Dimension-free $L^p$ estimates for vectors of Riesz transforms associated with orthogonal expansions

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by  Błażej Wróbel

DIMENSION-FREE Lp ESTIMATES FOR VECTORS OF RIESZ
TRANSFORMS ASSOCIATED WITH ORTHOGONAL EXPANSIONS
BL AZ˙EJ WRO´BEL
Abstract. An explicit Bellman function is used to prove a bilinear embedding the-
orem for operators associated with general multi-dimensional orthogonal expansions
7
1 on product spaces. This is then applied to obtain Lp, 1 < p < ∞, boundedness of
0 appropriatevectorialRiesztransforms,inparticularinthecaseofJacobipolynomials.
2 OurestimatesfortheLp normsoftheseRiesztransformsarebothdimension-freeand
linearinmax(p,p/(p−1)).Theapproachwepresentallowsustoavoidtheuseofboth
n
differential forms and general spectral multipliers.
a
J

7

] 1. Introduction
A
F The classical Riesz transforms on Rd are the operators
.
h R f(x)= ∂ ( ∆ )−1/2f(x), i= 1,...,d.
t i xi − Rd
a In [32] E. M. Stein proved that the vector of Riesz transforms
m
Rf = (R f,...,R f)
[ 1 d

has Lp bounds which are independent of the dimension. More precisely
1
v Rf 6 C f , 1< p < , (1.1)
9 k kLp(Rd) pk kLp(Rd) ∞
8 where C is independent of the dimension d. Note that (1.1) is formally the same as the
p
8
a priori bound
1
0 |∇f| Lp(Rd) 6 Cp (−∆)1/2f Lp(Rd).
.
1 Later it was realized tha(cid:13)t, for(cid:13)1 < p < 2 on(cid:13)e may tak(cid:13)e Cp 6 C(p 1)−1 in (1.1), see
0 (cid:13) (cid:13) (cid:13) (cid:13) −
[2], [11]. It is worth mentioning that the best constant in (1.1) remains unknown when
7
1 d 2; the best results to date are given in [3] and [16].
≥
: The main goal of this paper is to generalize (1.1) to product settings different than
v
i Rd = R R with the product Lebesgue measure. Our starting point is the obser-
X ×···×
vation that the classical Riesz transform can be written as R = δ ( d L )−1/2 where
r i i i=1 i
a δi = ∂xi, and Li = δi∗δi. The generalized Riesz transforms we pursPue are of the same
form
R = δ L−1/2, i = 1,...,d, (1.2)
i i
with δ being an operator on L2(X ,µ ),
i i i
d
L = δ∗δ , and L = L .
i i i i
i=1
X
Here the adjoint δ∗ is taken with regards to the inner product on L2(X ,µ ), where
i i i
µ is a non-negative Borel measure which is absolutely continuous with respect to the
i
Lebesgue measure. Throughout the paper we assume that each X , i = 1,...,d, is an
i
2010 Mathematics Subject Classification. 42C10, 42A50, 33C50.
Key words and phrases. Riesz transform, Bellman function, orthogonal expansion.
1
DIMENSION-FREE Lp ESTIMATES FOR VECTORS OF RIESZ TRANSFORMS 2
open interval in R, half-interval in R or is the real line; we also set X = X X
1 d
×···×
and µ = µ µ . We consider δ being given by
1 d i
⊗···⊗
δ f(x)= p (x )∂ +q (x ), x X ,
i i i xi i i i ∈ i
forsomereal-valuedfunctionsp C2(X )andq C1(X ).Weremarkthatasignificant
i i i i
∈ ∈
difference between the classical Riesz transforms and the general Riesz transforms (1.2)
lies in the fact that the operators δ and δ∗ do not need to commute.
i i
There are two assumptions which are critical to our results. Firstly, a computation,
see [28, p. 683], shows that the commutator [δ ,δ∗] is a function which we call v . We
i i i
assumethatv isnon-negative, cf.(A1). Secondly, itisnothardtoseethatL = d L
i i=1 i
may be written as L = L˜ +r, where L˜ is a purely differential operator (without a zero
P
order potential term) and r is the potential term. We impose that d q2 is controlled
i=1 i
from below by a constant times r, namely d q2 K r, for some K > 0, cf. (A2).
i=1 i ≤ · P
In several cases we will consider we can take K = 1 or K = 0. When 0 is not an L2
P
eigenvalue of L our main result can be summarized as follows.
