′ ( t ) dtt ≤ C j − j ) N − ℓN δ X m M (1 + | x m | ) − α (cid:13)(cid:13) χ g Q m ϕ k ( √ H ) f (cid:3)(cid:13)(cid:13) ≤ C j − j ) N − ℓN δ X m Z g Q m | χ g Q m ϕ k ( √ H ) f ( x ) | (1 + | x | ) − α dx ≤ C j − j ) N − ℓN δ Z R n | ϕ k ( √ H ) f ( x ) | (1 + | x | ) − α dx. Finally, recalling (3.28) and taking sum over ℓ yields the estimate (3.36). LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 23 Therefore, recalling δ ≤ δA ǫα,n ( δ ), we combine the estimates (3.29), (3.35), and(3.36) with (3.25) to obtain(3.38) Z k +2 k − E k,j ( t ) dtt ≤ C N ( j − j ) N δA ǫα,n ( δ ) Z R n | ϕ k ( √ H ) f ( x ) | dx (1 + | x | ) α for any N > j > k .3.4.2. Case 2: j ≤ k . In this case, the side length of each Q m is less than 4. Thus,(3.37) holds for any m ∈ Z n . Thus, the same argument in the proof of (3.36) workswithout modification. Similarly as before, we get Z k +2 k − E k,j ( t ) dtt ≤ C N ( j − j ) N δ Z R n | ϕ k ( √ H ) f ( x ) | dx (1 + | x | ) α (3.39)for any N > 0, which is stronger than (3.38).3.4.3. Completion of the proof of (3.5) . Finally, we are in position to completethe proof of (3.5). By the estimates (3.38) and (3.39) we now have the estimate(3.38) for any j ≥ j and k . Putting (3.38) in the right hand side of (3.20) andthen taking sum over j , we obtain Z R n | S hδ f ( x ) | (1 + | x | ) − α dx ≤ CδA ǫα,n ( δ ) Z R n X k | ϕ k ( √ H ) f ( x ) | (1 + | x | ) − α dx. Using Proposition 2.7 we get the estimate (3.5) and this completes the proof ofLemma 3.1. Remark 3.2. Let us generalize the square functions by setting S τ f ( x ) =: (cid:16) Z τ / (cid:12)(cid:12)(cid:12) φ (cid:16) δ − (cid:16) − Ht (cid:17)(cid:17) f ( x ) (cid:12)(cid:12)(cid:12) dtt (cid:17) / , S τ f ( x ) =: (cid:16) Z ∞ τ (cid:12)(cid:12)(cid:12) φ (cid:16) δ − (cid:16) − Ht (cid:17)(cid:17) f ( x ) (cid:12)(cid:12)(cid:12) dtt (cid:17) / for τ ≫ 1. By examining the proofs in the above one can obtain the bounds on S τ and S τ in the space L ( R n , (1 + | x | ) − α ). In fact, it is not difficulty to see that, for n ≥ α > k S τ k L ( R n , (1+ | x | ) − α ) → L ( R n , (1+ | x | ) − α ) ≤ Cδτ α +12 and k S τ k L ( R n , (1+ | x | ) − α ) → L ( R n , (1+ | x | ) − α ) ≤ Cδ − α τ − α . Optimization between these two estimates gives the choice τ = δ − .4. Sharpness of summability indices In this section we show the summability index for a.e. convergence in Theorem 1.1and that for boundedness of S λ ∗ ( H ) on L ( R n , (1 + | x | ) − α ) in Theorem 1.2 aresharp up to endpoint. For the purpose we only have to prove the following twopropositions. Proposition 4.1. Le n ≥ and n/ ( n − < p < ∞ . If sup R> | S λR ( H ) f | < ∞ a . e for all f ∈ L p ( R n ) , then we have λ ≥ λ ( p ) / . Since we are assuming λ ≥ 0, Proposition 4.1 shows the summability index inTheorem 1.1 is sharp up to endpoint. Proposition 4.2. Let ≤ α < n . Suppose that sup R> (cid:13)(cid:13) S λR ( H ) (cid:13)(cid:13) L (cid:0) R n , (1+ | x | ) − α ) → L ( R n , (1+ | x | ) − α (cid:1) < ∞ . (4.1) Then, we have λ ≥ max { , α − } . This clearly implies the necessity part of Theorem 1.2 becausesup R> (cid:13)(cid:13) S λR ( H ) f (cid:13)(cid:13) L ( R n , (1+ | x | ) − α ) ≤ k sup R> | S λR ( H ) f |k L ( R n , (1+ | x | ) − α ) . Proof of Proposition 4.1. To prove Proposition 4.1 we use a consequenceof the Nikishin-Maurey theorem (see for example, [18, 22, 40]). The following canbe deduced from Proposition 1.4 and Corollary 2.7 in [22, Ch. VI]. Theorem 4.3. Let ≤ p < ∞ and ( X, µ ) be a σ -finite measure space. Suppose that { T m } m ∈ N is a sequence of linear operators continuous from L p ( X, µ ) to L ( X, µ ) ofall measurable functions on ( X, µ ) (with the topology of convergence in measure).Assume that T ∗ f = sup m | T m f | < ∞ a.e. whenever f ∈ L p ( X, µ ) , then there existsa measurable function w > a.e. such that Z { x : | T ∗ f ( x ) | >α } w ( x ) dν ( x ) ≤ C (cid:18) k f k p α (cid:19) q , α > for all f ∈ L p ( X, µ ) with q = min( p, . In the remaining part of this subsection we mainly work with radial functions. Fora given function g on [0 , ∞ ), we set g rad ( x ) := g ( | x | ). Clearly, g rad ∈ L p ( R n ) ifand only if g ∈ L p ([0 , ∞ ) , r n − dr ). For the proof of Proposition 4.1 we first showProposition 4.4. Proposition 4.4. Let ≤ p < ∞ and w be a measurable function on (0 , ∞ ) with w > almost everywhere. Suppose that sup R> sup α> α Z(cid:8) x ∈ R n : | S λR ( H ) f ( x ) | >α (cid:9) w ( | x | ) dx ! / ≤ C k f k L p ( R n ) (4.2) for all radial functions f ∈ L p ( R n ) . Then, we have λ ≥ n (1 / − /p ) / − / . Proof of Proposition 4.1. Let R m be any enumeration of the rational numbers in(0 , ∞ ). Then we have S λ ∗ ( H ) f ( x ) = sup m | S λR m ( H ) f ( x ) | , which follows since, forevery x , S λR ( H ) f ( x ) is right continuous in R , 0 ≤ R < ∞ . LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 25 We now restrict the operator S λR m ( H ) to the set of radial functions. Since S λR ( H ) f is a radial function if f is radial, we may view S λR m ( H ) as an operator from L p ([0 , ∞ ) , r n − dr ) to itself. More precisely, let us denote by T m the mapping g S λR m ( H ) g rad for g ∈ L p ([0 , ∞ ) , r n − dr ). Since sup R> | S λR ( H ) f | < ∞ a.e. for all f ∈ L p ( R n ) byassumption, we clearly havesup m | T m g ( r ) | < ∞ , a.e. r ∈ [0 , ∞ ) . for all g ∈ L p ([0 , ∞ ) , r n − dr ). Now we take ( X, µ ) = ([0 , ∞ ) , r n − dr ). Clearly,each operator T m is continuous L p ( X, µ ) to itself and so is T m from L p ( X, µ )into L ( X, µ ). Then it follows from Theorem 4.3 that there exists a weight func-tion w > g sup m | T m g | is bounded from L p ([0 , ∞ ) , r n − dr ) to L , ∞ ([0 , ∞ ) , w ( r ) r n − dr ). So, we get a weight w which satisfies (4.2) for all ra-dial functions f ∈ L p ( R n ) because S λ ∗ ( H ) f ( x ) = sup m | S λR m ( H ) f ( x ) | for any f ∈ L p ( R n ). Therefore, by