arXiv:1702.06573v1 [math.PR] 21 Feb 2017
ON SQUARE FUNCTIONS AND FOURIER MULTIPLIERS FOR NONLOCAL
OPERATORS
RODRIGO BAÑUELOS AND DAESUNG KIM
Abstract. Using Itô’s formula for processes with jumps, we give a simple direct proof of the HardyStein identity proved in [5]. We extend the proof given in that paper to non-symmetric Lévy-Fourier
multipliers.
1. Introduction
Littlewood-Paley quadratic (square) functions have been of interest for many years with many
applications in harmonic analysis and probability. On the analysis side, these include the classical
square functions obtained from the Poisson semigroup as in [12] and more general heat semigroups
as in [11]. On the probability side, these correspond to the celebrated Burkholder-Gundy inequalities
which are the bread and butter of modern stochastic analysis. For a short review of some of this
literature, we refer the reader to [5]. In [5], the authors prove square function Lp , 1 < p < ∞, bounds
for non-local operators and then applied them to prove Lp bounds for certain Fourier multipliers that
arise from transformation of symmetric Lévy processes. The key to the proof in [5], outside of fairly
standard and well known uses of the Burkholder-Gundy inequalities and other arguments as in [12], is
a Hardy-Stein identity which is proved from properties of the semigroup. In the classical case of the
Laplacian, such Hardy-Stein identity follows from, essentially, Green’s theorem and the chain rule as
in Lemmas 1 and 2 in [12, pp. 86-87]. In the case of Brownian motion, a probabilisitic, BurkholderGundy type version of this Hardy-Stein identity, can also be proved as a simple application of Itô’s
formula. This was done in Theorem 1.1 in [3]; it is also a standard and well known argument. See for
example [9, p. 152]. As it turns out, the Itô formula proof can be adapted to processes with jumps to
prove the Hardy-Stein identity (Theorem 3.1) used in [5] for nonlocal operators. The purpose of this
note is to do exactly this and to show how to obtain a more general result on multipliers arising from
non-symmetric Lévy processes. It is also interesting to note that the needed form of the function F in
the Hardy-Stein identity is dictated by Itô’s formula itself.
Then paper is organized as follows. The Hardy-Stein identity via Itô’s formula is proved in §3
where we also note that it holds for more general martingales. In §4 we obtain, by symmetrization,
the multipliers for non-symmetric Lévy processes which extend the arguments in [5].
2. Preliminaries
The indicator function of a set A is denoted by 1A . For a, b ∈ R, we denote by a ∧ b = min{a, b}.
For a complex number ξ = x + iy, Re(ξ) denotes the real part of ξ. That is, Re(ξ) = x.
The d-dimensional stochastic process (Xt )t≥0 defined on the probability space (Ω, F , P) is called
a Lévy process if it has independent stationary increments with càdlàg paths. The celebrated LévyKhintchine theorem tells us that (Xt )t≥0 is characterized by its characteristic exponent ψ(ξ) given by
E[eiξ·Xt ] = e−tψ(ξ) , ξ ∈ Rd , t ≥ 0, and
Z
1
ψ(ξ) = ib · ξ + ξ · Aξ +
(1 − eiξ·y + iξ · y1{|y|≤1} )ν(dy),
2
Rd
2010 Mathematics Subject Classification. Primary 42B25; Secondary 60J75.
Key words and phrases. Itô’s formula, Hardy-Stein identity, Littlewood-Paley square functions, Fourier multiplier.
Authors supported in part by NSF grant #1403417-DMS; R. Bañuelos PI.
1
2
RODRIGO BAÑUELOS AND DAESUNG KIM
where b ∈ Rd , A is a positive semi-definite d × d matrix, and ν is a measure on Rd satisfying
Z
(1 ∧ |y|2 )ν(dy) < ∞.
Rd \{0}
The Lévy process (Xt )t≥0 is determined by the Lévy triplet (b, A, ν). The measure ν is called the Lévy
measure. For instance, the Brownian motion is the case b = 0, ν = 0 and A is the identity matrix. We
say that (Xt )t≥0 is a pure jump Lévy process if b = 0 and A = 0. We refer the reader to [1] for further
information.