Theorem. Set p∗ = max(p,p/(p 1)). Then the vectorial Riesz transform Rf =
−
(R f,...,R f) with R given by (1.2) satisfies the bounds
1 d i
Rf 6 24(1+√K)(p∗ 1) f , 1 < p < .
k kLp(X,µ) − k kLp(X,µ) ∞
In other words, introducing δf = (δ f,...,δ f), we have
1 d
δf 6 24(1+√K)(p∗ 1) L1/2f , 1 < p < .
| | Lp(X,µ) − Lp(X,µ) ∞
(cid:13) (cid:13)
(cid:13) (cid:13)
To prov(cid:13)e for(cid:13)mally this theorem we need so(cid:13)me ext(cid:13)ra technical assumptions. For the
(cid:13) (cid:13)
sake of clarity of the presentation we decided to concentrate on the case of orthogonal
expansions, when each of the operators L = δ∗δ has a decomposition in terms of some
i i i
orthonormal basis. Our precise setting is described in detail in Section 2. We follow the
approach of Nowak and Stempak from [28], in fact the present paper may be thought of
as an Lp counterpart for a large part of the L2 results from [28]. Adding the technical
assumptions(T1),(T2),and(T3)tothecrucialassumptions(A1)and(A2)westateour
main result as Theorem 2 in Section 3. In all the cases we will consider, the projection
Π appearing in Theorem 2 is either the identity operator or has its Lp norm bounded
by 2 for all 1 p . Moreover, we have Π = I if and only if 0 is not an L2 eigenvalue
≤ ≤ ∞
of L.
From Theorem 2 we obtain several new dimension-free bounds for vectors of Riesz
transforms connected with classical multi-dimensional orthogonal expansions. For more
details we refer to the examples in Section 5. For instance in Section 5.3 we obtain
the dimension-free boundedness for the vector of Riesz transforms in the case of Jacobi
polynomial expansions. This answers a question left open in Nowak and Sj¨ogren’s [26].
Moreover, the approach we present gives a unified way to treat dimension-free estimates
for vectors of Riesz transforms. In most of the previous cases separate papers were
written for each of the classical orthogonal expansions. More unified approaches were
recently presented by Forzani, Sasso, and Scotto in [12] and by the author in [36].
However, these papers treat only dimension-free estimates for scalar Riesz transforms
and not for the vector of Riesz transforms.
Let us remark that Theorem 2 formally cannot be applied to some cases where the
crucial assumptions on v and r continue to hold. This is true when L has a purely
i
continuous spectrum, for instance for the classical Riesz transforms on Rd (when v =0
i
DIMENSION-FREE Lp ESTIMATES FOR VECTORS OF RIESZ TRANSFORMS 3
and r = 0). However, it is not difficult to modify the proof of Theorem 2 so that it
remains valid for the classical Riesz transforms. We believe that a similar procedure
can be applied to other cases outside the scope of Theorem 2, as long as the crucial
assumptions (A1) and (A2) are satisfied.
We deduce Theorem 2 from a bilinear embedding theorem (see Theorem 4) together
withabilinearformula(seeProposition3). ThemaintoolthatisusedtoproveTheorem
4 is the Bellman function technique. This method was introduced to harmonic analysis
by Nazarov, Treil, and Volberg [23]. The proof of Theorem 4 is presented in Section 4
and is a consequence of a bilinear embedding theorem based on subtle properties of a
particular Bellman function. This approach was devised by Dragiˇcevi´c and Volberg in
[8, 9, 10]. Carbonaroand Dragiˇcevi´c developed the method further in [6]. The approach
from [6] was recently adapted by Mauceri and Spinelliin [20] to the case of the Laguerre
operator. Ourpapergeneralizessimultaneously[9](asweadmitanon-negativepotential
r) and [20] (as we consider general p in δ = p ∂ +q ).
