1. No category

Spectra of Bochner-Riesz means on Lp
Yang Chen
College of Mathematics and Computer Science
Hunan Normal University
Hunan 410081, P. R. China
E-mail: yang [email protected]
Qiquan Fang
School of Science
Zhejiang University of Science and Technology
Hangzhou, Zhejiang 310023, P. R. China
E-mail: [email protected]
Qiyu Sun
Department of Mathematics
University of Central Florida
Orlando, FL 32816, USA
E-mail: [email protected]
April 23, 2015
Abstract
The Bochner-Riesz mean Bδ , δ > 0, is shown to have the unit
interval [0, 1] as its spectrum on Lp when it is bounded on Lp .
1
Introduction and Main Results
Define Fourier transform fˆ of an integrable function f by
Z
ˆ
f (ξ) :=
f (x)e−ixξ dx
Rd
2010 Mathematics Subject Classification: Primary 47L80; Secondary 42B20.
Key words and phrases: Bochner-Riesz means, multiplier, spectra, spectral invariance, Wiener’s lemma, modulation-invariant space.
1
2
Y. Chen, Q. Fang and Q. Sun
and extend its definition to all tempered distributions as usual. Consider
Bochner-Riesz means Bδ , δ > 0, on Rd ,
2 δ ˆ
d
B
δ f (ξ) := (1 − |ξ| )+ f (ξ),
where t+ = max(t, 0), t ∈ R [Bo, Gr, St1, SW]. A conjecture is that the
Bochner-Riesz mean Bδ is bounded on Lp , the space of all p-integrable
functions on Rd , if and only if
1 1 1
(1.1)
δ > d − −
, 1 ≤ p < ∞.
p 2
2 +
The requirement (1.1) on the index δ is necessary for Lp boundedness of
the Bochner-Riesz mean Bδ [H]. The sufficiency is completely solved only
for dimension two [CP] and it is still open for high dimensions, see [Bou, L,
St2, TV1, TV2, TVV, Wo] and references therein.
Denote the identity operator by I. Consider spectra
σp (Bδ ) := C\{z ∈ C, zI − Bδ has bounded inverse on Lp }, 1 ≤ p < ∞,
of Bochner-Riesz means Bδ , δ > 0, on Lp . For p = 2,
σ2 (Bδ ) = closure of {(1 − |ξ|2 )δ+ , ξ ∈ Rd } = [0, 1],
as Bochner-Riesz means Bδ are multiplier operators with symbols (1−|ξ|2 )δ+ .
For p 6= 2, we have the following result.
Theorem 1.1. Let δ > 0 and 1 ≤ p < ∞. If the Bochner-Riesz mean Bδ
is bounded on Lp , then its spectrum σp (Bδ ) on Lp is the unit interval [0, 1].
From Theorem 1.1 we see that spectra of Bochner-Riesz means on different Lp spaces are the same.
The above spectral invariance on different Lp spaces holds for any multiplier operator Tm with its bounded symbol m satisfying the following
hypothesis,
(1.2)
|ξ|k |∇k m(ξ)| ∈ L∞ , 0 ≤ k ≤ d/2 + 1,
in the classical Mikhlin multiplier theorem, because in this case,
σ2 (Tm ) = closure of {m(ξ), ξ ∈ Rd },
and for any λ 6∈ σ2 (Tm ), the inverse of λI − Tm is a multiplier operator with
symbol (λ − m(ξ))−1 satisfying (1.2) too. Based on the above observations,
Bochner-Riesz means
3
we propose the following problem for multiplier operators: Under what conditions on symbol m does the corresponding multiplier operator Tm have its
spectrum on Lp independent on 1 ≤ p < ∞.
Spectral invariance for different function spaces is closely related to algebra of singular integral operators [CZ, FS, K, Su2] and Wiener’s lemma
for infinite matrices [GL, Su1, Su3]. It has been established for singular integral operators with kernels being H¨older continuous and having certain
off-diagonal decay [Ba, FS, FSS, ShS, Su2], but it is not well studied yet for
Calderon-Zygmund operators, oscillatory integrals, and many other linear
operators in Fourier analysis.
In this paper, we denote by S and D the space of Schwartz functions
and compactly supported C ∞ functions respectively, and we use the capital letter C to denote an absolute constant that may be different at each
occurrence.
2
Proofs
The proof of Theorem 1.1 reduces to showing
σp (Bδ ) ⊂ [0, 1]
(2.1)
and
[0, 1] ⊂ σp (Bδ ).
(2.2)
Denote by k · kp the norm on Lp , 1 ≤ p ≤ ∞. Given nonnegative integers
α0 and β0 , let Sα0 ,β0 contain all functions f with
kf kSα0 ,β0 :=
X
kxα ∂ β f (x)k∞ < ∞.
|α|≤α0 ,|β|≤β0
The inclusion (2.1) follows from the following theorem.