Throughout this paper, we assume that (Xt )t≥0 is a pure jump Lévy process with Lévy measure ν
and satisfying the so called Hartman-Wintner condition
(HW)
lim
|ξ|→∞
Re(ψ(ξ))
= ∞.
log(1 + |ξ|)
In [8, Theorem 1], V. Knopova and R. L. Schilling proved that (HW) holds if and only if for all t > 0,
the transition density pt (x) exists, pt ∈ C ∞ (Rd ) ∩ L1 (Rd ), and ∇pt ∈ L1 (Rd ). Thus, our assumption
guarantees the existence of a smooth transition density for (Xt )t≥0 .
Let S(Rd ) be the Schwartz space on Rd and f ∈ S(Rd ). We define the Fourier transform and the
inverse Fourier transform of f by
Z
f (x)e−ix·ξ dx,
F (f )(ξ) = fb(ξ) :=
Rd
Z
−1
∨
−d
F (f )(x) = f (x) = (2π)
f (ξ)eiξ·x dξ.
Rd
With our definition, Parseval’s formula takes the form
Z
Z
1
(2.1)
g (ξ)dξ,
fb(ξ)b
f (x)g(x)dx =
(2π)d Rd
Rd
for f, g ∈ L2 (Rd ).
The semigroup Pt associated to (Xt )t≥0 is defined by Pt f (x) = Ex [f (Xt )]. It is easy to see that
−tψ(ξ) b
d
P
f (ξ). The infinitesimal generator of Pt is denoted by L. One can see by [10, Theorem
t f (ξ) = e
31.5] that the infinitesimal generator L can be written as
Z
(2.2)
(f (x + y) − f (x) − y · ∇f (x) · 1{|y|<1} )ν(dy).
Lf (x) =
Rd
We note that the Hartman-Wintner condition (HW) ensures Pt f ∈ Lp (Rd )∩C ∞ (Rd ) for f ∈ Lp (Rd )
and 1 ≤ p < ∞. Since the transition density pt (x) exists, one see that the semigroup Pt is a Lp contraction for 1 ≤ p ≤ ∞, which means kPt f kp ≤ kf kp for any f ∈ Lp .
Next we recall Itô’s formula for a general stochastic process with jumps. Let (Zt )t≥0 be a stochastic
process defined by
Z tZ
Z tZ
e (ds, dx),
(2.3)
Zt = Z0 + Mt + At +
G(s, x)N (ds, dx) +
H(s, x)N
0
Rd
0
Rd
where Mt is a continuous square integrable local martingale, At is a continuous adapted process of
bounded variation with A0 = 0 and G(t, x) = (G1 (t, x), · · · , Gd (t, x)), H(t, x) = (H1 (t, x), · · · , Hd (t, x))
are predictable.
Theorem 2.1 (Itô’s formula [7, p. 66]). Let (Zt )t≥0 be a stochastic process given by (2.3) and ϕ a C 2
function on Rd . Furthermore, we assume that Gi (t, x) · Hj (t, x) = 0 for all 1 ≤ i, j ≤ d and
sup sup |H(t, x)| < ∞
0≤t≤T x∈Rd
ON SQUARE FUNCTIONS AND FOURIER MULTIPLIERS FOR NONLOCAL OPERATORS
3
for all T > 0 almost surely. Then we have
Z
Z t
Z t
1 t 2
(2.4) ϕ(Zt ) − ϕ(Z0 ) =
D ϕ(Zs ) · dhM is
∇ϕ(Zs ) · dAs +
∇ϕ(Zs ) · dMs +
2 0
0
0
Z tZ
(ϕ(Zs− + G(s, x)) − ϕ(Zs− )) N (ds, dx)
+
0
Rd
Z tZ
e (ds, dx)
+
(ϕ(Zs− + H(s, x)) − ϕ(Zs− )) N
0
Rd
Z tZ
+
(ϕ(Zs− + H(s, x)) − ϕ(Zs− ) − H(s, x) · ∇ϕ(Zs− )) ν(dx)ds.
0
Rd
We note that the Lévy-Itô decomposition (see [1, Theorem 2.4.16]) tells us that a pure jump Lévy
process (Xt )t≥0 can be written as
Z tZ
Z tZ
e (ds, dx)
Xt =
xN (ds, dx) +
xN
0
|x|≥1
0
|x|<1
where N (t, B) = ♯{0 ≤ s ≤ t : ∆Xs ∈ B} is the Poisson random measure that describes the jumps
e (t, B) = N (t, B) − tν(B) is its compensator. Thus, in Itô’s formula for (Xt )t≥0 we have
of Xt and N
Z0 = 0, M = A = 0, G = x · 1{|x|≥1} and H = x · 1{|x|<1} .