i i i xi i
In some applications of the Bellman function method the authors needed to prove
dimension-free bounds on Lp for particular spectral multipliers related to the consid-
ered operators, see [9] for such a situation. In other papers mentioned in the previous
paragraph they needed to consider operators acting on differential forms, cf. [6] and
[20]. One of the merits of our approach is that we avoid to use both general spectral
multipliers and differential forms. This is achieved by means of the bilinear formula
from Proposition 3. This formula relates the Riesz transform R with an integral where
i
only δ and two kinds of semigroups (one for L and one for L+v ) are present, see (3.2).
i i
For the sake of simplicity we use a real-valued Bellman function in Section 4. Thus
our main results Theorems 2 and 4 apply to real-valued functions. Of course they can
be easily extended to complex valued-functions with the constants being twice as large.
One may improve the estimates further by using a complex-valued Bellman function as
it was done in [8], [9], and [10].
Notations. We finish this section by introducing the general notations used in the
paper. ByNwedenotethesetofnon-negativeintegers. For N NandY beinganopen
subsetof RN thesymbolCn(Y),n N,denotes thespaceof rea∈l-valued functionswhich
∈
have continuous partial derivatives in Y up to the order n. In particular C0(Y) = C(Y)
denotes the space of continuous functions on Y equipped with the supremum norm. By
C∞(Y) we mean the space of infinitely differentiable functions on Y. Whenever we say
that ν is a measure on Y we mean that ν is a Borel measure on Y. Thesymbols f and
Hessf stand for the gradient and the Hessian of a function f: RN R. For a,b∇ RN,
we denote by a,b , the inner producton RN and set a 2 = a,a . T→he actual N s∈hould
h i | | h i
be clear from the context (in fact we always have N 1,d,d+1 .) For p (1, ) we
∈ { } ∈ ∞
set
p
p∗ = max p, .
p 1
(cid:18) − (cid:19)
2. Preliminaries
Our notations will closely follow that of [28]. All the functions we consider are real-
valued.
For i = 1,...,d, let X be either an open interval or an open half-interval in R of the
i
form
X = (σ ,Σ ), where 6 σ < Σ 6 .
i i i i i
−∞ ∞
DIMENSION-FREE Lp ESTIMATES FOR VECTORS OF RIESZ TRANSFORMS 4
Considerthemeasurespaces(X , ,µ ),where denotestheσ-algebraofBorelsubsets
i i i i
B B
of X and µ is a Borel measure on X . We impose that dµ (x ) = w (x )dx , wherew is
i i i i i i i i i
a positive C2 function in X . Note that in [28] the authors assumed that X = = X ;
i 1 d
···
this is however not needed in our paper. Throughout the article we let
X = X X , µ = µ µ ,
1 d 1 d
×···× ⊗···⊗
and abbreviate
Lp := Lp(X,µ), = , and = .
p Lp p→p Lp→Lp
k·k k·k k·k k·k
Thisnotation isalsousedforvector-valued functions. Namely, if g = (g ,...,g ): X
1 N
R, for some N N, then →
∈
1/p N 1/2
g = g(x)pdµ(x) , with g(x) = g (x)2 .
p i
k k | | | | | |
(cid:18)ZX (cid:19) (cid:18)i=1 (cid:19)
X
We shall also write f,g for f,g .
h iL2 h iL2(X,µ)
Let δ , i = 1,...,d, be the operators acting on C∞(X ) functions via
i c i
δ = p ∂ +q .
i i xi i
Here p and q are real-valued functions on X , with p C2(X ) and q C1(X ). We
i i i i i i i
∈ ∈
assume that p (x ) = 0, for x X . We shall also denote by p and q the exponents of
i i i i
6 ∈
Lp and Lq spaces. This will not lead to any confusion as the functions p and q will
i i
always appear with the index i =1,...,d.
