Rev. Acad. Canar. Cienc., XX (Núms. 1-2), 9-23 (2008) (publicado en septiembre de 2009)
SOME QUESTIONS RELATED TO FOURIER-BESSEL
EXPANSIONS
J. J. Betancor
ABSTRACT. In this note we present sorne questions related to Fourier-Bessel expansions that,
as far as we know, have not been studied. These questions are presented in two groups.
The first part is concerned to harmonic analysis and the second one refers to holomorphic
extensions of functions defined in (O, 1) and the image of LP(O, 1), 1 < p < oo, under the
Bessel heat semigroups on (O, 1).
l. INTRODUCTION
Assume that v > -1. By J,_, we denote the Bessel function of the first kind and order v.
For every n EN, >.n,v represents a positive zero of the function J,,,(z). Moreover, the sequence
{>.n,v}nEN is increasing. If, for every n EN, we define the function 'lf;~ by
where dn,v = vf2¡>.:/,~J11+1(>.n,v)l- 1 , the sequence {'lf;~}nEN is a complete and orthonormal
system in L2 (0, 1).
If f is a measurable function defined in (O, 1) we call the Fourier-Bessel expansion associated
with f to the series
00
L, <f,'l/J~>'l/J~,
n=O
where < ., . > denotes the inner product in L2 (0, 1), provided that the n-th Fourier-Bessel
coefficient < f, '1/1;. > of. f exists, for every n E N. In [59, Chapter XVII] we can encounter an
exhaustive study of Fourier-Bessel expansions.
In this note we present sorne questions related to Fourier-Bessel expansions. As far as we
know these questions have not been studied. We motívate each of the questions with a short
comment and with the suitable references. In Section 2 we present sorne problems related to
The author is partially supported by MTM2007/65609
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harmonic analysis operators associated with Fourier-Bessel expansions. The questions presented
in Section 3 are concerned to the holomorphic extension of functions defined in (O, 1)
and to the image of V'(O , 1), 1 < p < oo, under the Bessel heat semigroups on (O, 1).
The author will be very glad to discuss these and other questions about Fourier-Bessel
expansions with any interested reader.
2. HARMONIC ANALYSIS AND FOURIER-BESSEL EXPANSIONS
Harmonic analysis operators (multipliers, transplantation, Riesz transforms, LittlewoodPaley
g-functions, ... ) in the Fourier Bessel setting have been investigated recently by Ciaurri
and Stempak ([21], [22] and [23]) , Ciaurri and Roncal ([20]) and Roncal ([54]). We can say
that these studies have their starting point in [51].
The Bessel operator L,,, = - dd2
2 - 114;"2 can be factorized as follows X X
where ó,,, = --/fx + v+;12 and oi is the (formal) adjoint of ó,,, in L2 (0, 1). Following Stein's ideas
([56]) this fact suggests to define the Riesz transform associated with the Bessel operator L,,,
by
where the operator r;; 1!2 is defined by
00 1
L--;; 1! 2 f = L :X.- < f, 'ljJ~ > 'ljJ~, f E L2 (0, 1).
n=O n,v
The precise definition of R,,, in L2 (0 , 1) can be found in [22, p. 216-217]. In [22, Theorem 1]
weighted V'-boundedness properties of the Riesz transform R,,, were established. In particular,
Ciaurri and Stempak ([22]) proved that, for every 1 S p < oo, R,,, can be extended to LP(O, 1)
as a bounded operator from V'(O , 1) into itself, when 1 < p < oo, and from L1 (0, 1) into
L1•00 (0, 1). In order to show this result, Calderón-Zygmund singular integral theory is used.
In the first question, we purpose to get a pointwise representation for R,,, as a principal
value integral operator.
Q.1.- Is the following pointwise representation true?: For every f E V'(O, 1), 1 S p < oo,
(1) R,,,(f)(x) = lim f 1 R,,,(x,y)f(y)dy, a.e. x E (0,1),
E->O+ lo, lx-yl>E
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where
00
Rv(x, y)= L 1/i~(y)cp~(x), x, y E (O, 1),
n=O
and
Inspired by the techniques used in [14] and [15], where pointwise representations as principal
value integral operators for Riesz transforms in the Bessel on (O, oo) and ultraspherical settings
are obtained, we think that the kernel Rv(x, y) must be seen as a perturbation of the kernel
for the classical Riesz transform on the torus. Note that it is sufficient to establish (1) for
every f E Cg'°(O, 1) because from the results established in [22] it can be deduced that the
maximal operator
R~(f)(x) = sup 1 f 1 Rv(x, y)J(y)dyl
€>0 Ío, lx-yl>g
is bounded from LP(O, 1) into itself, for every 1 < p < oo, and from L1(0, 1) into L1,00 (0, 1).