Proposition 4.4, we conclude that λ ≥ n (1 / − /p ) / − / (cid:3) It remains to prove Proposition 4.4. For this we make use of estimates for theHermite and Laguerre functions. Recall that the Laguerre polynomials of type α are defined by the formula (see [50, 1.1.37]): e − x x α L αk ( x ) = 1 k ! d k dx k (cid:0) e − x x k + α (cid:1) , α > − . Define(4.3) L αk ( x ) = (cid:18) Γ( k + 1)Γ( k + α + 1) (cid:19) e − x x α L αk ( x ) . The functions { L αk } form an orthonormal family in L ( R + , dx ) where R + = (0 , ∞ ).Recall that P k is the Hermite spectral projection operator defined by (1.6) and set L nk ( r ) := L n/ − k ( r ) e − r . Lemma 4.5. [50, Corollary 3.4.1] If f ( x ) = f ( | x | ) on R n , then P k +1 ( f ) = 0 and P k ( f ) = R n/ − k ( f ) L nk ( r ) , where R n/ − k ( f ) = 2Γ( k + 1)Γ( k + n/ Z ∞ f ( r ) L nk ( r ) r n − dr. Lemma 4.6. [50, (i) in Lemma 1.5.4] Let α + β > − , α > − /q, and ≤ q ≤ .Then, if k is large enough, for β < /q − / we have k L α + βk ( x ) x − β/ k L q ( R + ) ∼ k q − − β . Though uniform boundedness of S λR ( H ) remains open, S λR ( H ) is clearly bounded on L p sincethere are finitely many Hermite functions appearing in S λR ( H ). Lemma 4.7. Let w be a measurable function on (0 , ∞ ) with w > almost every-where. If k ∈ N is large enough, we have sup β> β (cid:16) Z(cid:8) x ∈ R n : | L nk ( | x | ) | >β (cid:9) w ( | x | ) dx (cid:17) / ≥ C D k k − / (4.4) with a constant C > independent of k where D k = (Γ( k + n/ / Γ( k + 1)) / . To prove Lemma 4.7, we make use of the following asymptotic property of theLaguerre functions (see [39, 7.4, p.453]):(4.5) L αk ( r ) = (2 /π ) / ( νr ) / (cid:16) cos (cid:16) ( νr ) / − απ − π (cid:17) + O ( ν − / r − / ) (cid:17) , where ν = 4 k + 2 α + 2 and 1 /ν ≤ r ≤ Proof. Let us set E ( k ) := ( r ∈ (cid:2) , (cid:3) : (cid:12)(cid:12)(cid:12) cos (cid:16) √ νr − απ − π (cid:17)(cid:12)(cid:12)(cid:12) ≥ √ ) . We claim that there exists a constant C ∗ > 0, independent of k , such that | E ( k ) | ≥ C ∗ . (4.6)Assuming this for the moment, we proceed to show (4.4). By (4.3) and (4.5) wehave | L nk ( r ) | = | L n/ − k ( r ) r − ( n/ − D k | ≥ CD k k − / for all r ∈ E ( k ). On the other hand, since w ( r ) > r ∈ [1 / , F of [1 / , 1] and a constant c > | F | > / − C ∗ / w ( r ) ≥ c for all r ∈ F . Then, we have | E ( k ) ∩ F | ≥ | E ( k ) | + | F | − | [1 / , | ≥ C ∗ / β> β Z(cid:8) x ∈ R n : | L nk ( | x | ) | >β (cid:9) w ( | x | ) dx ≥ sup β> β Z(cid:8) E ( k ) ∩ F : | L nk ( r ) | >β (cid:9) w ( r ) r n − dr. Since w ( r ) ≥ c for r ∈ F , the left hand side of the above is bounded below by c (1 / n − sup β> β Z(cid:8) E ( k ) ∩ F : | L nk ( r ) | >β (cid:9) dr. Particularly, taking β = CD k k − / / | L nk ( r ) | ≥ CD k k − / for r ∈ E ( k ) and | E ( k ) ∩ F | ≥ C ∗ / 2, we seesup β> β Z(cid:8) x ∈ R n : | L nk ( | x | ) | >β (cid:9) w ( | x | ) dx ≥ c (1 / n − ( CD k k − / / | E ( k ) ∩ F |≥ c (1 / n +2 C C ∗ D k k − / , which implies that (4.4) holds for C = c (1 / n +2 C C ∗ .It now remains to prove (4.6), which is rather obvious. However, we include a prooffor the convenience of the reader. Note that if there exists m ∈ N such that2 mπ − π ≤ ν / r − απ − π ≤ mπ + π , LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 27 that is, ν − (2 mπ + απ/ ≤ r ≤ ν − (2 mπ +( α +1) π/ , then cos( ν / r − απ − π ) ≥√ / 2. Then there are at least [ √ ν π ] intervals ν − [(2 mπ + απ/ , (2 mπ + απ/ π ]and so [ √ ν π ] intervals ν − [(2 mπ + απ/ , (2 mπ + ( α + 1) π/ / , | E ( k ) | ≥ h √ ν π i π √ v ≥ / . So, (4.6) holds for C ∗ = 1 / 16. This completes the proof of Lemma 4.7. (cid:3) Lemma 4.8. Let ≤ q ≤ . Then we have the estimate k L nk k L q ([0 , ∞ ) , r n − dr ) ∼ D k k n (1 /q − / / . (4.7) Proof. By (4.3) we note that | L nk ( r ) | = D k | L n/ − k ( r ) r − ( n/ − | . We take α =2( n/ − /q and β = 2(1 / − /q )( n/ − 1) in Lemma 4.6 to obtain Z ∞ | L nk ( r ) | q r n − dr = D qk Z ∞ | L n/ − k ( r ) r − ( n/ − | q r n − dr ∼ D pk k nq ( q − ) . The proof of Lemma 4.8 is complete. (cid:3) In order to prove Proposition 4.4, we use the distributions χ ν − (see [28]) which isdefined by χ ν − = x ν − Γ( ν + 1) , Re ν > − , (4.8)where Γ is the Gamma function and x − = | x | if x ≤ x − = 0 if x > 0. ForRe ν > − 1, the distribution χ ν − is clearly well defined. see [21, p. 308] or [11, 20].For Re ν ≤ − χ ν − can extended by analytic continuation (see, for example, [28, ChIII, Section 3.2]). For compactly supported function F such that supp F ⊂ [0 , ∞ ),the Weyl fractional derivative of F of order ν is given by the formula(4.9) F ( ν ) = F ∗ χ − ν − − , ν ∈ C . Since F = F ( ν ) ∗ χ ν − − , we may write F ( H ) = ν ) R ∞ F ( ν ) ( t )( t − H ) ν − dt if F hascompact support in [0 , ∞ ). Thus, it follows that, for every ν ≥ , (4.10) F ( H ) = 1Γ( ν ) Z ∞ F ( ν ) ( R ) R ν − S ν − √ R ( H ) dR for all F compactly supported in [0 , ∞ ).Now we are ready to prove Proposition 4.4. Proof of Proposition 4.4. Let us set e w ( x ) = w ( | x | ). Using (4.10), we see that F ( H ) f ( x ) is qual to 1Γ( λ + 1) Z ∞ F ( λ +1) ( R ) R λ S λ √ R ( H ) f ( x ) dR. Since L , ∞ is normable, by Minkowski’s inequality and the assumption (4.2) wehave k F ( H ) f k L , ∞ ( e wdx, R n ) ≤ C sup R> k S λR ( H ) f k L , ∞ ( e wdx, R n ) Z ∞ | F ( λ +1) ( s ) | s λ ds ≤ C k f k L p ( R n ) Z ∞ | F ( λ +1) ( s ) | s λ ds for F compactly supported in [0 , ∞ ). Let η be a non-negative smooth function suchthat η (0) = 1 and supp η ⊂ [ − , F = η ( · − k ) in the above estimate,we get(4.11) k χ [2 k, k +1) ( H ) f k L , ∞ ( e wdx, R n ) ≤ Ck λ k f k L p ( R n ) because R ∞ | η ( λ +1) ( s − k ) | s λ ds ∼ k λ , and η ( H − k ) f = χ [2 k, k +1) ( H ) f = P k f. Now, let us set f k ( r ) := sign( L nk ( r )) | L nk ( r ) | / ( p − . Then, from Lemma 4.5 and Lemma 4.7, it follows that we obtain k χ [2 k, k +1) ( H ) f k ( | · | ) k L , ∞ ( e wdx, R