3. Hardy-Stein Identity
The purpose of this section is to show the Hardy-Stein identity using Itô’s formula, Theorem 2.4.
Theorem 3.1. Let 1 < p < ∞ and F (a, b; p) = |b|p − |a|p − pa|a|p−2 (b − a). If f ∈ Lp (Rd ), then we
have
Z
Z Z ∞Z
p
|f (x)| dx =
(3.1)
F (Pt f (x), Pt f (x + y); p)ν(dy)dtdx.
Rd
Rd
Rd
0
We note that the function F (a, b; p) is the second-order Taylor reminder of the function x 7→ |x|p
on R. Similarly, we define Fε (a, b; p) by
(3.2)
p
p
Fε (a, b; p) = (b2 + ε2 ) 2 − (a2 + ε2 ) 2 − pa(a2 + ε2 )
p−2
2
2
(b − a),
p
which is also the second-order Taylor reminder of the function x 7→ (x + ε2 ) 2 on R. By the convexity,
one can easily see that F (a, b; p) ≥ 0 and Fε (a, b; p) ≥ 0 for all a, b ∈ R.
Before we present the proof of Theorem 3.1, we give the following lemmas. The first lemma gives
basic properties of F and Fε which allow us to use a limiting argument when we consider the case
1 < p < 2. The lemmas is proved in [6].
Lemma 3.2 ([6, Lemma 6, p.198]). Let p > 1, F (a, b; p) = |b|p − |a|p − pa|a|p−2 (b − a) and K(a, b; p) =
(b − a)2 (|a| ∨ |b|)p−2 . Then we have
cp K(a, b; p) ≤ F (a, b; p) ≤ Cp K(a, b; p)
for some constants cp , Cp > 0. If 1 < p < 2, then we have
1
F (a, b; p)
0 ≤ Fε (a, b; p) ≤
p−1
for all ε, a, b ∈ R.
Next lemma is an application of Itô’s formula.
Lemma 3.3. Let T > 0 be fixed, f ∈ Lp (Rd ) for 1 ≤ p < ∞, and Pt f (x) = Ex [f (Xt )]. If we define
Yt := PT −t f (Xt ), then we have
Z tZ
e (ds, dy).
(PT −s f (Xs− + y) − PT −s f (Xs− ))N
Yt = Y0 +
0
Rd
4
RODRIGO BAÑUELOS AND DAESUNG KIM
Proof. Let ϕ(s, x) = PT −s f (x) (which is C 2 by our (HW) assumption), G(s, x) = x · 1{|x|≥1} , and
H(s, x) = x · 1{|x|<1} and apply Itô’s formula to get
Z t
∂
ϕ(s, x)|x=Xs− ds
ϕ(t, Xt ) − ϕ(0, X0 ) =
0 ∂s
Z tZ
+
(ϕ(s, Xs− + y) − ϕ(s, Xs− )) N (ds, dy)
+
+
0
|y|≥1
0
|y|<1
0
|y|<1
Z tZ
Z tZ
e (ds, dy)
(ϕ(s, Xs− + y) − ϕ(s, Xs− )) N
(ϕ(s, Xs− + y) − ϕ(s, Xs− ) − y · ∇ϕ(s, Xs− )) ν(dy)ds.
∂
Since we have ∂s
ϕ(s, x) = −LPT −s f (x), the formula (2.2) yields
Z t
Z t
∂
LPT −s f (Xs− )ds
ϕ(s, x)|x=Xs− ds = −
0 ∂s
0
Z tZ
= −
ϕ(s, Xs− + y) − ϕ(s, Xs− ) − y · ∇ϕ(s, Xs− ) · 1{|y|<1} ν(dy)ds
0
Rd
Z tZ
= −
(ϕ(s, Xs− + y) − ϕ(s, Xs− ) − y · ∇ϕ(s, Xs− )) ν(dy)ds
−
0
|y|<1
0
|y|≥1
Z tZ
(ϕ(s, Xs− + y) − ϕ(s, Xs− )) ν(dy)ds.
e (ds, dy) = N (ds, dy) − ν(dy)ds that
It then follows from N
Z tZ
e (ds, dy),
(ϕ(s, Xs− + y) − ϕ(s, Xs− )) N
ϕ(t, Xt ) − ϕ(0, X0 ) =
Rd
0
as desired.
x
Lemma 3.4. The semigroup Pt defined by Pt f (x) := E [f (Xt )] is ultracontractive, meaning that for
every t > 0, there exists a constant Ct > 0 such that
1
kPt f k∞ ≤ Ctp kf kp
(3.3)
for f ∈ Lp (Rd ) and p ≥ 1 and Ct → 0 as t → ∞.