Suppose now that B is a Banach space. If 1 :::; p < oo, we denote by L~(JR) the space
constituted by all the strongly measurable B-valued function f defined on lR such that
ll!llL~(IR) = (k_11J(x)ll~dx)11p <oo.
The weak L~(JR)-space, that is Li¡00 (1R), is defined in the natural way for 1 :::; p < oo.
By H we represent as usual the Hilbert transform given by
H(f)(x) = lim { f(y) dy, a.e. x E JR,
g-;Q+ Í ¡x-yl>ó X - y
for every f E LP(JR) and 1 :S: p < oo.
If 1 :S: p < oo, n E N, f; E LP(JR) and b; E B, i = 1, ... , n, we define
n
Hf = Lb;Hf;,
i=l
where f = L;~ 1 b;f;. When f takes this form we say that f E LP(JR) ®B.
The notion of UMD Banach space was introduced by Burkholder ([18]) in a probabilistic
context. For instance, all LP(μ)-spaces where 1 < p < oo or all reflexive Orlicz spaces are
UMD spaces ([31]). UMD spaces were characterized by using Hilbert transform (see [17] and
[18]).
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Theorem l. {(17] and [18]) Let B be a Banach space. The following conditions are equivalent.
(i) B is a UMD Banach space.
(ii) The Hilbert transform H can be extended to Lj¡(JR) as a bounded operator from Lj¡(JR)
into itself, far some (far all) 1 < p < oo.
(iii) The Hilbert transform H can be extended to Lk(R) as a bounded operator from Lk(R)
into L ii00 (JR).
This result connects harmonic analysis with the geometry of Banach spaces (see [17]).
Recently, other singular integrals different to the Hilbert transform have been used to describe
the UMD property. Versions of Theorem 1 where the Hilbert transform is replaced by the
Riesz transform associated with
• the Ornstein-Uhlenbeck operator ([34]),
• the harmonic oscillator operator ([1]),
• Bessel operators on (O, oo) ([11])
have been established.
In this point it is natural to make the following question:
Q.2.- Can the UMD property of a Banach space B be characterized in terms of the Lj¡(O, 1)boundedness
of the Riesz transform Rv associated with the orthonormal sequence { 1/J~}nE N? .
In arder to answer this question we suggest to analyze the kernel of the Riesz transform
Rv in the local region, that is, clase to the diagonal D = {(x , x) : x E (O, l)}. We can
observe that in the proof of [22, Theorem 1.1] the asymptotic expansion of the Bessel function
Jv far big values of the argument ([22, (2.2)]) is a key point. We think that the first term
of that asymptotic expansions will allow to compare the Riesz transform Rv with the usual
conjugation operator, and then, we will can connect the Lj¡-boundedness properties of Rv
with the UMD property of the Banach space B.
Konig [38] and Konig and Nielsen [39] investigated vector valued Lagrange interpolation
and mean convergence of Hermite and Jacobi series.
We denote by {p~'¡3 }nEN the sequence ofnormalized Jacobi polynomials in L2((-1, 1), dwa,¡3),
where dwa,¡3(x) = (1-x)ª(l+x)i3 anda, /3 > -1 ([57]). The followingvector valued expansion
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theorem for Jacobi polynomials generalizes the classical result of Pollard [52] and Muckenhoupt
[50].
Theorem 2. {[39, Theorem 2]) Let B be a Banach space, 1 < p < oo and a, f3 > - l. Then,
the following assertions are equivalent.
{i) For all f E Lii((-1,1),dwa,¡J ),
n L < f , P~'¡J > L1i((-1, l) ,dw0 ,J3) P~'¡J ---> f , as n ---> oo,
k=O
in Lii((-1, 1), dwa,¡J).