Theorem 2.1. Let B be a Banach space of tempered distributions with S
being dense in B. Assume that there exist nonnegative integers α0 and β0
such that any convolution operator with kernel K ∈ Sα0 ,β0 is bounded on B,
(2.3)
kK ∗ f kB ≤ CkKkSα0 ,β0 kf kB for all f ∈ B.
If the Bochner-Riesz mean Bδ is bounded on B, then (λI − Bδ )−1 is bounded
on B for any λ ∈ C\[0, 1].
4
Y. Chen, Q. Fang and Q. Sun
Proof. Take λ ∈ C\[0, 1] and r0 ∈ (0, min(|λ|1/δ , 1)/2). Let ψ1 and ψ2 ∈ D
satisfy
ψ1 (ξ) = 1 when |ξ| ≤ 1 − r0 , ψ1 (ξ) = 0 when |ξ| ≥ 1 − r0 /2;
and
ψ2 (ξ) = 1 − ψ1 (x) if |ξ| ≤ 1 + r0 /2, ψ2 (ξ) = 0 if |ξ| > 1 + r0 .
Define m(ξ) := (λ − (1 − |ξ|2 )δ+ )−1 , m1 (ξ) := m(ξ)ψ1 (ξ) and m2 (ξ) :=
m(ξ)ψ2 (ξ). Then m(ξ) is the symbol of the multiplier operator (λI − Bδ )−1
and
m(ξ) = m1 (ξ) + m2 (ξ) + λ−1 (1 − ψ1 (ξ) − ψ2 (ξ)).
As m1 , ψ1 , ψ2 ∈ D, multiplier operators with symbols m1 and ψ1 + ψ2 are
bounded on B by (2.3). Therefore the proof reduces to establishing the
boundedness of the multiplier operator with symbol m2 ,
k(m2 fˆ)∨ kB ≤ Ckf kB , f ∈ B.
(2.4)
where f ∨ is the inverse Fourier transform of f .
Take an integer N0 > α0 /δ. Write
m2 (ξ) = λ−1
N0
X
n=0
+
∞
X
(λ−1 )n ((1 − |ξ|2 )δ+ )n ψ2 (ξ) =: m21 (ξ) + m22 (ξ),
n=N0 +1
and denote multiplier operators with symbols m21 and m22 by T21 and T22
respectively. Observe that
−1
T21 = λ Ψ2 +
N0
X
λ−n−1 (Bδ )n Ψ2 ,
n=1
where Ψ2 is the multiplier operator with symbol ψ2 . Then T21 is bounded
on B by (2.3) and boundedness assumption for the Bochner-Riesz mean Bδ ,
(2.5)
kT21 f kB ≤ Ckf kB
for all f ∈ B.
Recall that ψ2 ∈ D is supported on {ξ, 1 − r0 ≤ |ξ| ≤ 1 + r0 }. Then the
inverse Fourier transform Kn of (1 − |ξ|2 )nδ
+ ψ2 (ξ) satisfies
kKn kSα0 ,β0 ≤ Cnα0 (2r0 )nδ , n ≥ N0 + 1.
Therefore the convolution kernel
K(x) := λ
−1
∞
X
n=N0 +1
λ−n Kn (x)
Bochner-Riesz means
5
of T22 belongs to Sα0 ,β0 . This together with (2.3) proves
kT22 f kB ≤ Ckf kB
(2.6)
for all f ∈ B.
Combining (2.5) and (2.6) proves (2.4). This completes the proof.
ˆ
Let f and K be Schwartz functions with f (0) = 1 and K(0)
= 0, and
−d
set fN (x) = N f (x/N ), N ≥ 1. Then for any positive integer α ≥ d + 1
there exists a constant Cα such that
Z
Z
+
|K(x − y)| |fN (y) − fN (x)|dy
|K ∗ fN (x)| ≤
√
√
|x−y|≤ N
|x−y|> N
Z
≤
|K(x − y)| |fN (y)|dy + Cα N −d−1/2 (1 + |x/N |)−α
√
|x−y|> N
Z
−1/2
(2.7)
(1 + |x − y|)−α |fN (y)|dy + N −d (1 + |x/N |)−α .
≤ Cα N
Rd
This implies that
kK ∗ fN kp
= 0, 1 ≤ p < ∞.
N →∞
kfN kp
lim
Then the inclusion (2.2) holds by the following theorem.
Theorem 2.2. Let B be a Banach space of tempered distributions with S
being dense in B. Assume that for any ξ0 ∈ Rd there exists ϕ0 ∈ D such
that ϕ
b0 (0) = 1 and
∨
k(mfd
N,ξ0 ) kB
lim
=0
N →∞
kfN,ξ0 kB
(2.8)
for all Schwartz functions m with m(ξ0 ) = 0, where fd
N,ξ0 (ξ) = ϕ0 (N (ξ−ξ0 )).