Proof. Although not explicitly written, this result follows from [8]. Since its proof is quite simple,
we present it here for completeness. Note that e−tψ(ξ) = E0 [eiξ·Xt ] = (2π)d F −1 (pt (·))(ξ). Since pt is
in L1 (Rd ), one sees that e−tψ(ξ) ∈ L∞ (Rd ). We claim that e−tψ(ξ) is in L1 (Rd ) for each t > 0. To
see this, it suffices to show e−t Re ψ(ξ) ∈ L1 (Rd ). Let f : Rd → R be a function such that Re ψ(ξ) =
log(1 + |ξ|)f (ξ). Then the Hartman-Wintner condition (HW) tells us that f (ξ) → ∞ as |ξ| → ∞. Fix
t > 0 and choose R = Rt > 0 so that tf (ξ) > d + 1 whenever |ξ| ≥ R. Let B = B(0, R) be a ball
centered at 0, of radius R. If we write
Z
Z
Z
e−t Re ψ(ξ) dξ =
e−t Re ψ(ξ) dξ +
e−t Re ψ(ξ) dξ =: I + II,
Rd
Rd \B
B
it follows from the construction of f that
Z
Z
1
1
dξ < ∞.
I=
dξ
≤
d+1
tf
(ξ)
|ξ|≥R (1 + |ξ|)
|ξ|≥R (1 + |ξ|)
Since we have
(3.4)
e
−t Re ψ(ξ)
= |e
−tψ(ξ)
Z
| = Rd
e
iξ·x
pt (x)dx ≤ 1,
ON SQUARE FUNCTIONS AND FOURIER MULTIPLIERS FOR NONLOCAL OPERATORS
5
it is easy to see II ≤ |B| < ∞, which implies that e−t Re ψ(ξ) ∈ L1 (Rd ) as desired. Since pt ∈
L1 (Rd ) ∩ L∞ (Rd ), we can apply the Fourier inversion formula to obtain
Z
1
1
−tψ(ξ)
F
(e
)
=
e−tψ(ξ) e−ix·ξ dξ.
pt (x) =
(2π)d
(2π)d Rd
Define
Ct =
1
(2π)d
Z
e−t Re ψ(ξ) dξ,
Rd
then it is obvious to see that Ct is finite for each t > 0 and |pt (x)| ≤ Ct for all x ∈ Rd . Using Jensen’s
inequality, we obtain
Z
Z
p1
1
p
|Pt f (x)| = f (y)pt (x, y)dy ≤ |f (y)| pt (x, y)dy ≤ Ctp kf kp
Rd
Rd
for any x ∈ Rd , which yields (3.3).
Now, we prove the second assertion that Ct → 0 as t → ∞. First, we note that Re ψ(ξ) is nonnegative
by (3.4) so that e−t Re ψ(ξ) tends to 0 as t → ∞ for every ξ. Since e−t Re ψ(ξ) is integrable for all t ≥ 1
and bounded by e− Re ψ(ξ) , it follows from the dominated convergence theorem that
Z
1
e−t Re ψ(ξ) dξ = 0.
lim Ct = lim
t→∞
t→∞ (2π)d Rd
Proof of Theorem 3.1. We begin with the case p ≥ 2. Define ϕ(x) = |x|p , Yt := PT −t f (Xt ) and
H(s, y) = PT −s f (Xs− + y) − PT −s f (Xs− ) for fixed T > 0. Let 0 < T0 < T , then it follows from
Lemma 3.3 that
Z tZ
e (ds, dy)
Yt = Y0 +
H(s, y)N
0
Rd
for any 0 ≤ t ≤ T0 . Note that H(s, y) is uniformly bounded in s ∈ [0, T0 ] and y ∈ Rd by Lemma 3.4 .