{ii) Bisan UMD space and m(a, (3) < p < m(a, (3)', where m(a, (3) = max{l, 4~~!;), 4~~!~)}
and m( a, (3)' is the conjugate of m( a, (3).
The characterization of UMD spaces corresponding to Theorem 2 for Hermite was proved
in [38]. In [38, Theorem 3], the mean convergence due to Askey and Wainger [2] for the
expansions of the functions in V'(IR), 4/3 < p < 4, into Hermite functions was extended to
vector valued setting for UMD Banach spaces.
Mean convergence results for Fourier-Bessel expansions can be found in [4] and [5]. As far
as we know a version of Theorem 2 for the system of Bessel functions { ?f ~}nEN has not been
studied.
Q.3.- To characterize the UMD Banach spaces in terms of the mean convergence inL1I:i(O, 1)
of the expansions associated with the sequence {1/i~}nEN·
The Poisson semigroup {Pt"}t2-:o associated with the orthonormal sequence {1/i~}nEN is defined
by
00
P((f) = L e-t>.n,v < f, 1fl:, > 1/1~, f E L2(0, 1),
n=O
for every t 2: O.
In [20] Ciaurri and Roncal studied Littlewood-Paley g-functions for the Poisson semigroup
{ Pt}t2':0 defined as usual by
( roo 1 EJk 12dt)1/2 g~ (f)(x) = Jo tk EJtk P((f)(x) t , k EN and x E (0, 1).
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In [20] it was proved that, for every k E N and 1 < p < oo, there exists C > O such that
bll!llLP(0,1) :S: llgk(f)lb(o,1) :S: CllJllLP(0,1), f E IJ'(O, 1).
These inequalities have been showed by using vector valued Calderón-Zygrnund theory developed
by Rubio de Francia, Ruiz and Torrea [55].
If { Pth2'.o denotes the classical Poisson sernigroup on the torus T we consider the Littlewood
Paley g-function of the first arder given by
( roo 1 {) 12dt)1/2 g(f)(x)= Jo t 8tPt(f)(x) t , xET.
It is well-known that for every 1 < p < 00 there exists e> o for which
(2)
1
z;ll!llLP(T) ::::: llg(f)llLP(T) ::::: Cllflb(T)• f E IJ'(T).
Kwapien [40] established that the two inequalities in (2) hold for every f E L1J0 (T) and for all
(for sorne) 1 < p < oo if, and only if, B is isornorphic to a Hilbert space.
The following question can be studied.
Q.4.- Are the following assertions equivalent for a Banach space B ?,
(i) B is isornorphic to a Hilbert space.
(ii) For every (equivalently, for sorne) 1 < p < oo, there exists C >O such that
Xu ([60]) considered the generalized 9q, q > 1, Littlewood-Paley functions associated with
{ Pt}t2'.0 defined by
( roo 1 {) lq dt) l /q
gq(f)(x)= Jo t 8tP1(f)(x) t , xET,
for every q > 1, and he introduced the notions of Lusin type and cotype of a Banach space.
Let B be a Banach space. If f is a B-valued function defined on T it defines
gq(f)(x) = ( Jroo o 11 t f{J) tPt(f)(x) llqB td t) 1/q, X ET,
for every 1 < q < oo. If q 2: 2, B has the Lusin cotype q when, for sorne 1 < p < oo, there
exists e > o such that
(3) llgq(f)llLP(0,1)::::: CllJllL~(0, 1)> f E L'~(O, 1).
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If (3) holds for sorne 1 < p < oo, then (3) holds for every 1 < p < oo, possibly for other
constants e.
If 1 < q :S 2, B has the Lusin type q when, for sorne 1 < p < oo, there exists C > O such
that
(4) llJllL~(O,l) :S Cllgq(f)llLP(O,l) l f E L1IJ(O, 1).
If ( 4) is true for sorne 1 < p < oo, then ( 4) is also true for every 1 < p < oo, where the
constant e > o depends on p.
With this notation the Kwapien's result can be rewritten in the following way: A Banach
space Bis isornorphic to a Hilbert space if, and only if, B has Lusin type 2 and Lusin cotype
2.
The rnartingale type and cotype of a Banach space B were studied by Bourgain ([17]).
