Various sharp estimates for semi-discrete Riesz transforms of the second order

0.3 MB·

by  Komla Domelevo

VARIOUS SHARP ESTIMATES FOR SEMI-DISCRETE
RIESZ TRANSFORMS OF THE SECOND ORDER
7
1
K.DOMELEVO, A.OSE¸KOWSKI,ANDS.PETERMICHL
0
2

n
a
J Abstract. Wegiveseveralsharpestimatesforaclass ofcombinations

5 of second order Riesz transforms on Lie groups = × that
x y
1 are multiply connected, composed of a discrete abGelian cGompoGnent
x
and a connected component y endowed with a biinvariant measuGre.
] TheseestimatesincludenewshGarpLpestimatesviaChoitypeconstants,
A
dependinguponthemultipliersoftheoperator. Theyalsoincludeweak-
C type, logarithmic and exponential estimates. Wegive an optimal Lq →
. Lp estimate as well.
h
t ItwasshownrecentlybyArcozzi-Domelevo-Petermichlthatsuch
a
second order Riesz transforms applied to a function may be written as
m
conditional expectation of a simple transformation of a stochastic inte-
[ gral associated with thefunction.

Theproofsofourtheoremscombinethisstochasticintegralrepresen-
1
tation with a number of deep estimates for pairs of martingales under
v
strong differential subordination by Choi,Banuelos and Ose¸kowski.
6
0 Whentwocontinuousdirectionsareavailable,sharpnessisshownvia
1 the laminates technique. We show that sharpness is preserved in the
4 discrete case using Lax-Richtmyertheorem.
0
.
1 1. Introduction
0
7 Sharp,classical Lp norminequalities forpairsofdifferentially subordinate
1 martingales date back to the celebrated work of Burkholder [15] in 1984
:
v where the optimal constant is exhibited. See also from the same author
i [17][18]. The relation between differentially subordinate martingales and
X
CZ (i.e. Cald´eron–Zygmund) operators is known at least since Gundy–
r
a Varopoulos [32]. Banuelos–Wang [12] were the first to exploit this
connection to prove new sharp inequalities for singular intergrals. This in-
tersection of probability theory withclassical questions inharmonicanalysis
has lead to much interest and a vast literature has been accumulating on
this line of research.
In this article we state a number of sharp estimates that hold in the very
recent, new direction concerning the semi–discrete setting, applying it to a
family of second order Riesz transforms on multiply–connected Lie groups.
Date: February 12, 2017.
Adam Osekowski is supported by Narodowe Centrum Nauki Poland (NCN), grant
DEC-2014/14/E/ST1/00532.
Stefanie Petermichl is supported by ERC project CHRiSHarMa DLV-862402.
1
2 K.DOMELEVO,A.OSE¸KOWSKI,ANDS.PETERMICHL
We recall their representation through stochastic integrals using jump pro-
cesses on multiply–connected Lie groups from [3]. In this representation
formula jump processes play a role, but the strong differential subordina-
tion holds between the martingales representing the test function and the
operator applied to the test function.
The usual procedure for obtaining (sharp) inequalities for operators of
Calder´on–Zygmund type from inequalities for martingales is the following.
Starting with a test function f, martingales are built using Brownian mo-
tion or background noise and harmonic functions in the upper half space
+ × n. Through the use of Itøformula, it is shown that the martin-
R R
gale arising in this way from Rf, where R is a Riesz transform in n, is
R
a martingale transform of the martingale arising from f. The two form
a pair of martingales that have differential subordination and (in case of
Hilbert or Riesz transforms) orthogonality. One then derives sharp martin-
gale inequalities under hypotheses of strong differential subordination (and
orthogonality) relations.
In the case of Riesz transforms of the second order, the use of heat exten-
sions in the upper half space instead of Poisson extensions originated in the
context of aweighted estimate inPetermichl–Volberg[44]andwas used
to prove Lp estimates for the second order Riesz transforms based on the
results of Burkholder in Nazarov–Volberg [50] as part of their best-at-
time estimate for the Beurling–Ahlfors operator, whose real and imaginary
parts themselves are second order Riesz transforms. We mention the re-
cent version on discrete abelian groups Domelevo–Petermichl [24] also
using a type of heat flow. These proofs are deterministic. The technique
of Bellman functions was used. This deterministic strategy does well when
no orthogonality is present and when strong subordination is the only im-
portant property. Stochastic proofs (aside from giving better estimates in
some situations) also have the advantage that once the integral representa-
tion is known, the proofs are a very concise consequence of the respective
statements on martingales.