We can apply Itô’s formula to ϕ(Yt ) and obtain
Z tZ
e (ds, dy)
(3.5) ϕ(Yt ) − ϕ(Y0 ) =
(ϕ(Ys− + H(s, y)) − ϕ(Ys− )) N
0
Rd
Z tZ
+
(ϕ(Ys− + H(s, y)) − ϕ(Ys− ) − H(s, y) · ∇ϕ(Ys− )) ν(dy)ds,
0
Rd
for all 0 ≤ t ≤ T0 . Note that Ys− + H(s, y) = PT −s f (Xs− + y) and Ys− = PT −s f (Xs− ). If we take
expectation of both sides in (3.5), the first integral on the right hand side will be zero because it is a
martingale. We also note that
ϕ(Ys− + H(s, y)) − ϕ(Ys− ) − H(s, y) · ∇ϕ(Ys− )
=
|PT −s f (Xs− + y)|p − |PT −s f (Xs− )|p
=
−pPT −s f (Xs− )|PT −s f (Xs− )|p−2 (PT −s f (Xs− + y) − PT −s f (Xs− ))
F (PT −s f (Xs− + y), PT −s f (Xs− ); p).
By putting t = T0 and taking expectation of both sides, we have
"Z
#
T0 Z
p
x
p
x
x
(3.6) E |YT0 | − E |Y0 | = E
F (PT −s f (Xs− + y), PT −s f (Xs− ); p)ν(dy)ds .
0
Rd
6
RODRIGO BAÑUELOS AND DAESUNG KIM
R
R
Since we have Rd Ex |YT0 |p dx = kPT −T0 f kpp and Rd Ex |Y0 |p dx = kPT f kpp , taking integral of both sides
in (3.6) we see that
"Z
#
Z
T0 Z
F (PT −s f (Xs− + y), PT −s f (Xs− ); p)ν(dy)ds dx
Ex
kPT −T0 f kpp − kPT f kpp =
Rd
=
Z
Rd
=
Z
Rd
0
Z
Z
Rd 0
Z T0 Z
T0 Z
Rd
F (PT −s f (z + y), PT −s f (z); p)ps (x, z)ν(dy)dsdzdx.
Rd
F (PT −s f (z + y), PT −s f (z); p)ν(dy)dsdz.
Rd
0
Since F (a, b; p) is nonnegative and kPT −T0 f kpp → kf kpp as T0 → T , the monotone convergence theorem
yields
Z Z TZ
kf kpp − kPT f kpp =
(3.7)
F (Ps f (z + y), Ps f (z); p)ν(dy)dsdz.
Rd
0
Rd
By Lemma 3.4, one sees that PT f (x) → 0, as T → ∞ for each x ∈ Rd . This and the fact that
kPT f kp ≤ kf kp for all T > 0 gives that limT →∞ kPT f kpp = 0 and since F (a, b; p) is nonnegative, the
right hand side of (3.7) converges to the desired limit and this proves the result for p ≥ 2.
We now consider the case 1 < p < 2. Let ε > 0. If we follow the same argument as in the case
p
p > 2 with the function ϕ(x) = (|x|2 + ε2 ) 2 , we arrive at
Z Z TZ
Z p
p
Fε (Ps f (z + y), Ps f (z); p)ν(dy)dsdz,
(|f (x)|2 + ε2 ) 2 − (|PT f (x)|2 + ε2 ) 2 dx =
Rd
Rd
0
Rd
where Fε is the function in (3.2).
p
Let us look first look at the left hand side of this identity. Since the function x 7→ x 2 is p2 -Hölder
p
continuous on [0, ∞) for 1 < p < 2, we have (|f (x)|2 + ε2 ) 2 − εp ≤ Cp |f (x)|p and (|PT f (x)|2 +
p
ε2 ) 2 − εp ≤ Cp |PT f (x)|p . Thus the left hand side converges to kf kpp − kPT f kpp , as ε → 0, by the
dominated convergence theorem. On the other hand, 0 ≤ Fε (a, b; p) → F (a, b; p), as ε → 0, and
1
F (a, b; p), by Lemma 3.2. Since the integral
0 ≤ Fε (a, b; p) ≤ p−1
Z Z TZ
I(ε, T ) :=
Fε (Ps f (z + y), Ps f (z); p)ν(dy)dsdz
Rd
Rd
0
is bounded for each ε > 0, Fatou’s lemma and the dominated convergence theorem give (see [6, p.199])
that
Z Z TZ
I(ε, T ) →
F (Ps f (z + y), Ps f (z); p)ν(dy)dsdz,
Rd
Rd
0
as ε → 0. We now finish the proof by letting T → ∞.