If the Bochner-Riesz mean Bδ is bounded on B, then
(2.9)
inf
f 6=0
k(λI − Bδ )f kB
=0
kf kB
for all λ ∈ [0, 1].
Proof. The infimum in (2.9) is obvious for λ = 0. So we assume that λ ∈
(0, 1] from now on. Select ξ0 ∈ Rd so that (1 − |ξ0 |2 )δ+ = λ. Then for
sufficiently large N ≥ 1,
(2.10)
∨
(λI − Bδ )fN,ξ0 = (mξ0 f[
N,ξ0 ) ,
where mξ0 (ξ) = (λ − (1 − |ξ|2 )δ+ )ψ(ξ − ξ0 ) and ψ ∈ D is so chosen that
ψ(ξ) = 1 for |ξ| ≤ (1 − |ξ0 |)/2 and ψ(ξ) = 0 for |ξ| ≥ 1 − |ξ0 |. Observe that
6
Y. Chen, Q. Fang and Q. Sun
mξ0 ∈ D satisfies mξ0 (ξ0 ) = 0. This together with (2.8) and (2.10) proves
that
k(λI − Bδ )fN,ξ0 kB
lim
= 0.
N →∞
kfN,ξ0 kB
Hence (2.9) is proved for λ ∈ (0, 1].
Remark 2.3. For ξ ∈ Rd , define modulation operator Mξ by
Mξ f (x) = eixξ f (x).
We say that a Banach space B is modulation-invariant if for any ξ ∈ Rd
there exists a positive constant Cξ such that
kMξ f kB ≤ Cξ kf kB ,
f ∈ B.
Such a Banach space with modulation bound Cξ being dominated by a
polynomial of ξ was introduced in [Ta] to study oscillatory integrals and
Bochner-Riesz means. Modulation-invariant Banach spaces include weighted
α
α
, Herz spaces Kpα,q ,
, Besov spaces Bp,q
Lp spaces, Triebel-Lizorkin spaces Fp,q
and modulation spaces M p,q , where α ∈ R and 1 ≤ p, q < ∞ [G, Gr, LYH,
T]. For functions fN,ξ0 , N ≥ 1, in Theorem 2.2,
fN,ξ0 (x) = eixξ0 (ϕ0 (N ·))∨ (x)
and
(mfN,ξ0 )∨ (x) = eixξ0 (mξ0 ϕ0 (N ·))∨ (x),
where mξ0 (ξ) = m(ξ + ξ0 ) satisfies mξ0 (0) = 0. Then for a modulationinvariant space B, the limit (2.8) holds for any ξ0 ∈ Rd if and only if it is
true for ξ0 = 0. Therefore we obtain the following result from Theorem 2.2.
Corollary 2.4. Let B be a modulation-invariant Banach space of tempered
distributions with S being dense in B. Assume that there exists ϕ0 ∈ D
such that ϕˆ0 (0) = 1 and limN →∞ k(mϕ0 (N ·))∨ kB /k(ϕ0 (N ·))∨ kB = 0 for all
Schwartz functions m(ξ) with m(0) = 0. If the Bochner-Riesz mean Bδ is
bounded on B, then its spectrum on B contains the unit interval [0, 1].
3
Remarks
In this section, we extend conclusions in Theorem 1.1 to weighted Lp spaces,
Triebel-Lizorkin spaces, Besov spaces, and Herz spaces.
Bochner-Riesz means
3.1
7
Spectra on weighted Lp spaces
Let 1 ≤ p < ∞ and Q contain all cubes Q ⊂ Rd . A positive function w is
said to be a Muckenhoupt Ap -weight if
1 Z
1 Z
p−1
−1/(p−1)
w(x)dx
w(x)
dx
≤ C, Q ∈ Q
|Q| Q
|Q| Q
for 1 < p < ∞, and
1
|Q|
Z
w(x)dx ≤ C inf w(x), Q ∈ Q
Q
x∈Q
for p = 1 [GF, M]. For δ > (d − 1)/2, convolution kernel of the BochnerRiesz mean Bδ is dominated by a multiple of (1 + |x|)−δ−(d+1)/2 and hence
it is bounded on weighted Lp space Lpw for all 1 ≤ p < ∞ and Muckenhoupt Ap -weights w. For δ = (d − 1)/2, complex interpolation method was
introduced in [SS] to establish Lpw -boundedness of Bδ for all 1 < p < ∞
and Muckenhoupt Ap -weights w. The reader may refer to [GF, LS, Lo] and
references therein for Lpw -boundedness of Bochner-Riesz means with various
weights w. In this subsection, we consider spectra of Bochner-Riesz means
on Lpw .