In[3]theauthorsprovedsharpLpestimatesforsemi–discretesecondorder
Riesz transforms R2 using stochastic integrals. There is an array of Riesz
α
transforms of the second order that are treated, indexed my a matrix index
α (see below for precisions on acceptable α). The following representation
formula of semi-discrete second order Riesz transforms R2 a` la Gundy–
α
Varopoulos (see [32]) is instrumental:
Theorem. (Arcozzi–Domelevo–Petermichl, 2016) The second order
Riesz transform R2f of a function f ∈ L2( ) as defined in (1.1) can be
α
G
written as the conditional expectation
α,f
(M |Z = z).
E 0 0
α,f f
Here M is a suitable martingale transform of a martingale M associated
t t
to f, and Z is a suitable random walk on
t
G
SEMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 3
We remark that the Lp estimates of the discrete Hilbert transform on the
integers are still open. It is a famous conjecture that this operator has the
same norm as its continuous counterpart.
These known Lp norm inequalities use special functions found in the re-
sults of Pichorides [45], Verbitsky[18], Ess´en [28], Banuelos–Wang
[12] when orthogonality is present in addition to differential subordination
or Burkholder [15][16][17], Wang [51] when differential subordination is
the only hypothesis.
The aim of the present paper is to establish new estimates for semi–
discrete Riesz transforms by using the martingale representation above to-
gether with recent martingale inequalities found in the literature.
Here is a brief description of the new results in this paper.
• In the case where the function f is real valued, we can obtain better
estimates for R2 than in the general case. These estimates depend
α
uponthemakeofthematrixindexα. Theprecisestatementisfound
in Theorem 1.2.
• Weprovearefinedsharpweaktypeestimateusingaweaktypenorm
defined just before the statement of Theorem 1.3 .
• We prove logarithmic and exponential estimates, in a sense limiting
(in p) cases of the classical sharp Lp estimate. See Theorem 1.4.
• We consider the norm estimates of the R2 : Lq → Lp, spaces of
α
different exponent. The statement is found in Theorem 1.5.
1.1. Differential operators and Riesz transforms.
First order derivatives and tangent planes. We will consider Lie
groups := × , where is a discrete abelian group with a fixed
x y x
G G G G
set G of m generators, and their reciprocals, and is a connected, Lie
y
G
group of dimension n endowed with a biinvariant metric. The choice of the
set G of generators in corresponds to the choice of a bi-invariant metric
x
G
structureon . Wewilluseon themultiplicativenotationforthegroup
x x
G G
operation. We will define a product metric structure on , which agrees
G
with the Riemannian structure on the first factor, and with the discrete
“word distance” on the second. We will at the same time define a “tangent
space” T for at a point z = (x,y) ∈ ( × ) = . We will do this
z x y
G G G G G
in three steps.
First, since is an n-dimensional connected Lie group with Lie alge-
y
bra G . We caGn identify each left-invariant vector field Y in G with its
y y
value at the identity e, G ≡ T . Since is compact, it admits a bi-
y e y
G G
invariant Riemannian metric, which is unique up to a multiplicative factor.
We normalize it so that the measure µ associated with the metric satisfies
y
µ ( ) = 1. The measure µ is also the normalized Haar measure of the
y y y
G
group. We denote by < ·,· > be the corresponding inner product on T
y y y
and by ∇ f(y) the gradient at y ∈ of a smooth function f : →G.
y y y
G G R
4 K.DOMELEVO,A.OSE¸KOWSKI,ANDS.PETERMICHL
Let Y ,...,Y be a orthonormal basis for G . The gradient of f can be
1 n y
written ∇ f = Y (f)Y +...+Y (f)Y .
y 1 1 n n
Second, in the discrete component , let G = (g ) be a set
x x i i=1,...,m
G
of generators for , such that for i 6= j and σ = ±1 we have g 6= gσ.
Gx i j
The choice of a particular set of generators induces a word metric, hence, a
geometry, on . Any two sets of generators induce bi-Lipschitz equivalent
x
G
metrics.