Let δ∗ be the formal adjoint of δ with regards to the inner product on L2(X ,µ ), i.e.
i i i i
1
δ∗f = ∂ p w f +q f, f C∞(X ).
i −w xi i i i ∈ c i
i
A simple calculation, see [28, p. 6(cid:0)83], sh(cid:1)ows that when p C2(X ) and q C1(X )
i i i i
∈ ∈
then the commutator
w′ ′
[δ ,δ∗]= δ δ∗ δ∗δ = p 2q′ p i p′′ =:v (2.1)
i i i i − i i i i − iw − i i
(cid:18) (cid:18) i(cid:19) (cid:19)
is a locally integrable function (0-order operator). Most of the assumptionsmade in this
section are of a technical nature. The assumption that is crucial to our results is the
following:
the functions v ,i = 1,...,d, are non-negative. (A1)
i
The property (A1) has been (explicitly or implicitly) instrumental for establishing the
main results in [15], [20], [26], [34]. It is also explicitly stated by Forzani, Sasso, and
Scotto as Assumption H1 c).
For a scalar a 0 we let L and L to be given on C∞(X) by
i ≥ i c
d
L := δ∗δ +a , L = L .
i i i i i
i=1
X
HereeachL canbeconsideredtoacteitheronC∞(X )oronC∞(X),thusthedefinition
i c i c
of L makes sense. Note that both L and L are symmetric on C∞(X) with respect to
i c
the inner product on L2. We assume that for each i = 1,...,d, there is an orthonormal
basis {ϕiki}ki∈N which consists of L2 eigenvectors of Li that correspond to non-negative
eigenvalues {λiki}ki∈N, i.e.
L ϕi = λi ϕi .
i ki ki ki
DIMENSION-FREE Lp ESTIMATES FOR VECTORS OF RIESZ TRANSFORMS 5
Then,itmustbethat λ a ,fork Nand i= 1,...,d. We requirethatthesequence
ki ≥ i i ∈
{λiki}ki∈N is strictly increasing and that limki→∞λiki = ∞. Since Li is hypoelliptic we
have ϕi C∞(X ). Setting, for k = (k ,...,k ) Nd,
ki ∈ i 1 d ∈
ϕ = ϕ1 ϕd , , (2.2)
k k1 ⊗···⊗ kd
we obtain an orthonormal basis of eigenvectors on L2 for the operator L = L + +L .
1 d
···
The eigenvalue corresponding to ϕ is
k
λ := λ1 + +λd ,
k k1 ··· kd
so that Lϕ = λ ϕ . We consider the self-adjoint extension of L (still denoted by the
k k k
same symbol) given by
Lf = λ f,ϕ ϕ
k h kiL2 k
kX∈Nd
on the domain
Dom(L) = f L2: λ 2 f,ϕ 2 < .
{ ∈ | k| |h kiL2| ∞}
kX∈Nd
We assume that the eigenfunctions ϕi , i = 1,...,d, are such that
ki
δ ϕi ,δ ϕi = L ϕi ,ϕi , (T1)
i ki i mi L2(Xi,µi) i ki mi L2(Xi,µi)
for i = 1,...,d, and (cid:10)k ,m N, c(cid:11)f. [28, eq. (2(cid:10).8)]. The co(cid:11)ndition (T1) implies that the
i i
∈
functions
δ ϕ = ϕ1 δ ϕi ϕd (2.3)
i k k1 ⊗···⊗ i ki ⊗···⊗ kd
are pairwise orthogonal on L2 and
δ ϕ ,δ ϕ = λi a .
h i k i kiL2 ki − i
cf. [28, Lemma 5,6]. Moreover, since ϕ C∞(X) we also see that δ ϕ C∞(X).
k i k
∈ ∈
We also impose a boundary condition on the functions ϕi and δ ϕi . Namely, we
ki i ki
require that for each i = 1,...,d, if z σ ,Σ then,
i i i
∈ { }
lim (1+ ϕi s1 + δ ϕi s2)(p2w ∂ ϕi ) (x ) = 0,
xi→zi | ki| | i ki| i i xi ki i (T2)
lim ((cid:2)1+ ϕi s1 + δ ϕi s2)(p2w ∂ δ ϕi )(cid:3)(x ) = 0,
xi→zi | ki| | i ki| i i xi i ki i
for all k N and s ,s(cid:2) > 0. Condition (T2) is close to the as(cid:3)sumption H1 a) from [12].
i 1 2
∈
Observe that the term ϕi s1 + δ ϕi s2 in (T2) is significant only when the functions
| ki| | i ki|
ϕi and δ ϕi are unbounded on X .
ki i ki i
Let
A= a + +a , Λ = λ1+ +λd.