These properties are related to the geornetry of the Banach spaces ( convexity and srnoothness
properties) . Xu ([60]) established that a Banach space B has Lusin type (respectively, Lusin
cotype) q, if and only if, B has rnartingale type (respectively, rnartingale cotype) q.
Lusin type and Lusin cotype (equivalently, rnartingale type and cotype) of Banach spaces
have been characterized by using Littlewood-Paley g-functions associated with Poisson sernigroup
for
• the Ornstein-Uhlenbeck operator ([34] and [43]) ,
• Laguerre operators ([13]),
• Bessel operators in (O,oo) ([11]) .
In [43] Martínez, Torrea and Xu have studied Lusin type and cotype of a Banach space
by using gq-functions for Poisson sernigroups that are subordinated to general diffusion sernigroups.
If we define the generalized g~,q' 1 < q < oo, Littlewood-Paley functions for the Poisson
sernigroup {Pf'}t::::o on B-valued functions in the obvious way, we can consider the next
question
Q.5.- If B is a Banach space, then is the following property true?,
B has the Lusin type q with 1 < q :S 2 (respectively, Lusin cotype q with 2 :S q < oo) if,
and only if, for sorne (equivalently, for every) 1 < p < oo, there exists C > O such that
llJllL~(O,l) :S Cll9~,q (f)llLP(O, l)l f E L1IJ(O, 1).
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( respectively,
119~,q(f)llLP(O,l) s; CllJllL~(O,l)l f E Lj¡(O, 1).).
In order to deal the questions Q.4 and Q.5 an idea is to show that 9~,q and 9q differ, at
least and in acertain sense in a neighborhood of the diagonal, by a bounded operator from
Lj¡(O, 1) into LP(O, 1). To get this property we must use the asymptotic representation for t he
Bessel function Jv when the argument is large ([22, (2.2)]).
The heat semigroup {Wl}t2'.0 for { 7/l~ }nEN is given by
00
W((f) = L e->.~,vt < f, 'I/;~ > 1/;~, f E L2(0, 1).
n=O
The maximal operator W,!' for { 1/;~ }nEN is given by
W,'.'(f) = sup IW((f)I.
t>O
LP-boundedness properties for W,!' were obtained in [32]. Motivated by the results obtained
by Macías, Segovia and Torrea ([41] and [42]) and Nowak and Sjogren ([48]) we think t hat the
results established in [32] can be improved studying weak and restricted weak LP-boundedness
with power weights.
Q.6.- To describe the pencil phenomenon for the power weighted LP-boundedness of the
maximal operator W,!'.
Dziubanski and Zienkiewicz ([28], [29] and [30]) have investigated Hardy spaces associated
with Schrodinger operators L = - !:!. +V, where the potential V satisfies, for instance, sorne
kind of converse HOider inequality. Also, Dziubanski ([24] and [26]) studied Hardy spaces for
Laguerre operators. In the Bessel setting on (O, oo) Hardy spaces were described by using
maximal operators and Riesz transforms in [10] .
Assume that v > - 1/2. W,!' is bounded from LP(O, 1) into itself, for every 1 < p < oo, and
from L1(0, 1) into L1,00 (0, 1). We can define the Hardy space H~ associated with {1/;~}nEN as
follows
H~ = {f E L1(0, 1) : W,'.'(j) E L1(0, 1)}.
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Classical results and the ones presented in [25] and [10] for Hardy spaces in the Bessel setting
on (O, oo) justify the study of the space H~. Also, Miyachi's papers ([44],[45] and [46]) are
remarkable in this context.
Q.7.- To characterize the Hardy space H~ by using atomic representations, Riesz transforms
and Littlewood-P.aley g-functions.
Dziubanski, Garrigós, Martínez, Torrea and Zienkiewicz [27] defined the subspace BMOL(JRn)
of the space BMO(JRn) of bounded mean oscillation functions in JRn . BMOL(JRn) is the dual
space of the Hardy space associated with the Schri:idinger operator L defined in [28]. In [27] it
is investigated the behaviour of maximal operators, fractional integrals and Littlewood-Paley
functions on BMOL(JRn). Bongioanni, Harboure and Salinas [16] have studied the Riesz
transform in the Schri:idinger setting on B MOL (JRn).