At any point x ∈ , and given a direction i∈ {1,...,m}, we can define
x
G
the right and the left derivative at x in the direction i:
(∂+f/∂x )(x,y) := f(x+g ,y)−f(x,y) := (∂+f)(x,y)
i i i
(∂−f/∂x )(x,y) := f(x,y)−f(x−g ,y) := (∂−f)(x,y).
i i i
Comparing with the continuous component, this suggests that the tangent
planeTˆ atapointxofthediscretegroup mightactually besplitinto
x x x
G G
a “right” tangent plane T+ and a “left” tangent plane T− , according
x x x x
G G
to the direction with respect to which discrete differences are computed.
We consequently define the augmented discrete gradient ∇ f(x), with a
x
hat, as the 2m–vector of Tˆ := T+ ⊕ T− accounting for all the
x x x x x x
G G G
local variations of the function f in the direct vicinity ofbx; that is, the
2m–column–vector
m
∇ f(x) := (X+f,X+f,...,X−f,X−f,...)(x) = Xτf(x)
x 1 2 1 2 i
i=1τ=±
XX
b
with Xτ ∈ Tˆ , where we noted the discrete derivatives X±f := ∂±f and
i xGx i i
introduced the discrete 2m–vectors X± as the column vectors of 2m
i Z
X+ = (0,...,1,...,0)×0 , X− = 0 ×(0,...,1,...,0).
i m i m
Here the 1’s in X± are located at the i–th position of respectively the first
i
or the second m–tuple. Notice that those vectors are independent of the
point x. The scalar product on Tˆ := T+ ⊕T− is defined as
x x x x x x
G G G
m
1
(U,V) := UτVτ.
Tˆx x 2 i i
G i=1τ=±
XX
We chose to put a factor 1 in front of the scalar product to compensate for
2
the fact that we consider both left and right differences.
Finally, for a function f defined on the cartesian product := × ,
x y
G G G
the (augmented) gradient ∇ f(z) at the point z = (x,y) is an element of
z
the tangent plane Tˆ := Tˆ ⊕T , that is a (2m+n)–column–vector
z x x y y
G G G
b
m n
∇ f(z) := Xτf(z)Xˆτ + Y f(z)Yˆ (z)
z i i j j
i=1τ=± j=1
XX X
b = (X+f,X+f,...,X−f,X−f,...,Y f,Y f,...)(z)
1 2 1 2 1 2
SEMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 5
where Xˆτ and Yˆ (z) can be identified with column vectors of size (2m+n)
i j
with obvious definitions and scalar product (·,·) .
Tˆz z
Letdµ := dµ dµ ,dµ beingthecountingmeasGureon anddµ being
z x y x x y
G
the Haar measure on . The inner product of ϕ,ψ in L2( ) is
y
G G
(ϕ,ψ) := ϕ(z)ψ(z)dµ (z).
L2( ) z
G
Z
G
Finally, we make the following hypotheses
Hypothesis. We assume everywhere in the sequel:
1. The discrete component of the Lie group is an abelian group
x
G G
2. The connected component of the Lie group is a Lie group that
y
G G
can beendowed with a biinvariant Riemannian metric, so that the family
(Y ) commutes with ∆ .
j j=1,...,n y
Notice that this includes compact Lie groups since those can be en-
y
G
dowed with a biinvariant metric. It also includes the usual Euclidian spaces
since those are commutative.
Riesz transforms. Following [1][2], recall first that for a compact Rie-
mannian manifold without boundary, one denotes by ∇ , div and
∆ := div ∇ reMspectively the gradient, the divergence anMd theMLapla-
M M M
cian associated with . Then −∆ is a positive operator and the vector
Riesz transform is defiMned as the linMear operator
R := ∇ ◦(−∆ )−1/2
M M M
acting on L2( ) (L2 functions with vanishing mean). It follows that if f is
a function d0efiMned on and y ∈ then R f(y) is a vector of the tangent
M M M
plane T .