1 ··· d 0 0 ··· 0
Then Λ is the smallest eigenvalue of L. We set
0
Nd, Λ > 0
Nd = 0
Λ Nd (0,...,0) , Λ = 0.
0
(cid:26) \{ }
and define
Πf = f,ϕ ϕ .
h kiL2 k
kX∈NdΛ
DIMENSION-FREE Lp ESTIMATES FOR VECTORS OF RIESZ TRANSFORMS 6
Then in the case Λ > 0 we have Π = I, while in the case Λ = 0 the operator Π is the
0 0
projection onto the orthogonal complement of the vector ϕ . The Riesz transforms
(0,...,0)
studied in this paper are formally of the form
R := δ L−1/2Π,
i i
while the rigorous definition of R is
i
−1/2
R f = λ f,ϕ δ ϕ (2.4)
i k h kiL2 i k
kX∈NdΛ
In many of the considered cases Π 0 so that R = δ L−1/2 and Nd = Nd.
≡ i i Λ
It was proved in [28, Proposition 1] that the vector of Riesz transforms
Rf = (R f,...,R f)
1 d
satisfies
Rf 6 f .
2→2 p
k k k k
The main goal of this paper is to prove similar estimates for p in place of 2. We aim at
these estimates being dimension-free and linear in p∗.
To state and prove our main results we need several auxiliary objects. Firstly, we let
d
= p ∂ . (2.5)
i i xi
d
That is, is the ’differential’ part of δ . In many (though not all) of our applications
i i
we will have q 0 and thus δ d . The formal adjoin of d on L2(X ,µ ) is
i i i i i i
≡ ≡
1
d∗f = ∂ p w f , f C∞(X ). (2.6)
i −w xi i i ∈ c i
i
A computation shows that L = d∗d +(cid:0) r , w(cid:1)hith
i i i i
w′
r = a + q2 p q′ p′q p q i . (2.7)
i i i − i i− i i− i iw
(cid:18) i(cid:19)
We shall also need
d n
L˜ := d∗d = L r, where r := r .
i i − i
i=1 i=1
X X
Then L˜ is the potential-free component of L and the potential r is a locally integrable
function on X. We assume that
d
there is a constant K > 0 such that q2(x ) K r(x), (A2)
i i ≤ ·
i=1
X
for almost every x X. In many of our examples we shall have q = 0 and thus (A2)
∈
holding with K = 0.
Next we define
M := δ∗δ +δ δ∗ = L+[δ ,δ∗]= L+v ,
i j j i i i i i
j6=i
X
cf.[28, (eq. 5.1)], and set
ci = δ ϕ −1,
k k i kk2
if δ ϕ = 0 and ci = 0 in the other case. Then ciδ ϕ (excluding those of ciδ ϕ
i k 6 k { k i k}k∈Nd k i k
which vanish) is an orthonormal system of eigenvectors of M such that M (ciδ ϕ )
i i k i k
equals λ ciδ ϕ .
k k i k
DIMENSION-FREE Lp ESTIMATES FOR VECTORS OF RIESZ TRANSFORMS 7
We denote
= lin ϕ : k Nd , = δ [ ] = lin δ ϕ : k Nd ,
k i i i k
D { ∈ } D D { ∈ }
and make the technical assumption that
both and , i = 1,...,d, are dense subspaces of Lp, 1 p < . (T3)
i
D D ≤ ∞
In most of our applications the condition (T3) will follow from [12, Lemma 7.5], which
is itself a consequence of [4, Theorem 5].
Lemma 1 ([12, Lemma 7.5]). Assume that ν is a measure on X such that, for some
ε> 0 we have
d
exp ε y dν(y) < .
i
| | ∞
ZX (cid:18) i=1 (cid:19)
X
Then, for each 1 6 p < , multivariable polynomials on are dense in Lp(X,ν).