Recently, Betancor, Chicco Ruiz, Fariña and Rodríguez-Mesa ([8] and [9]) have describe the
dual of the Hardy space in the Bessel setting on (O, oo) and they have analyzed the behavior
of the harmonic analysis operators (maximal operators, Riesz transforms and g-functions) on
the new BMO type space.
Hence, when the last question Q. 7 is sol ved, the following problem can be dealt.
Q.8.- To describe the dual space of H~ as a certain class of functions with bounded mean
oscillation on (O, 1) and then, to study the behavior of the harmonic analysis operators on the
new space.
Nowak and Stempak [49] and Nowak and Sji:igren [47] sorne investigated sorne aspects of
the harmonic analysis for multidimensional Laguerre expansions. In [19], Nowak, Ciaurri and
Stempak studied LP-boundedness properties of transplantation operators in the n-dimensional
Jacobi setting. Betancor, Castro and Curbelo ([6] and [7]) have analyzed LP-boundedness
properties for maximal operators, Riesz transforms, multipliers of Laplace transform type,
and Littlewood-Paley g-functions for the multidimensional Bessel operators on (O, oor. The
multidimensional harmonic analysis for the Bessel operators on (O, l)n has not been studied.
Q.9.- To developed the harmonic analysis in the multidimensional Bessel setting on (O, 1r.
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3. HOLOMORPHIC EXTENSION AND FOURIER-BESSEL EXPANSIONS
In [37] Karp deals with the following problem: to find necessary and sufficient conditions
to impose to a function f for its Fourier coefficients {fk}kEN in terms of classical orthogonal
polynomials to satisfy the inequality
00
(5) L lfkl2Bk < oo,
k=O
with B > l. Karp ([37]) considered this problem for the system of Hermite, Laguerre and
Jacobi polynomials. In [37] we also can find a big list of references about this question.
Suppose that B > l. We represent by Eo the ellipse
Eo = {z E <C: Jz - lJ + Jz + ll < 0112 + e-112}.
By 8Eo we denote the boundary of E9. The weighted Szego space AL2(8Eo) consists of ali
those holomorphic functions f in Eo such that f has nontangential boundary values for almost
every point in 8Eo and that
r lf(z)l2 ldzl <OO. ÍaE8
In the Jacobi setting the Karp's results are the following.
Theorem 3. {[37, Theorem 4]) Let a, (3 ;::: - 1/2 and B > - l. Assume that f is a measurnble
function on (-1, 1). Far every k EN, fk denotes the k-th Fourier (a, (3) -Jacobi coefficient of
f. Then, the f ollowing assertions are equivalent.
{i) 2:~1 lfkl2Bk < oo,
{ii) f is the restriction to the interval ( -1, 1) of a function in AL2 ( 8Eo).
Corollary l. {[37, Corollary 4.1]) Let a, (3 ;::: -1/2 and B > - l. Assume that f is a measurable
function on (-1, 1). Far every k EN, fk denotes the k-th Fourier (a, (3) -Jacobi coefficient
off. Then, f is the restriction to (-1, 1) of an entire function if and only if the condition {i)
in Theorem 3 holds far every B > l.
This problem has not been studied for Bessel expansions.
Q.10.- To describe, in terms of holomorphy, the class of measurable functions f on (O, 1)
that satisfy (5) for sorne or for ali B > 1, when, for every k EN, Ík =< f , 1/J'k >.
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In [36] Karp established other type of results about analytic continuation of functions defined
on a finite interval by using Chebyshev polynomials.
It is well known that the Segal-Bargmann transform ([3]) is an isomorphism from L2 (JRn)
into the Fock space, in other words, for every t > O, the mapping
is an isometry between the corresponding Hilbert spaces. As usual { e-tb. }t;:-:o represents the
heat semigroup associated to the Euclidean Laplacian and O(en) denotes the space of entire
functions in en. Bargmann's results have been extended to other contexts. In particular, in
[58] Thangavelu characterizes the image of L2 by the heat semigroup defined by the Hermite
and Laguerre expansions as weighted Bergman spaces.
We consider, for every t >O, the function
Ut(X + iy) = 2n(sinh(4t))-nl2exp(tanh(2t)lxl2 - coth(2t)lyl2), x, y E JRn.
The space Ht(en) , t > O, is constituted by ali the entire functions Fon en such that
llFll1, = r IF(x + yi)l 2Ut(X + iy)dxdy <OO.