y
M
Similarly on = , we define ∇ := ∇ as before, and then we define
z
thedivergenceoMperatGorasitsformalGadjoint, thatis−div = −div := ∇∗,
z z
with respect to the natural L2 inner produbct of vector fieGlds:
c b
(U,V) := (U(z),V(z)) dµ (z)
L2(Tˆ ) Tˆz z
G Z G
G
We have the L2-adjoints (X±)∗ = −X∓ and Y∗ = −Y . If U ∈ Tˆ is
i i j j G
defined by
m n
U(z) = Uτ(z)Xˆτ + U (z)Yˆ ,
i i j j
i=1τ=± j=1
XX X
we define its divergence ∇∗U as
z
m n
∇∗U(z) :=b−1 X−τUτ(z)− Y U (z).
z 2 i i j j
i=1τ=± j=1
XX X
b
6 K.DOMELEVO,A.OSE¸KOWSKI,ANDS.PETERMICHL
The Laplacian ∆ is as one might expect:
G
∆ f(z) := −∇∗∇ f(z)= −∇∗∇ f(z)−∇∗∇ f(z)
z z z x x y y
m n
= bXb−X+f(z)+b bY2f(z) b b
i i j
i=1 j=1
X X
m n
= X2f(z)+ Y2f(z)
i j
i=1 j=1
X X
=: ∆ f(z)+∆ f(z)
x y
where we denoted X2 := X+X− = X−X+. We have chosen signs so that
i i i i i
−∆ > 0 as an operator. The Riesz vector (Rˆ f)(z) is the (2m + n)–
z
coluGmn–vector of the tangent plane Tˆ defined as the linear operator
z
G
Rˆ f := ∇ f ◦(−∆ f)−1/2
z z z
(cid:16) (cid:17)
We also define transforms along the coordinate directions:
b
R± = X±◦(−∆ )−1/2 and R = Y ◦(−∆ )−1/2.
i i z j j z
Plan of the paper. In the next two sections, we present successively
the main results of the paper and recall the weak formulations involving
second order Riesz transforms and semi-discrete heat extensions. Section 2
introduces the stochastic setting for our problems. This includes in Subsec-
tion 2.1 semi-discrete random walks, martingale transforms and quadratic
covariations. Subsection2.2 presents a setof martingale inequalities already
known in the literature. Finally, in Section 3 we give the proof of the main
results.
1.2. Main results. In this text, we are concerned with second order Riesz
transformsandcombinations thereof. WefirstdefinethesquareRiesztrans-
form in the (discrete) direction i to be
R2 := R+R− = R−R+.
i i i i i
Then, given α := ((αx) ,(αy ) ) ∈ m× n×n, we define R2 to
i i=1...m jk j,k=1...n C C α
be the following combination of second order Riesz transforms:
m n
(1.1) R2 := αx R2+ αy R R ,
α i i jk j k
i=1 j,k=1
X X
where the first sum involves squares of discrete Riesz transforms as defined
above,andthesecondsuminvolves productsofcontinuousRiesztransforms.
This combination is written in a condensed manner as the quadratic form
R2 = Rˆ ,A Rˆ
α z α z
(cid:16) (cid:17)
SEMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 7
where A is the (2m+n)×(2m+n) block matrix
α
Ax 0
(1.2) A := α
α 0 Ay
(cid:18) α (cid:19)
with
Ax = diag(αx,...,αx ,αx,...,αx ) ∈ 2m×2m,Ay =(αy ) ∈ n×n.
α 1 m 1 m C α jk j,k=1...n C
In the theorems below, we assume that is a Lie group and R2 is a
α
G
combination of second order Riesz transforms as defined above. The first
application of the stochastic integral formula, Theorem 1.1 was done in [3],
while the other applications, Theorems 1.2 1.3 1.4 and 1.5 are new.
Theorem 1.1. (Arcozzi–Domelevo–Petermichl, 2016) For any 1 <
p < ∞ we have
kR2k 6 kA k (p∗−1),
α p α 2
where, as previously, p∗ = max{p,p/(p−1)}.
Above, we have set:
kA k = max(kAxk ,kAyk )= max(|αx|,...,|αx |,kAyk ).
α 2 α 2 α 2 1 m α 2
In the case where = only consists of the discrete component, this was
x
G G
provedin[25][24]usingthedeterministicBellmanfunctiontechnique. Inthe
case where = is a connected compact Lie group, this was proved in [8]
y
G G
using Brownian motions defined on manifolds and projections of martingale
transforms.