∞
In what follows we consider the self-adjoint extension of M given by
i
M f = λ f,ci δ ϕ ci δ ϕ , (2.8)
i k k i k L2 k i k
kX∈Nd (cid:10) (cid:11)
on the domain
Dom(M ) = f L2: λ 2 f,ciδ ϕ 2 < .
i { ∈ | k| | k i k L2| ∞}
kX∈Nd (cid:10) (cid:11)
KeepingthesymbolM for thisself-adjoint extension is aslight abuseof notation, which
i
however will not lead to any confusion. Finally, we shall need the semigroups
Pt := e−tL1/2 and Qit := e−tMi1/2.
These are formally defined on L2 as
Ptf = e−tλ1k/2hf,ϕkiL2ϕk, Qitf = e−tλ1k/2 f,cikδiϕk L2cikδiϕk.
kX∈Nd kX∈Nd (cid:10) (cid:11)
Note that for t > 0 we have P [ ] and Qi[ ] , i = 1,...,d.
t D ⊆ D t Di ⊆ Di
3. General results for Riesz transforms
Recallthatweareinthesettingoftheprevioussection. Inparticulartheassumptions
(A1), (A2), and the technical assumptions (T1), (T2), (T3), are in force. The following
is the main result of our paper.
Theorem 2. For each 1< p < we have
∞
Rf 6 24(1+√K)(p∗ 1) Πf , f Lp. (3.1)
k kp − k kLp ∈
Remark. InalltheexamplesweconsiderinSection5theprojectionΠsatisfies Π 6
p→p
k k
2, 16 p 6 . In fact in many of the examples Π equals the identity operator.
∞
For the proof of Theorem 2 we need two ingredients. The first of these ingredients
is a bilinear formula that relates the Riesz transform with an integral in which both P
t
and Qi are present.
t
Proposition 3. Let i= 1,...,d. Then the formula
∞
R f,g = 4 δ P Πf,∂ Qig tdt, (3.2)
h i iL2 − i t t t L2
Z0
holds for f and g . (cid:10) (cid:11)
i
∈ D ∈ D
DIMENSION-FREE Lp ESTIMATES FOR VECTORS OF RIESZ TRANSFORMS 8
Before proving the proposition let us make two remarks.
Remark 1. Formulas similar to (3.2) were proved before, though, depending on the
context,theymayhaveinvolvedspectralmultipliersoftheoperatorL. However,treating
these spectral multipliers appropriately was achieved with variable success. A way of
avoiding multipliers was first devised in [6] for Riesz transforms on manifolds. In such
a setting, the above formula is a special case of the identity (3) there. The approach
in [6] was adapted in [20] to the case of Hodge-Laguerre operators. In the case of
Laguerre polynomial expansions (see Section 5.2) the formula (3.2) is a special case of
[20, eq. (5.1)]. We note that both in [6] and [20] the authors needed to consider the
Riesz transform as well as formula (3.2) for differential forms; this is not needed in our
approach.
Remark 2. Note that if the operators δ and δ∗ commute, then Qi = P and the formula
i i t t
(3.2)can beformally obtained viathespectraltheorem. Theproblemis thatoften these
operators do not commute. A way to overcome this non-commutativity problem was
devised by Nowak and Stempak in [30]. They introduced a symmetrization T of δ that
i i
does commute with its adjoint, i.e. satisfies T∗ = T . These symmetrization is defined
i − i
on L2(X˜), where
X˜ = (X ( X )) (X ( X )).
1 1 d d
∪ − ×···× × −
Set T = n T2 and let S = e−tT1/2. The formula (3.2) for T is then formally
− i=1 i t i
∞
P T T−1/2f,g = 4 T S f,∂ S g tdt. (3.3)
D i EL2(X˜) − Z0 h i t t t iL2(X˜)
This leads to a proof of (3.2) different then the one presented in our paper. Namely,
a computation shows that applying (3.3) to functions f: X˜ R and g: X˜ R which
→ →
are both even in all the variables we arrive at (3.2).