Remark 3.5. One can easily see that the proof also holds for a general martingale. Suppose that
a predictable process H(t, x) is uniformly bounded in t ∈ [0, T ] and x ∈ Rd for each T > 0 and a
martingale Mt is given by a stochastic integral
Z tZ
e (ds, dy).
Mt = M0 +
H(s, y)N
0
Rd
Then, the same argument yields the following Hardy-Stein type identity for Mt that
Z ∞Z
E|M∞ |p − E|M0 |p =
E[F (Ms− , Ms− + H(s, y); p)]ν(dy)ds
0
when 1 < p < ∞.
Rd
ON SQUARE FUNCTIONS AND FOURIER MULTIPLIERS FOR NONLOCAL OPERATORS
7
4. Fourier multipliers and square functions
The main application of the results in [5] were to show the boundedness of the Fourier multipliers
introduced in [4] on Lp , 1 < p < ∞, without appealing to martingale transforms. Of course, a
disadvantage of these proofs is that we do not obtained the sharp bounds given in [4] which follow
from Burkholder’s sharp inequalities. In this section, we define the Fourier multiplier of the nonsymmetric Lévy process Xt . To detour the difficulties that arise from the asymmetry of ν, we rely
on the symmetrization scheme used in the earlier work [4]. Then, we introduce appropriate square
functions as in [5].
We start with the statement of the main result. Let φ : (0, ∞) × Rd → R be a bounded function and
1 < p, q < ∞ with p1 + q1 = 1. We define Λφ (f, g) for f ∈ L2 (Rd ) ∩ Lp (Rd ) and g ∈ L2 (Rd ) ∩ Lq (Rd ) by
Z Z ∞Z
Λφ (f, g) =
(Pt f (x + y) − Pt f (x))(Pt g(x + y) − Pt g(x))φ(t, y)ν(dy)dtdx.
Rd
0
Rd
d
Let m : R → C be a function. We define a Fourier multiplier operator Mm with a multiplier m by
F (Mm f )(ξ) = m(ξ)fb(ξ). By Parseval’s formula (2.1) , we have
Z
Z
1
Mm f (x)g(x)dx =
hMm f, gi =
F (Mm f )(ξ)F (g)(ξ)dξ
(2π)d Rd
Rd
Z
1
g (ξ)dξ
=
m(ξ)fb(ξ)b
(2π)d Rd
Theorem 4.1. The bilinear operator Λφ (f, g) is absolutely convergent for f ∈ L2 (Rd ) ∩ Lp (Rd ) and
g ∈ L2 (Rd )∩Lq (Rd ). Furthermore, there is a unique linear operator Sφ on Lp (Rd ) such that Λφ (f, g) =
hSφ (f ), gi and Sφ = Mmφ with the Fourier multiplier mφ given by
Z ∞Z
|eiξ·y − 1|2 e2t Re(ψ(ξ)) φ(t, y)ν(dy)dt.
mφ (ξ) =
Rd
0
bt )t≥0 be a càdlàg stochastic process on (Ω, F , P) which has the same finite dimensional
Let (X
bt )t≥0 is said to be the dual process
distribution as (−Xt )t≥0 , independent of (Xt )t≥0 . The process (X
of (Xt )t≥0 . Note that it is also a Lévy process and its Lévy triplet is (0, 0, ν(−dx)). We define the
bt )]. The reason why we call (X
bt )t≥0 the dual is due to
corresponding semigroup by Pbt f (x) := Ex [f (X
the fact that for Borel function f and g, we have
Z
Z
f (x)Pbt g(x)dx.
Pt f (x)g(x)dx =
Rd
Rd
et := X t + X
b t , then it follows from the independence
We introduce the symmetrization of Xt . Let X
2
2
bt that
of Xt and X
e
E[eiξ·Xt ] = E[e
bt)
iξ·(X t +X
2
2
] = E[e
iξ·X t
2
]E[e
bt
iξ·X
2
t
t
] = e− 2 ψ(ξ) e− 2 ψ(−ξ) = e−t Re(ψ(ξ)) .
If we define νe(B) = 12 (ν(B) + ν(−B)) for any measurable set B, then we have
Z
Z
Re(ψ(ξ)) =
(1 − cos(ξ · y))ν(dy) =
(1 − cos(ξ · y))e
ν (dy)
Rd
Rd
Z
(1 − eiξ·y + iξ · y1|y|≤1 )e
ν (dy)
=
Rd
e
=: ψ(ξ).
et a Lévy process but it also has a Lévy triplet (0, 0, νe) and the characThis shows that not only is X
e
et the symmetrization of Xt . Define Pet f (x) = Ex [f (X
et )]. Note that
teristic exponent ψ(ξ).