In the case where the function f is real valued, we can obtain better
estimates. For any real numbers a < b and any 1< p < ∞, let C be the
a,b,p
constants introduced in Ban˜uelos and Ose¸kowski [10].
Theorem 1.2. Assume that aI 6 A 6 bI in the sense of quadratic forms.
α
Then R2 : Lp( , ) → Lp( , ) enjoys the norm estimate kR2k 6 C .
α α p a,b,p
G R G R
We should point out here that the constants C appear in earlier works
a,b,p
of Burkholder [15] (for a = −b: then C = b(p∗−1)), and in the paper
a,b,p
[19] by Choi (in the case when one of a, b is zero). The Choi constants are
not explicit; an approximation of C is known and writes as
0,1,p
C = p + 1log 1+e−2 + β2 +....,
0,1,p 2 2 2 p
(cid:16) (cid:17)
2
with β = log2 1+e−2 + 1log 1+e−2 −2 e−2 .
2 2 2 2 1+e−2
Coming back(cid:16)to com(cid:17)plex-valu(cid:16)ed fun(cid:17)ctions(cid:16), we w(cid:17)ill also establish the fol-
lowing weak-type bounds. We consider the norms
||f|| = sup µ (E)1/p−1 fdµ ,
Lp,∞( , ) z z
GC (cid:26) ZE (cid:27)
where the supremum is taken over the class of all measurable subsets E of
of positive measure.
G
8 K.DOMELEVO,A.OSE¸KOWSKI,ANDS.PETERMICHL
Theorem 1.3. For any 1< p < ∞ we have
1−1/p
1 2p−1
Γ if 1< p ≤ 2,
2 p−1
kRα2kLp(G,C)→Lp,∞(G,C) 6 kAαk2·(cid:18)pp−(cid:18)1 1/p (cid:19)(cid:19)
 if p ≥ 2.
2
(cid:18) (cid:19)



We will also prove the followinglogarithmic and exponential estimates,
which can be regarded as versions of Theorem 1.1 for p = 1 and p = ∞.
Consider the Young functions Φ, Ψ : [0,∞) → [0,∞), given by Φ(t) =
et−1−t and Ψ(t) = (t+1)log(t+1)−t.
Theorem 1.4. Let K > 1 be fixed.
(i) For any measurable subset E of and any f on we have
G G
µ (E)
|R2f|dµ 6 kA k · K Ψ(|f|)dµ + z .
α z α 2 z 2(K −1)
ZE (cid:18) Z (cid:19)
G
(ii) For any f : → bounded by 1,
G C
Φ K|RkAα2f|k dµz 6 2||Kf|(|LK1(G−,C1)).
Z (cid:18) α 2(cid:19)
G
Ourfinalresultconcerns another extension of Theorem 1.1, which studies
the action of R2 between two different Lp spaces. For 1 ≤ p < q < ∞, let
α
C be the constant defined by Ose¸kowski in [40].
p,q
Theorem 1.5. For any 1 ≤ p < q < ∞, any measurable subset E of and
G
any f ∈ Lq( ) we have
G
kR2fk 6 C kA k kfk µ (E)1/p−1/q.
α Lp(E, ) p,q α 2 Lq( , ) z
C GC
An interesting feature is that all the estimates in the five theorems above
are sharp when the group = × and dim( )+dim∞( ) > 2,
x y y x
where dim∞( ) denotes thGe numGber oGf infinite comGponents of G.
x x
G G
1.3. Weak formulations. Let f : → be given. The heat extension
f˜(t) of f is defined as f˜(t) := et∆zfG=: PCf. We have therefore f˜(0) = f.
t
Theaim of this section is to derive weak formulations for second order Riesz
transforms. We start with the weak formulation of the identity operator I,
that is obtained by using semi-discrete heat extensions (see [3] for details).
SEMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 9
Assume f in L2( ) and g in L2( ). Let f¯be the average of f on if
G G G G
has finite measure and zero otherwise. Then
(If,g) = (f,g)
L2( )
G
∞
= f¯g¯+2 ∇ P f,∇ P g dt
z t z t
Z0 (cid:16) (cid:17)L2(TˆG)
∞ b 1 bm
= f¯g¯+2 (XτP f)(z)(XτP g)(z) +
2 i t i t
Z0 Zz∈G( Xi=1τX=±
n
(Y P f)(z)(Y P g)(z) dµ (z)dt
j t j t z

Xj=1 
and the sums and integrals that arise converge absolutely.