n ) = R n/ − k ( f k ) k L nk ( | · | ) k L , ∞ ( e wdx, R n ) ≥ CD − k k − / Z ∞ f k ( r ) L nk ( r ) r n − dr. By our choice of f k it is clear that R ∞ f k ( r ) L nk ( r ) r n − dr = k L nk k p ′ L p ′ ([0 , ∞ ) , r n − dr ) = k f k k L p ([0 , ∞ ) , r n − dr ) k L nk k L p ′ ([0 , ∞ ) , r n − dr ) . Thus, using Lemma 4.8, we get k χ [2 k, k +1) ( H ) f k ( | · | ) k L , ∞ ( e wdx, R n ) ≥ Ck n (1 / − /p ) / − / k f k ( | · | ) k L p ( R n ) . Then we combine this with (4.11) where we take f = f k ( | · | ) to obtain k n (1 / − /p ) / − / k f k ( | · | ) k L p ( R n ) ≤ Ck λ k f k ( | · | ) k L p ( R n ) (4.12)with C independent of k . Obviously 0 < k f k k pL p ( R n ) < ∞ . Thus, (4.12) implies k n (1 / − /p ) / − / ≤ Ck λ with C independent of k . Letting k tend to infinity, weget λ ≥ n (1 / − /p ) / − / (cid:3) Proof of Proposition 4.2. The proof of Proposition 4.2 is based on thefollowing weighted estimates of the normalized Hermite functions. Lemma 4.9. Let α ≥ . Then, if k ∈ N is large enough, we have Z ∞−∞ h k ( x )(1+ | x | ) α dx ≥ Ck α/ , (4.13) Z ∞−∞ h k ( x )(1 + | x | ) − α dx ≥ C max { k − α/ , k − / } . (4.14)To prove the lower bounds (4.13) and (4.14), we make use of the following asymp-totic property of the Hermite function (see [50, 1.5.1, p. 26]):(4.15) h k ( x ) = (cid:0) π (cid:1) (cid:0) N − x (cid:1) − cos (cid:18) N (2 θ − sin θ ) − π (cid:19) + O (cid:16) N ( N − x ) − (cid:17) , where N = 2 k + 1, 0 ≤ x ≤ N / − N − / and θ = arccos( xN − / ). LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 29 Proof. We begin with showing that there exists a constant C > N, | E ( N ) | ≥ C √ N , (4.16)where E ( N ) := ( x ∈ " √ N , √ N √ : cos (cid:18) N (2 θ − sin θ ) − π (cid:19) ≥ √ ) , and θ = arccos( xN − / ). For (4.16), it is enough to show(4.17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( t ∈ (cid:20) , √ (cid:21) : cos N (2˜ θ − sin ˜ θ ) − π ! ≥ √ )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C, ˜ θ = arccos( t )with C independent of N , which is equivalent to (4.16) as is easy to see by change ofvariables. In order to show (4.17), we make change of variables y = 2˜ θ − sin ˜ θ . Thecondition t ∈ [1 / , / √ 2] implies that ˜ θ ∈ [ π/ , π/ 3] and y ∈ [ π/ − √ / , π/ −√ / − √ < dydt = − − t √ − t < − −√ for t ∈ [1 / , / √ C > N suchthat (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( y ∈ " π − √ , π − √ : cos (cid:18) N y − π (cid:19) ≥ √ )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C, but this is clear from an elementary computation.Once we have (4.16), the desired estimate (4.13) follows because h k ( x ) ≥ CN − / , x ∈ E by (4.15). Clearly, we also have the following estimate R ∞−∞ h k ( x )(1 + | x | ) − α dx ≥ Ck − α/ . To complete the proof it remains to show Z ∞−∞ h k ( x )(1 + | x | ) − α dx ≥ Ck − / . In the similar manner as before it is easy to show (see also [4, Lemma 3.4]) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( x ∈ [0 , 1] : cos (cid:18) N (2 θ − sin θ ) − π (cid:19) ≥ √ )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C with C independent of N . Combining this with (4.15) we get R h k ( x ) dx ≥ CN − / ≥ Ck − / and hence the desired estimate. (cid:3) Lemma 4.10. Let α ≥ . Then, for all f ∈ L , we have the estimate k α/ k χ [ k,k +1) ( H ) f k ≤ C k χ [ k,k +1) ( H ) f k L ( R n , (1+ | x | ) α ) . (4.18) Proof. As in the proof of Lemma 1.5, we decompose f = P nj =1 f j such that f , . . . , f n are orthogonal to each other and, for 1 ≤ i ≤ n , µ i ≥ | µ | /n when-ever h f i , Φ µ i 6 = 0 and, additionally, the supports of the maps µ → h f i , Φ µ i aremutually disjoint. So, we have k χ [ k,k +1) ( H ) f k = n X j =1 k χ [ k,k +1) ( H ) f j k , and there is a j ∈ { , · · · , n } such that k χ [ k,k +1) ( H ) f j k ≥ n − k χ [ k,k +1) ( H ) f k .Thus, it is sufficient for (4.18) to show that(4.19) k α/ k χ [ k,k +1) ( H ) f j k ≤ C k χ [ k,k +1) ( H ) f k L ( R n , (1+ | x | ) α ) . Without loss of generality, we may assume j = 1.We proceed to show (4.19) for j = 1. Since (1 + | x | ) α ≥ (1 + | x j | ) α , we have k χ [ k,k +1) ( H ) f k L ( R n , (1+ | x | ) α ) ≥ Z R n X | µ | + n = k X | ν | + n = k c ( µ ) c ( ν )Φ µ ( x )Φ ν ( x )(1+ | x | ) α dx, where c ( µ ) = h f, Φ µ i . Clearly the right hand side of the above is equal to X | µ | + n = k X | ν | + n = k c ( µ ) c ( ν ) Z ∞−∞ h µ ( x ) h ν ( x )(1 + | x | ) α dx n Y i =2 h h µ i , h ν i i . By orthonormality of the Hermite functions and the relation 2 | µ | + n = k = 2 | ν | + n this is again identical to X | µ | + n = k | c ( µ ) | Z ∞−∞ h µ ( x )(1 + | x | ) α dx . Therefore, using (4.13) in Lemma 4.9, we have k χ [ k,k +1) ( H ) f k L ( R n , (1+ | x | ) α ) ≥ X | µ | + n = k | c ( µ ) | µ α/ . This yields the desired estimate (4.19) for j = 1 because µ ≥ | µ | /n whenever h f i , Φ µ i 6 = 0. Indeed, it follows from the construction of f i and c ( µ, i ) that X | µ | + n = k | c ( µ ) | µ α/ ≥ X | µ | + n = k |h f , Φ µ i| µ α/ ∼ k α/ k χ [ k,k +1) ( H ) f k . Therefore, we get (4.19). (cid:3) Proof of Proposition 4.2. Since we are assuming that (4.1) holds, by duality wehave the equivalent estimatesup R> (cid:13)(cid:13) S λR ( H ) (cid:13)(cid:13) L ( R n , (1+ | x | ) α ) → L ( R n , (1+ | x | ) α ) < ∞ . We combine this and (4.10) with ν = λ + 1 to obtain k F ( H ) f k L ( R n , (1+ | x | ) α ) ≤ C sup R> k S λR ( H ) f k L ( R n , (1+ | x | ) α ) Z ∞ | F ( λ +1) ( s ) | s λ ds ≤ C k f k L ( R n , (1+ | x | ) α ) Z ∞ | F ( λ +1) ( s ) | s λ ds for F compactly supported in supp F ⊂ [0 , ∞ ). Similarly as before, we take F ( t ) = η ( t − k ) in the above where η is a non-negative smooth function with η (0) = 1and supp η ⊂ [ − , R ∞ | η ( λ +1) ( s − k ) | s λ ds ∼ k λ and η ( H − k ) = χ [ k,k +1) ( H ) , it follows that k χ [ k,k +1) ( H ) f k L ( R n , (1+ | x | ) α ) ≤ Ck λ k f k L ( R n , (1+ | x | ) α ) . (4.20) LMOST EVERYWHERE