We say X
e
the Fourier transform of Pt f is given by
e
F (Pet f )(ξ) = e−tψ(ξ) fb(ξ) = e−t Re(ψ(ξ)) fb(ξ).
8
RODRIGO BAÑUELOS AND DAESUNG KIM
et is a pure jump symmetric Lévy process and the measure νe satisfies all the assumptions that
Since X
we impose on ν before, for example (HW) condition, it leads us to apply the result of [5] for the
et . In particular, we obtain two side estimates for the square functions of X
et .
symmetrized process X
et by
We define the square functions of the symmetrized process X
Z ∞ Z 12
2
e
e )(x) =
G(f
(4.1)
,
Pt f (x + y) − Pet f (x) νe(dy)dt
Rd
0
Z
e ∗ (f )(x) =
G
∞
0
Z
A(t,x,f )
2
e
Pt f (x + y) − Pet f (x) νe(dy)dt
! 12
where A(t, x, f ) := {y ∈ Rd : |Pet f (x)| > |Pet f (x + y)|}. The following lemma is found in [5, Theorem
4.1, Corollary 4.4 ].
Lemma 4.2. Let 2 ≤ p < ∞ and f ∈ Lp (Rd ). Then there are constants cp and Cp depending only on
p such that
e )kp ≤ Cp kf kp .
cp kf kp ≤ kG(f
If 1 < p < ∞ and f ∈ Lp (Rd ), then we have
e ∗ (f )kp ≤ Dp kf kp .
dp kf kp ≤ kG
for some dp and Dp depending only on p.
Before going through the proof of Theorem 4.1, we address a measure theoretic lemma which plays
an important role in the symmetrization argument. Define ν(B) = 21 (ν(B)−ν(−B)) for any measurable
sets B ⊆ Rd .
Lemma 4.3. There is a measurable function r(y) such that ν(dy) = r(y)e
νs (dy). Furthermore, the
function r(y) is bounded νe-a.s. with krk∞,eν ≤ 1.
Proof. The existence of such measurable function follows from
R the Radon-Nikodym theorem. Let us
explain this in detail. Since we assume that ν({0}) = 0 and Rd (1 ∧ |x|2 )ν(dx) < ∞, the measure ν is
σ-finite. So are νe and ν obviously. Suppose B ⊆ Rd is a measurable set such that νe(B) = 0. Since
ν is a positive measure, we have ν(B) = ν(−B) = 0, which implies ν(B) = 0. Thus ν is absolutely
continuous with respect to νe. The Radon-Nikodym theorem tells us now that there is a measurable
ν (y).
function r(y) such that ν(dy) = r(y)e
To see r(y) is bounded, we consider the set Bε := {y ∈ Rd : |r(y)| > 1 + ε} for an arbitrary ε > 0.
From the relation ν(dy) = r(y)e
ν (y) obtained above, we have ν(Bε ) > (1 + ε)e
ν (Bε ). It then yields
εν(Bε ) + (2 + ε)ν(−Bε ) < 0 so that ν(Bε ) = ν(−Bε ) = 0. Therefore, r(y) is bounded νe-a.s. and
krk∞,eν ≤ 1.
Proof of Theorem 4.1. The first argument is directly obtained by Theorem 3.1. Indeed, we define
Z ∞ Z
12
2
G(f )(x) =
|Pt f (x + y) − Pt f (x)| ν(dy)dt
.
0
Rd
It then follows from the Cauchy-Schwartz inequality that
Z Z ∞Z
|Λφ (f, g)| ≤ kφk∞
|Pt f (x + y) − Pt f (x)||Pt g(x + y) − Pt g(x)|ν(dy)dtdx
Rd
≤
0
Rd
kφk∞ kG(f )k2 kG(g)k2 .
Since f, g ∈ L2 (Rd ), Theorem 3.1 implies that Λφ (f, g) is absolutely convergent.