} JR2n
Thangavelu proved the following result.
Theorem 4. n58, Theorem 3.1] Let t > O. An entire function F on en belongs to Ht(en)
if and only if F =cm¡, for some f E L2 (JRn). Moreover, if F = e-m¡, then ll/llL2(1Rn) =
llFllH,. Here, IHr denotes the harmonic oscillator or Hermite operator.
In [58, Theorem 3.6] it characterizes the image of L2 (0, oo) by the heat semigroup for
Laguerre operators in terms of certain weighted Bergman spaces. The corresponding result
for the heat semigroup associated with Bessel operators on (O, oo) can be found in [58, Theorem
3.5].
Q.11.- To characterize the image of L2 (0, 1) under the heat semigroup W[, t > O, defined
by the sequence { 'l/i~}nEN in terms of weighted Bergman spaces.
Hall [33] investigated the problem of describing the image of V'(JRn) under the classical heat
semigroup ctb. , t > O, for every 1 < p < oo. Recently, Radha and Venku Naidu [53] have
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studied this problem when { e-tt>. h2':o is replaced by the Hermite polynomial heat semigroup
( Ornstein-Uhlenbeck semigroup).
Q.12.- To define the weighted Bergman type space that is the image of LP(O, 1) by the heat
semigroup W(, t >O, associated with the system {7/i~}nEN·
REFERENCES
[1] l. Abu-Falaha and J.L. Torrea, Hermite function expansions versus Hermite polynomial expansions, Glasgow
Math. J., 48 (2) (2006), 203-215.
[2] R. Askey and S. Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math.,
87 (1965), 695-708.
[3] V. Bargmann, On a Hilbert space of analytic functions andan associated integral transform, Comm. Pure
and Applied Mathematics, 14 (1961), 187-214.
[4] A. Benedek and R. Panzone, Note on mean convergence of eigenfunctions expansions, Rev. Un. Mat.
Argentina, 25 (1970), 167-184.
[5] A. Benedek and R. Panzone, On mean convergence of Fourier-Bessel series of negative order, Studies in
Appl. Math., 50 (1971), 281-292.
[6] J.J. Betancor, A. Castro and J. Curbelo, Harmonic analysis operators associated with multidimensional
Bessel operators, preprint, 2009.
[7] J.J. Betancor, A. Castro and J. Curbelo, Multidimesional Hankel multipliers of Laplace transform type,
preprint, 2009.
[8] J.J. Betancor, A. Chicco Ruiz, J.C. Fariña and L. Rodríguez-Mesa, Odd BMO(IR) functions and Carleson
measures in the Bessel setting, Int. Equat. Oper. Th. (to appear).
[9] J.J. Betancor, A. Chicco Ruiz, J.C. Fariña and L. Rodríguez-Mesa, Maximal operators, Riesz transforms
and Littlewood-Paley functions associated with Bessel operators on BMO, J. Math. Anal. Appl. (to appear).
[10] J.J. Betancor, J. Dziubanski and J.L. Torrea, On Hardy spaces associated with Bessel operators, J. Anal.
Math., 107 (2009), 195-219.
[11] J.J. Betancor, J.C. Fariña, M. Martínez and J.L. Torrea, Riesz transform and g-function associated with
Bessel operators and their appropriate Banach spaces, Israel J. Math., 157 (2007) , 259-282.
[12] J.J. Betancor, J.C. Fariña, L. Rodríguez and A. Sanabria, Higher order Riesz transforms for Laguerre
expansions, preprint, 2009.
[13] J.J. Betancor, J.C. Fariña, L. Rodríguez, A. Sanabria and J.L. Torrea, Lusin type and cotype for Laguerre
g-functions,Israel J. Math. (to appear).
[14] J.J. Betancor, J.C. Fariña, M. Martínez and L. Rodríguez-Mesa, Higher order Riesz transforms associated
with Bessel operators, Ark. Mat., 46 (2) (2008), 219-250.
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[15] J.J. Betancor, J.C. Fariña, L. Rodríguez-Mesa and R. Testoni, Higher order Riesz transforms in the
ultraspherical setting as principal value integral operators, preprint, 2009.