CONVERGENCE OF BOCHNER-RIESZ MEANS 31 We now consider specific functions g k , G k which are given by g k ( x ) = h k ( x ) h ( x ) · · · h ( x n ) , G k ( x ) = g k ( x )(1 + | x | ) − α , and claim that(4.21) (cid:13)(cid:13) χ [ k,k +1) ( H ) G k (cid:13)(cid:13) L ( R n , (1+ | x | ) α ) ≤ Ck λ min (cid:8) k α/ , k / (cid:9) k χ [ k,k +1) ( H ) G k k with C independent of k . Indeed, since k g k k = 1, we have k χ [ k,k +1) ( H ) G k k ≥h χ [ k,k +1) ( H ) G k , g k i = h G k , χ [ k,k +1) ( H ) g k i . Thus, noting that χ [ k,k +1) ( H ) g k = g k from our choice of g k and g k ( x )(1 + | x | ) − α/ = G k ( x )(1 + | x | ) α/ , we get k χ [ k,k +1) ( H ) G k k ≥ h G k , g k i = k (1 + | x | ) α/ G k k k (1 + | x | ) − α/ g k k . Since k g k k L ( R n , (1+ | x | ) − α ) ≥ R ∞−∞ | h ˜ k ( x ) | (1+ | x | ) − α dx (cid:0) R | h ( t ) | dt (cid:1) n − , by theestimate (4.14) it follows that k g k k L ( R n , (1+ | x | ) − α ) ≥ C max { k − α/ , k − / } . Combining this with the above inequality yields k χ [ k,k +1) ( H ) G k k ≥ C max { k − α/ , k − / }k G k k L ( R n , (1+ | x | ) α ) . We also have k λ k G k k L ( R n , (1+ | x | ) α ) ≥ C k χ [ k,k +1) ( H ) G k k L ( R n , (1+ | x | ) α ) using theestimate (4.20). Thus we have the estimate (4.21).We apply the estimate (4.18) to the function G k and combine the consequent esti-mate with (4.21) to get k α/ k χ [ k,k +1) ( H ) G k k ≤ Ck λ min (cid:8) k α/ , k / (cid:9) k χ [ k,k +1) ( H ) G k k (4.22)with C independent of k . Since h G k , Φ µ i = R R n | h k ( x ) h ( x ) · · · h ( x n ) | (1 + | x | ) − α dx = 0 for µ = ( k, , . . . , k χ [ k,k +1) ( H ) G k k = 0. Thus,(4.22) implies k α/ ≤ Ck λ min (cid:8) k α/ , k / (cid:9) with C independent of k . Letting k →∞ gives λ ≥ max (cid:8) ( α − / , (cid:9) as desired. (cid:3) Remark 4.11. Proposition 4.4 can be used to give another proof of Proposition4.2 provided that 1 < α < n . Indeed, to the contrary, suppose that (4.1) holdswith some 1 < α < n and λ < ( α − / 4. Now, for given 1 < α < n and λ < ( α − / 4, we can choose a p such that p > n/ ( n − α > n (1 − /p ), and( α − / > ( n (1 − /p ) − / > λ . By our choice of p and H¨older’s inequalitywe have that f ∈ L ( R n , (1 + | x | ) − α ) if f ∈ L p . From this, (4.1) implies that (4.2)holds for all f ∈ L p and w ( x ) = (1 + | x | ) − α . Applying Proposition 4.4, we get λ ≥ n (1 / − /p ) / − / , which is a contradiction. Acknowledgments. P. Chen was supported by NNSF of China 11501583. X.T.Duong was supported by the Australian Research Council (ARC) through the re-search grant DP190100970. D. He was supported by NNSF of China (No. 11701583).S. Lee was supported by NRF (Republic of Korea) grant No. NRF2018R1A2B2006298.L. Yan was supported by the NNSF of China, Grant No. 11521101 and 11871480,and by the Australian Research Council (ARC) through the research grant DP190100970.P. Chen and L. 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