ON SQUARE FUNCTIONS AND FOURIER MULTIPLIERS FOR NONLOCAL OPERATORS
9
Let us show the second assertion. We firstly use Parseval’s formula (2.1) so as to obtain
=
=
=
Λφ (f, g)
Z ∞Z Z
1
F (Pt f (· + y) − Pt f (·))(ξ)F (Pt g(· + y) − Pt g(·))(ξ)φ(t, y)dξν(dy)dt
(2π)d 0
Rd Rd
Z ∞Z Z
1
(eiξ·y − 1)e−tψ(ξ) fb(ξ)(eiξ·y − 1)e−tψ(ξ) gb(ξ)φ(t, y)dξν(dy)dt
(2π)d 0
Rd Rd
Z ∞Z Z
1
|eiξ·y − 1|2 e−2t Re(ψ(ξ)) fb(ξ)b
g (ξ)φ(t, y)dξν(dy)dt
(2π)d 0
Rd Rd
where ψ(ξ) is the characteristic exponent of (Xt )t≥0 . Notice that we have used the fact that
F (Pt f (· + y) − Pt f (·))(ξ) = (eiξ·y − 1)F (Pt f )(ξ) = (eiξ·y − 1)e−tψ(ξ) fb(ξ).
Using the identity ν = νe + ν, we have
Z ∞Z Z
1
Λφ (f, g) =
g (ξ)φ(t, y)dξe
ν (dy)dt
|eiξ·y − 1|2 e−2t Re(ψ(ξ)) fb(ξ)b
(2π)d 0
Rd Rd
Z ∞Z Z
1
g (ξ)φ(t, y)dξν(dy)dt
+
|eiξ·y − 1|2 e−2t Re(ψ(ξ)) fb(ξ)b
(2π)d 0
d
d
R
R
where νe(B) = 21 (ν(B) + ν(−B)) and ν(B) = 21 (ν(B) − ν(−B)) for any measurable set B. By Lemma
4.3, there is a measurable function r(y) such that ν(dy) = r(y)e
ν (dy) with krk∞,eν ≤ 1. If we define
η(t, y) = φ(t, y)(1 + r(y)), then η is bounded νe-a.s. Thus we now obtain
Z ∞Z Z
1
Λφ (f, g) =
|eiξ·y − 1|2 e−2t Re(ψ(ξ)) fb(ξ)b
g (ξ)η(t, y)dξe
ν (dy)dt.
(2π)d 0
Rd Rd
et and Pet , the symmetrization of Xt and Pt . Since the characteristic exponent of X
et is
We consider X
e
the real part of ψ(ξ), ψ(ξ) = Re(ψ(ξ)), and
F (Pet f (· + y) − Pet f (·))(ξ) = (eiξ·y − 1)F (Pet f )(ξ) = (eiξ·y − 1)e−t Re(ψ(ξ)) fb(ξ),
thereby obtaining by Parseval’s formula (2.1) that
Z Z ∞Z
Λφ (f, g) =
(Pet f (x + y) − Pet f (x))(Pet g(x + y) − Pet g(x))η(t, y)e
ν (dy)dtdx
Rd
Rd
0
e η (f, g).
=: Λ
(4.2)
To show the boundedness of Λφ (f, g), we will use square functions defined in (4.1). It is enough to
show the case p > 2 and 1 < q < 2. Note that kηk∞,eν is finite and kηk∞,eν ≤ 2kφk∞ . It then follows
from Cauchy-Schwartz and Hölder inequalities that
Z Z ∞Z
e η (f, g)| ≤ kηk∞,eν
|Λ
|Pet f (x + y) − Pet f (x)||Pet g(x + y) − Pet g(x)|e
ν (dy)dtdx
Rd 0
Rd
Z Z ∞Z
≤ 2kηk∞,eν
|Pet f (x + y) − Pet f (x)||Pet g(x + y) − Pet g(x)|e
ν (dy)dtdx
Rd 0
A(t,x,g)
Z
e )(x)G
e∗ (g)(x)dx
≤ 2kηk∞,eν
G(f
Rd
≤
e )kp kG
e ∗ (g)kq
2kηk∞,eν kG(f
where A(t, x, g) := {y ∈ Rd : |Pet g(x)| > |Pet g(x + y)|}. By Lemma 4.2 and (4.2), we obtain
Λφ (f, g) ≤ 4Cp Dq kφk∞ kf kp kgkq .
Therefore, the Riesz representation theorem yields that there is a unique linear operator Sφ satisfying
Λφ (f, g) = hSφ (f ), gi.
10
RODRIGO BAÑUELOS AND DAESUNG KIM
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Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 479072067, USA
E-mail address: [email protected]
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 479072067, USA
E-mail address: [email protected]