[16] B. Bongioanni, E. Harboure and O. Salinas, Riesz transforms related to Schrodinger operators acting on
BMO type spaces, J. Math. Anal. Appl., 357 (1) (2009), 115-131.
[17] J. Bourgain, Sorne rernarks on Banach spaces in which rnartingale difference sequences are unconditional,
Ark. Mat., 21 (1983), 163-168.
[18] D.L. Burkholder, A geornetrical condition that irnplies the existence of certain singular integrals of Banach
valued functions, Proc. Conf. Harm. Anal., Univ. of Chicago, 271-286, 1981.
[19] O. Ciaurri, A. Nowak and K. Stempak, Jacobi transplantation revisited, Math. Z. , 257 (2) (2007), 355-380.
[20] O. Ciaurri and L. Roncal, Littlewood-Paley-Stein gk-functions far Fourier-Bessel expansions, preprint,
2009.
[21] O. Ciaurri and K. Stempak, Transplantation and rnultiplier theorems far Fourier-Bessel expansions, Trans.
Amer. Math. Soc., 358 (10) (2006), 4441-4465.
[22] O. Ciaurri and K. Stempak, Conjugacy far Fourier-Bessel expansions, Studia Mathematica, 176 (3) (2006),
215-247.
[23] O. Ciaurri and K. Stempak, Weighted transplantationfor Fourier-Bessel series, J. Anal. Math., 100 (2006) ,
133-156.
[24] J. Dziubanski, Hardy spaces associated with sernigroups generated by Bessel potentials, Houston J. Math.,
34 (1) (2008), 205-234.
[25] J . Dziubanski, Hardy spaces far Laguerre expansions, Constr. Approx., 27 (3) (2008), 269-287.
[26] J . Dziubanski, Atornic decornposition of Hardy spaces associated with certain Laguerre expansions, J.
Fourier Anal. Appl., 15 (2) (2009), 129-152.
[27] J. Dziubanski, G. Garrigós, M. Martínez, J.L. Torrea and J. Zienkienwicz, BMO spaces related to
Schrodinger operators with potentials satisfying a reverse Holder inequality, Math. Z., 249 (2) (2005),
329-356.
[28] J. Dziubanski and J. Zienkienwicz, Hardy space H 1 associated to Schrodinger operator with potential
satisfying reverse Holder inequality, Rev. Mat. Iberoamericana, 15 (2) (1999), 279-296.
[29] J. Dziubanski and J. Zienkienwicz, HP spaces associated with Schrodinger operators with potential frorn
reverse Holder classes, Coll. Math., 98 (1) (2003), 5-38.
[30] J. Dziubanski and J . Zienkienwicz, Hardy spaces H 1 for Schrodinger with certain potentials, Studia Math. ,
164 (1) (2004), 39-53.
[31] D.L. Fernández and J.B. García, Interpolation of Orlicz-valued function spaces and UMD property, Studia
Math., 99 (1991), 23-40.
[32] J.E. Gilbert, Maxirnal theorerns for sorne orthogonal series, ! , Trans. Amer. Math. Soc., 145 (1969) , 495-
515.
[33] B.C. Hall, Bounds on the Segal-Bargrnann transforms of LP functions, J. Fourier Anal. Appl., 7 (6) (2001),
553-569.
21
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
[34] E. Harboure, J.L. Torrea and B. Viviani, Vector valued extensions of operators related to OrnsteinUhlenbeck
semigroup, J. Anal. Math., 91 (2003), 1-29.
[35] D. Karp, Holomorphic spaces related to orthogonal polynomials and ·analytic continuation of functions,
Analytic extension formulas and their applications (Fukuoka 1999/Kyoto 2000) 169-187, Int. Soc. Anal.
Appl. Comput., 9, Kluwer Acad. Pub!., Dordrecht, 2001.
[36] D. Karp, Hypergeometric reproducing kernels and analytic continuation from a half-line, Int. Transforms
Spec. Funct., 14 (6) (2003), 485-498
[37] D. Karp, Square summability with geometric weight for classical orthogonal expansions, Advances in Analysis,
Ed. H.G.W. Begehr et al, World Scientific, Singapore (2005), 407-421.
[38] H. Kónig, Vector valued Lagrange interpolation and mean convergence of Hermite series, Functional analysis
(Essen, 1991), 227-247, Lecture Notes in Pure and Appl. Math., 150, Decker, New York, 1994.