Proof of Proposition 3. We start with proving (3.2) for f = ϕ and g = δ ϕ , with some
k i n
k,n Nd. If k = 0 and Λ = 0 then both sides of (3.2) vanish. Thus we can assume
0
∈
that λ > 0. A computation shows that
k
δ L−1/2f,g = λ−1/2 δ f,g
i L2 k h i iL2
D E
and
∞ ∞
−4Z0 DδiPtf,∂tQ(tj)gEL2 tdt = −4Z0 D∞e−tλ1k/2δif,−λ1n/2e−tλ1n/2gEL2tdt
= 4λ1n/2 e−t(λ1k/2+λ1n/2)tdt·hδif,giL2
Z0
1/2
4λ
n
= δ f,g ,
(λ1/2+λ1/2)2 ·h i iL2
k n
hence
∞
δ L−1/2f,g +4 δ P f,∂ Q(j)g tdt
i i t t t
D E Z0 D EL2 (3.4)
1/2
4λ
−1/2 n
= λ δ f,g .
k − (λ1/2 +λ1/2)2!·h i iL2
k n
Now δ f is also an L2 eigenvector for M corresponding to the eigenvalue λ . Conse-
i i k
quently, since eigenspaces for M corresponding to different eigenvalues are orthogonal,
i
DIMENSION-FREE Lp ESTIMATES FOR VECTORS OF RIESZ TRANSFORMS 9
δ f,g is nonzero only if λ = λ . Coming back to (3.4) we obtain (3.2) for f = ϕ and
i n k k
h i
g = δ ϕ .
i n
Finally, by linearity (3.2) holds also for f and g . (cid:3)
i
∈ D ∈ D
The second ingredient we need to prove Theorem 2 is a bilinear embedding, as was
thecase in [6,9,10, 19]. For N N(the cases interesting to usbeingN = 1 andN = d)
we take F = (f ,...,f ): X ∈(0, ) RN and set
1 N
× ∞ →
d N d
F 2 := r F 2+ ∂ F 2+ d F 2 = r F 2+ ∂ F 2+ d F 2 . (3.5)
| |∗ | | | t | | i | | | | t k| | i k|
Xi=1 Xk=1(cid:16) Xi=1 (cid:17)
The absolute values in (3.5) denote the Euclidean norm on RN. Below we only state
|·|
our bilinear embedding. The proof of it is presented in the next section.
Theorem 4. Let f : X R and g = (g ,...,g ) : Xd Rd and assume that f
1 d
→ → ∈ D
and g , for i = 1,...,d. Denote
i i
∈ D
F(x,t) = P Πf(x) and G(x,t) = Q g = Q1g ,...,Qdg .
t t t 1 t d
Then
(cid:0) (cid:1)
∞
F(x,t) G(x,t) dµ(x)tdt 6 6(p∗ 1) Πf g . (3.6)
∗ ∗ p q
| | | | − k k k k
Z0 ZX
Remark. The theorem can be slightly generalized, at least at a formal level. Namely in
Theorem 4, we do not need that v = [δ ,δ∗]. It is enough to have any v 0 and take
j j j i ≥
Qt = e−tMi with Mi = L+vi.
Our main theorem is an immediate corollary of Proposition 3 and Theorem 4.
Proof of Theorem 2. Itis enoughtoprove thatforeach f Lp andg Lq,i = 1,...,d,
i
∈ ∈
the absolute value of d R f,g does not exceed
i=1h i ii
P6(4+√K)(p∗ 1) Πf g 2 1/2 .
p i
− k k | |
(cid:13) i (cid:13)q
(cid:13)(cid:0)X (cid:1) (cid:13)
A density argument based on the assumption T(cid:13)3 allows us to(cid:13)take f and g ,
(cid:13) (cid:13) ∈ D i ∈ Di
i= 1,...,d. From Proposition 3 we have
1 ∞ ∞
R f,g = d P Πf,∂ Qig tdt+ q P Πf,∂ Qig tdt
−4h i iiL2 i t t t i L2 i t t t i L2
Z0 Z0
and thus, assumption (A2) g(cid:10)ives (cid:11) (cid:10) (cid:11)
d
R f,g
(cid:12) h i iiL2(cid:12)
(cid:12)Xi=1 (cid:12)
(cid:12) (cid:12)
(cid:12) ∞ (cid:12) d 1/2
(cid:12)6 4 √(cid:12)K r(x)P Πf(x) + d P Πf(x)2 G(x,t) dµ(x)
t i t ∗
| | | | | |
Z0 ZX (cid:18)i=1 (cid:19) !
p X
∞
4(√K +1) F(x,t) G(x,t) dµ(x)
∗ ∗
≤ | | | |
Z0 ZX
Now, Theorem 4 completes the proof. (cid:3)
4. Bilinear embedding theorem
This section is devoted to the proof of our embedding theorem - Theorem 4. We shall
follow closely the resoning from [6] and [20].