In order to pass to the weak formulation for the squares of Riesz trans-
forms, we first observe that the following commutation relations hold
Y ◦∆ = ∆ ◦Y
j z z j
Xτ ◦∆ = ∆ ◦Xτ, τ ∈ {+,−}
i z z i
ThisisaneasyconsequenceofthehypothesismadeontheLiegroup. Fol-
lowing [3], the following weak formulation for second order Riesz transforms
holds
Assume f in L2( ) and g in L2( ), then
G G
∞
(R2f,g) = −2 A ∇ P f,∇ P g dt
α L2(G) Z0 (cid:16) α z t z t (cid:17)L2(TˆG)
∞ b 1 m b
= −2 αx (XτP f)(z)(XτP g)(z)
2 i i t i t
Z0 Zz∈G( Xi=1τX=±
n
y
+ α (Y P f)(z)(Y P g)(z) dµ (z) dt
jk j t k t z
)
j,k=1
X
and the sums and integrals that arise converge absolutely.
2. Stochastic integrals and martingale transforms
In what follows, we assume that we have a complete probability space
(Ω,F, ) with a c`adla`g (i.e. right continuous left limit) filtration (Ft)t>0
P
of sub-σ–algebras of F. We assume as usual that F contains all events
0
of probability zero. All random walks and martingales are adapted to this
filtration.
We define below a continuous-time random process Z with values in ,
G
Z := (X ,Y ) ∈ × , having infinitesimal generator L = ∆ . Thepure-
t t t x y z
G G
jumpcomponent X is a compound Poisson jump process on the discrete set
t
, wheras the continuous component Y is a standard brownian motion on
x t
G
themanifold . Then,Itˆo’s formulaensuresthatsemi-discrete “harmonic”
y
G
functions f : +× → solving the backward heat equation (∂ +∆ )f =
t z
R G C
10 K.DOMELEVO,A.OSE¸KOWSKI,ANDS.PETERMICHL
f
0 give rise to martingales M := f(t,Z ) for which we define a class of
t t
martingale transforms.
2.1. Stochastic integrals, Martingale transforms and quadratic co-
variations.
Stochastic integrals on Riemannian manifolds and Itoˆ integral.
Following Emery [26][27], see also Arcozzi [1][2], we define the Brownian
motionY on ,acompactRiemannianmanifold,astheprocessY : Ω→
t y t
G
(0,T)× such that for all smooth functions f : → , the quantity
y y
G G R
1 t
(2.1) f(Y )−f(Y )− (∆ f)(Y ) ds=:(I )
t 0 2 y s dyf t
Z0
is an –valued continuous martingale. For any adapted continuous process
R
Ψ with values in the cotangent space T∗ of , if Ψ (ω) ∈ T∗ for
Gy Gy t Yt(ω)Gy
all t > 0 and ω ∈ Ω, then one can define the continuous Itˆo integral I of
Ψ
Ψ as
t
(I ) := hΨ ,dY i
Ψ t s s
Z0
so that in particular
t
(I ) := hd f(Y ),dY i
dyf t y s s
Z0
The integrand above involves the 1–form of T∗
y y
G
d f(y):= (Y f)(y) Y∗.
y j j
j
X
A pure jump process on . We will now define the discrete m–
x
G
dimensional process X on the discrete abelian group as a generalized
t x
G
compound Poisson process. In order to do this we need a number of inde-
pendent variables and processes:
First, for any given 1 6 i 6 m, let Ni be a c`adla`g Poisson process of
t
parameter λ, that is
(λt)n
∀t, (Ni = n) = e−λt.
P t n!
The sequence of instants where the jumps of the Ni occur is noted (Ti) ,
t k k∈
with the convention Ti = 0. N
0
Second, we set
m
N = Ni
t t
i=1
X
Almost surely, for any two distincti and j, we have {Ti} ∩{Tj} = ∅.
k k∈ k k∈
Lettherefore{T } = ∪m {Ti} betheorderedsequeNnceofinstaNntsof
k k∈ i=1 k k∈
jumpsofN andletiN≡ i (ω)betheNindexofthecoordinatewherethejump
t t t