[39] H. Kónig and N.J. Nielsen, Vector-valued LP-convergence of orthogonal series and Lagrange interpolation,
Forum Math., 6 (2) (1994), 183-207.
[40] S. Kwapien, Isomorphic characterizations of inner product spaces by orthogonal series with vector-valued
coefficients, Studia Mathematica, 44 (1972), 583-595.
[41] R. Macías, C. Segovia and J.L. Torrea, Heat diffusion maximal operators for Laguerre semigroups with
negative parameters, J. Funct. Anal., 229 (2) (2005), 300-316.
[42] R. Macías, C. Segovia and J.L. Torrea, Weighted norm estimates for the maximal operator of the Laguerre
functions heat diffusion semigroup, Studia Math., 172 (2) (2006), 149-167.
[43] M. Martínez, J.L. Torrea and Q. Xu, Vector-valued Littlewood-Paley-Stein theory for semigroups, Adv.
Math., 203 (2) (2006), 430-475.
[44] A. Miyachi, A transplantation theorem for Jacobi series in weighted Hardy spaces, Adv. Math., 184 (1)
(2004), 177-202.
[45] A. Miyachi, Weighted Hardy spaces on an interval and Poisson integrals associated with ultraspherical
series, J. Func. Anal., 239 (2) (2006) , 446-496.
(46] A. Miyachi, Weighted Hardy spaces and Jacobi series, Selected papers on analysis and related topics, 37-52,
Amer. Math. Soc. Transl. Ser. 2, 223, Amer. Math. Soc., Providence, RI, 2008.
[47] A. Nowak and P. Sjógren, Weak type (1, 1) estimates for maximal operators associated with various multidimensional
systems of Laguerre functions, Indiana Univ. Math. J., 56 (1) (2007), 417-436.
[48] A. Nowak and P. Sjógren, The multidimensional pencil phenomenon for Laguerre heat-diffusion maximal
operators, Math. Ann., 344 (1) (2009), 213-248.
[49] A. Nowak and K. Stempak, Riesz transforms for multi-dimensional Laguerre function expansions, Adv.
Math., 215 (2) (2007), 642-678.
[50] B. Muckenhoupt, Mean convergence of Jacobi series, Proc. Amer. Amer. Math., 23 (1969), 306-310.
[51] B. Muckenhoupt and E. Stein, Classical expansions and their relation to conjugate harmonic functions,
Trans. Amer. Math. Soc., 118 (1965), 17-92.
[52] H. Pollard, The mean convergence of orthogonal series, JI, Trans. Amer. Math. Soc., 63 (19,48), 355-367.
22
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
(53] R. Radha and D. Venku Naidu, Image of LP(llr) under the Hermite semigroup, Inter. J. Math. Math. Sci.,
2008, Hindawi Publishing Corporation.
[54] L. Roncal, Weighted boundedness of operators related to the Fourier-Bessel expansions, Ph. D. Thesis,
Department of Mathematics and Computation, University of La Rioja, 2009.
(55] J.L. Rubio de Francia, F. Ruiz and J.L. Torrea, Calderón-Zygmund theory for operator-valued kernels,
Adv. Math., 62 (1986), 7-48.
(56] E. Stein, Tapies in harmonic analysis related to the Littlewood-Paley theory, Annals of Mathematics Studies,
No. 63, Princeton University Press, Princeton, N.J., 1970.
[57] G. Szeg6, Orthogonal polynomials, AMS, Providence, 1959.
[58] S. Thangavelu, Hermite and Laguerre semigroups:some recent developments, Technical Report No. 2006/7,
Department of Mathematics, Indian Institute of Sciences, Bangalore.
(59] G.N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1958.
(60] Q. Xu, Littlewood-Paley theory for functions with values in uniformly convex spaces, J. Reine Angew.
Math., 504 (1998), 195-226.
DEPARTAMENTO DE ANÁLISIS MATEMÁTICO, UNIVERSIDAD DE LA LAGUNA, CAMPUS DE ANCHIETA, AVDA.
ASTROFÍSICO FRANCISCO SÁNCHEZ, S/N, 38271 LA LAGUNA (STA. CRUZ DE TENERIFE), 8PAIN
E-mail address: jbetanco©ull. es
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