Theorem 3.1. Let δ > 0, 1 ≤ p < ∞, and w be a Muckenhoupt Ap -weight.
If the Bochner-Riesz mean Bδ is bounded on Lpw , then its spectrum on Lpw
is the unit interval [0, 1].
Proof. Denote the norm on Lpw by k · kp,w . By Theorems 2.1 and 2.2, and
modulation-invariance of Lpw , it suffices to prove
(3.1)
kK ∗ f kp,w ≤ CkKkSd+1,0 kf kp,w for all f ∈ Lpw ,
and
(3.2)
k(mϕ0 (N ·))∨ kp,w
=0
N →∞ k(ϕ0 (N ·))∨ kp,w
lim
for all Schwartz functions ϕ0 and m with ϕ
b0 (0) = 1 and m(0) = 0.
Observe that |K(x)| ≤ CkKkSd+1,0 (1 + |x|)−d−1 . Then (3.1) follows from
the standard argument for weighted norm inequalities [GF].
Recall that any Ap -weight is a doubling measure [GF]. This doubling
property together with (2.7) leads to
(3.3)
k(mϕ0 (N ·))∨ kpp,w ≤ CN −p/2 k(ϕ0 (N ·))∨ kpp,w + CN −(d+1/2)p w([−N, N ]d ).
8
Y. Chen, Q. Fang and Q. Sun
On the other hand, there exists 0 > 0 such that |ϕ∨0 (x)| ≥ |ϕ∨0 (0)|/2 6= 0
for all |x| ≤ 0 . This implies that
(3.4)
k(ϕ0 (N ·))∨ kpp,w ≥ CN −dp w([−0 N, 0 N ]d ).
Combining (3.3), (3.4) and the doubling property for the weight w, we
establish the limit (3.2) and complete the proof.
3.2
Spectra on Triebel-Lizorkin spaces and Besov spaces
Let φ0 and ψ ∈ S be so chosen that φb0 is supported in {ξ, |ξ| ≤ 2}, ψˆ
supported in {ξ, 1/2 ≤ |ξ| ≤ 2}, and
φb0 (ξ) +
∞
X
ˆ −l ξ) = 1, ξ ∈ Rd .
ψ(2
l=1
α
For α ∈ R and 1 ≤ p, q < ∞, let Triebel-Lizorkin space Fp,q
contain all
tempered distributions f with
α
kf kFp,q
∞
X
1/q lαq
q
2 |ψl ∗ f |
:= kφ0 ∗ f kp + < ∞,
p
l=1
α
where ψl = 2ld ψ(2l ·), l ≥ 1. Similarly, let Besov space Bp,q
be the space of
tempered distributions f with
α := kφ0 ∗ f kp +
kf kBp,q
∞
X
2lαq kψl ∗ f kqp
1/q
< ∞.
l=1
Next is our results about spectra of Bochner-Riesz means on Triebel-Lizorkin
spaces and on Besov spaces.
Theorem 3.2. Let δ > 0, α ∈ R and 1 ≤ p, q < ∞. If the Bochner-Riesz
α
α
α
mean Bδ is bounded on Fp,q
(resp. Bp,q
), then its spectrum on Fp,q
(resp. on
α
Bp,q ) is the unit interval [0, 1].
Proof. For λ 6∈ [0, 1] and δ > 0, both Bδ and (λI − Bδ )−1 − λ−1 I are
multiplier operators with compactly supported symbols. Therefore Bδ (resp.
α
(λI − Bδ )−1 ) is bounded on the Triebel-Lizorkin space Fp,q
if and only if it
α
is bounded on the Besov space Bp,q if and only if it is bounded on Lp . The
above equivalence together with Theorem 1.1 yields our desired conclusions
for spectra of Bochner-Riesz means on Triebel-Lizorkin spaces and on Besov
spaces.
Bochner-Riesz means
3.3
9
Spectra on Herz spaces
For α ∈ R and 1 ≤ p, q < ∞, let Herz space Kpα,q contain all locally pintegrable functions f with
kf k
Kpα,q
:= kf χ|·|≤1 kp +
∞
X
lαq
2
kf χ2l−1 <|·|≤2l kqp
1/q
< ∞,
l=1
where χE is the characteristic function on a set E. The boundedness of
Bochner-Riesz means on Herz spaces is well studied, see for instance [LYH,
Wa]. Following the argument used in the proof of Theorem 3.1, we have
Theorem 3.3. Let δ > 0, 1 ≤ p, q < ∞ and α > −d/p. If the BochnerRiesz mean Bδ is bounded on Kpα,q , then its spectrum on Kpα,q is the unit
interval [0, 1].
Acknowledgements
The project is partially supported by National Science Foundation (DMS1412413) and NSF of China (Grant Nos. 11426203).
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