DIMENSION-FREE Lp ESTIMATES FOR VECTORS OF RIESZ TRANSFORMS 10
4.1. The Bellman function. Before proceeding to the proof of Theorem 4 we need to
introduce its most important ingredient: the Bellman function.
Choose p > 2. Let q =p/(p 1) and
−
q(q 1)
γ = γ(p) = −
8
and define β : [0, )2 [0, ] by
p
∞ → ∞
s2s2−q ; sp 6 sq
1 2 1 2
p q
βp(s1,s2)= s1+s2+γ 2 sp+ 2 1 sq ; sp > sq. (4.1)
 p 1 q − 2 1 2
(cid:18) (cid:19)
For m = (m ,m ) N2 the Nazarov-Treil Bellman function corresponding to p,m is
1 2 
∈
the function
B = B : Rm1 Rm2 [0, )
p,m
× → ∞
given, for any ζ Rm1 and η Rm2, by
∈ ∈
1
B (ζ,η) = β (ζ , η ). (4.2)
p,m p
2 | | | |
The function B bears its origins in the article [22] by F. Nazarov and S. Treil. It was
employed (and simplified) in [5, 6, 8, 9, 10]. Note that B is C1(Rm1+m2) and is C2
everywhere except on the set
(ζ,η) Rm1 Rm2: η = 0 or ζ p = η q .
{ ∈ × | | | | }
To remedy the non-smoothness of B we consider the regularization
B = B := B ψ ,
κ,p,m κ ∗Rm1+m2 κ
where
− 1 1
ψ(x) = cme 1−|x|2χBm1+m2(x) and ψκ(x) = κm1+m2ψ(x/κ),
with c such that ψ (x)dx = 1. Since both B and ψ are bi-radial also B is
m Rm1+m2 κ κ κ
bi-radial. Hence, there is β = β acting from [0, ]2 to R such that
R κ κ,p ∞
1
B (ζ,η) = β (ζ , η ), ζ Rm1,η Rm2.
κ κ
2 | | | | ∈ ∈
We shall need some properties of B that were essentially proved in [6], [9], and [19],
κ
[20].
Proposition 5. Let κ (0,1). Then, for s > 0, i = 1,2, we have
i
∈
(i) 0 6 β (s ,s ) 6 (1+γ(p))((s +κ)p+(s +κ)q),
κ 1 2 1 2
(ii) 0 6 ∂ β (s) 6 C max((s +κ)p−1),s +κ) and 0 6 ∂ β (s) 6 C (s +κ)q−1,
s1 κ p 1 2 s2 κ p 2
with C being a positive constant.
p
Moreover, B C∞(Rm1+m2), and for any ξ = (ζ,η) Rm1+m2 there exists a positive
κ
τ =τ (ζ , η )∈such that for ω = (ω ,ω ) Rm1+m2 w∈e have
κ κ 1 2
| | | | ∈
(iii) Hess(B )(ξ)ω,ω > γ(p) τ ω 2+τ−1 ω 2 .
h κ i 2 κ| 1| κ | 2|
(iv) ( B )(ξ),ξ > γ(p) τ ζ 2+τ−1 η 2 κE (ξ).
h ∇ κ i 2 κ| (cid:0)| κ | | − κ(cid:1)
(v) The function E (ξ) is continuous on Rd+1 and satisfies
κ (cid:0) (cid:1)
lim E (ξ)= 0, ξ Rd+1, E (ξ) C (ζ p−1+ η + η q−1+κq−1).
κ κ d,p
κ→0+ ∈ | | ≤ | | | | | |