1. A transference result of the Lp continuity from
Jacobi Riesz transform to the Gaussian and
Laguerre Riesz transforms
Eduard Navas (UNEFM, Coro)
Wilfredo Urbina (Roosevelt University, Chicago)
Western Spring Sectional Meeting University of New Mexico, Albuquerque, NM
April 4-6, 2014

2. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Jacobi case
The Jacobi case.
Let us consider the fP(;

3. )
n gn2N; which are orthogonal polynomials with
respect to the Jacobi measure ;

4. in (1; 1), with ;

5. 1;
;

6. (dx) = !;

7. (x)dx = (1;1)(x)
(1 x)(1 + x)

8. 2+

9. +1B( + 1;

10. + 1)
dx (1)
= ;

11. (1;1)(x)(1 x)(1 + x)

12. dx;
2+

13. +1B(+1;

14. +1) = (+

15. +2)
2+

16. +1(+1)(

17. +1) .
with ;

18. = 1
i.e.
Z 1
1
n n;m = ^ hn
P(;

19. )
n (y)P(;

20. )
m (y) ;

21. (dy) = ;

22. h(;

23. )
(;

24. )
n;m; (2)

25. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Jacobi case.
where n;m = 0; 1; 2; ,
h(;

26. )
n =
2+

27. +1
(2n + +

28. + 1)
(n + + 1) (n +

29. + 1)
(n + 1) (n + +

30. + 1)
; (3)
and
^ hn
(;

31. )
=
1
(2n + +

32. + 1)
( +

33. + 2) (n + + 1) (n +

34. + 1)
( + 1)(

35. + 1) (n + 1) (n + +

36. + 1)
n k22
= kP(;

37. )
;(;

38. )

39. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Jacobi case.
The Jacobi polynomial P(;

40. )
n is also a polynomial solution of the Jacobi
differential equation, with parameters ;

41. ; n,
1 x2
y00 + [

42. ( +

43. + 2) x] y0 + n (n + +

44. + 1) y = 0; (4)
thus, P(;

45. )
n is an eigenfunction of the (one-dimensional) Jacobi differential
operator
L;

46. = (1 x2)
d2
dx2 (

47. ( +

48. + 2)x)
d
dx
;
associated with the eigenvalue +

49. n = n(n++

50. +1). Moreover, L;

51. is
a diffusion since it is a second order differential p
operator with non-constant
term. Observe that if we choose ;

52. =
1 x2 d
dx ; and consider its formal
L2(;

53. )-adjoint,
;

54. =
p
1 x2 d
dx
+ [( +
1
2
)
r
1 + x
1 x
(

55. +
1
2
r
1 x
1 + x
)
]I;
then L;

56. =
;

57. ;

58. : The differential operator ;

59. is considered the
“natural” notion of derivative in the Jacobi case.

60. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Jacobi case.
The operator semigroup associated to the Jacobi polynomials is defined for
positive or bounded measurable Borel functions of (1; 1), as
T;

61. t f (x) =
Z 1
1
p;

62. (t; x; y)f (y);

63. (dy): (5)
where
p(;

64. )(t; x; y) =
X
k
ek(k++

65. +1)t
^ hk
(;

66. ) P(;

67. )
k (x)P(;

68. )
k (y):
The explicit representation of p(;

69. )(t; x; y) is very complicated since the
eigenvalues n are not linearly distributed and was obtained by G. Gasper.
fT;

70. t g is called the Jacobi semigroup and can be proved that is a Markov
semigroup.
The Jacobi-Poisson semigroup fP;

71. t g can be defined, using Bochner’s
subordination formula,
e1=2t =
1
p
Z 1
0
eu
p
u
et2
4u du;

72. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Jacobi case.
as the subordinated semigroup of the Jacobi semigroup,
P(;

73. )
t f (x) =
1
p
Z 1
0
eu
p
u
T(;

74. )
t2=4u f (x)du:
For a function f 2 L2
[1; 1] ; (;

75. )
let us consider its Fourier- Jacobi
expansion
f =
1X
k=0
hf ; P;

76. k i
^ hk
(;

77. ) P;

78. k ; (6)
where
hf ; P;

79. k i =
Z 1
1
f (y)P;

80. k (y);

81. (dy):
Then the action of Tt and Pt can be expressed using the Jacobi expansion as
Ttf =
1X
k=0
hf ; P;

82. k i
^ hk
(;

83. ) ek tP;

84. k ; and Ptf =
1X
k=0
hf ; P;

85. k i
^ hk
(;

86. ) e
p
k tP;

87. k ;

88. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Jacobi case.
Following the classical case, the Jacobi-Riesz transform can be define
formally as
R;

89. = ;

90. (L;

91. )1=2 =
p
1 x2 d
dx
(L;

92. )1=2; (7)
where (L;

93. )=2 is the Jacobi-Riesz potential of order =2. (L;

94. )=2
can be represented as
(L;

95. )=2f =
1
()
Z 1
0
t1P(;

96. )
t fdt;
and then for f 2 L2
[1; 1] ; (;

97. )
, R;

98. f has Jacobi expansion
R;

99. f (x) =
1X
k=1
hf ; P(;

100. )
k i
^ hk
1=2
(;

101. ) k
(k + +

102. + 1)
2
p
1 x2P(+1;

103. +1)
k1 (x);
(8)
where k = k(k + +

104. + 1).

105. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Jacobi case.
The Lp-continuity of the Riesz-Jacobi transform R;

106. , was proved by Zh. Li
(J. Approx. Theory 86 (1996), no. 2, 179196; MR1400789 (98g:42044)) and
L. Caffarelli (Sobre conjugaci—n y sumabilidad de series de Jacobi, Univ.
Buenos Aires, 1971) in the case d = 1. In the case d 1
R;

107. i ; i = 1; ; d; are defined analogously, using partial differentiation in
(7), and their Lp-continuity was proved by A. Nowak and P. Sjogren.
Theorem
Assume that 1 p 1 and ;

108. 2 [1=2;1)d. There exists a constant cp
such that
kR;

109. i f kp;(;

110. ) cpkf kp;(;

111. ): (9)
for all i = 1; ; d.

112. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Hermite case
The Hermite case.
Now consider the Hermite polynomials fHngn, which are defined as the
orthogonal polynomials associated with the Gaussian measure in R,
(dx) = ex2
p
dx; i.e.
Z 1
1
Hn(y)Hm(y)
(dy) = 2nn!n;m; (10)
n;m = 0; 1; 2; , with the normalization
H2n+1(0) = 0; H2n(0) = (1)n (2n)!
n!
: (11)
We have
H0
n(x) = 2nHn1(x); (12)
H00
n (x) 2xH0
n(x) + 2nHn(x) = 0: (13)

113. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Hermite case.
thus Hn is an eigenfunction of the one dimensional Ornstein-Uhlenbeck
operator or harmonic oscillator operator,
L =
1
2
d2
dx2 + x
d
dx
; (14)
associated with the eigenvalue n = n. Observe that if we choose
= p1
2
d
dx ; and consider its formal L2(
)-adjoint,
=
1
p
2
d
dx
+
p
2xI
then L =
: The differential operator
is considered the “natural”
notion of derivative in the Hermite case.

114. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Hermite case.
The Gaussian-Riesz transform can be defined formally, as
R
=
L1=2 =
1
p
2
d
dx
L1=2: (15)
Therefore, for f 2 L2 (R;
) with Hermite expansion
f =
1X
k=1
hf ;Hki
2kk!
Hk;
its Gaussian-Riesz transform has Hermite expansion
R
f (x) =
1X
k=1
hf ;Hki
2kk!
p
2kHk1(x): (16)
The Lp continuity of the of the Gaussian-Riesz transform was proved by B.
Muckenhoupt in 1969, in the case d = 1 (Trans. Amer. Math. Soc. 139
(1969), 243260; MR0249918 (40 3159)). In the case d 1
R
i
; i = 1; ; d; are defined analogously, using partial differentiation in
(15), and their Lp continuity has been proved by very different ways, using
analytic and probabilistic tools, by P. A. Meyer, R. Gundy, S. P´erez and F.
Soria, G. Pissier, C. Guti´errez and W. Urbina.

115. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Hermite case.
Theorem
Assume that 1 p 1. There exists a constant cp such that
kR
i
f kp;
cpkf kp;
: (17)
for all i = 1; ; d.

116. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Laguerre case.
The Laguerre case.
Finally, analogously for the Laguerre polynomials fL
k g, 1 which are
defined as the orthogonal polynomials associated with the Gamma measure
on (0;1), (dx) = (0;1)(x) xex
(+1) dx; i.e.
Z 1
0
L
n (y)L
m(y) (dy) =
n +
n
n;m =
(n + + 1)
( + 1)n!
n;m; (18)
n;m = 0; 1; 2; . We have
k (x))0 = L+1
(L
k1 (x): (19)
x(L
k (x))00 + ( + 1 x)(L
k (x))0 + kL
k (x) = 0: (20)
thus L
k is an eigenfunction of the (one-dimensional) Laguerre differential
operator
L = x
d2
dx2 ( + 1 x)
d
dx
;
associated with the eigenvalue k = k.Observe that if we choose =
p
x d
dx ;
and consider its formal L2(
)-adjoint,

117. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Laguerre case.
=
p
x
d
dx
+ [
+ 1=2
p
x
+
p
x]I
then L =
: The differential operator is considered the “natural”
notion of derivative in the Laguerre case.
The Laguerre-Riesz transform can be defined formally, as
R = (L)1=2 =
p
x
d
dx
(L)1=2: (21)
Therefore for f 2 L2 ((0;1); ) with Laguerre expansion
f =
1X
k=0
( + 1)k!
(k + + 1)
hf ; L
k iL
k
its Laguerre-Riesz transform has Laguerre expansion
Rf (x) =
1X
k=1
( + 1)k!
(k + + 1)
p
k)1p
xhf ; L
(
k iL+1
k1 (x): (22)

118. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
The Laguerre case.
The Lp continuity of the Laguerre-Riesz transform was proved by B.
Muckenhoupt, for the case d = 1 (Trans. Amer. Math. Soc. 147 (1970),
403418; MR0252945 (40 6160)). In the case d 1 Ri
; i = 1; ; d; are
defined analogously, using partial differentiation in (21), and their Lp
continuity was proved by A. Nowak, using Littlewood-Paley theory.
Theorem
Assume that 1 p 1 and 2 [1=2;1)d. There exists a constant cp
such that
kRi
f kp; cpkf kp;: (23)
for all i = 1; ; d;.

119. The Classical orthogonal polynomials, their semigroup and Riesz Transforms.
Asymptotic relations
Asymptotic relations.
Now, it is well know the asymptotic relations of the Jacobi polynomials with
other classical orthogonal polynomials:
i) The asymptotic relation with the Hermite polynomials is
lim
!1
n=2C
p
) =
n (x=
Hn(x)
n!
: (24)
ii) The asymptotic relation with the Laguerre polynomials is
lim

120. !1
P(;

121. )
n (1 2x=

122. ) = L
n (x): (25)

123. Main Results
In the case of the Riesz transforms, we have proved (see ArXiv:1202.5728)
that the Lp-continuity of the Gaussian-Riesz transform and the Lp-continuity
of the Laguerre-Riesz transform can be obtained from the Lp-continuity of
the Jacobi-Riesz transform using the asymptotic relations.
Theorem
Let ;

124. 1 and 1 p 1, then the Lp(;

125. ) boundedness for the
Jacobi-Riesz transform
kR;

126. f kp;(;

127. ) Cpkf kp;(;

128. ) (26)
implies
i) the Lp(
)-boundedness for the Gaussian-Riesz transform
kR
f kp;
Cpkf kp;
: (27)
ii) the Lp()-boundedness for the Laguerre-Riesz transform
kRf kp; Cpkf kp;: (28)

129. Main Results
For the proof of the theorem we need the following technical results,
Proposition
i) Let f(x) = f (
p
x)1[1;1](x) where f 2 L2(R;
): Then
f 2 L2([1; 1]; ) and
lim
!1
kfk2; = kf k2;
ii) Similarly, Let f

130. (x) = f

131. 2
(1 x)
1[1;1](x) where
f 2 L2(R; ): Then f

132. 2 L2([1; 1]; (;

133. )) and
lim

134. !1
kf

135. k2;(;

136. ) = kf k2;

137. Main Results.
and
Proposition
i) Let f 2 L2(R;
): Then
lim
!1
hf; k=2C
k i = hf ;
Hk
k!
i
ii) Similarly, Let f 2 L2(R; ): Then
lim

138. !1
hf

139. ; P(;

140. )
k i = hf ; L
k i

141. Main Results.
Idea of the proof for the theorem of the Riesz transform.
Let us consider first the case p = 2 then by Parseval’s identity, for
f 2 L2 ([1; 1] ; )
R(;

142. )f
2
2;(;

143. )
=
1X
k=1
j
hf ; P;

144. k i
^ hn
(;

145. )
j2 (k + +

146. + 1)2
4k
p
1 x2P(+1;

147. +1)
k1
2
2;(;

148. )
and
p
1 x2P(+1;

149. +1)
k1
2
2;(;

150. )
=
4( + 1)(

151. + 1)
( +

152. + 3)( +

153. + 2)
P(+1;

154. +1)
k1
2
2;(+1;

155. +1)
=
4k
(k + +

156. + 1)
P(;

157. )
k
2
2;(;

158. )

159. The Main Result.
and therefore,
R(;

160. )f
2
2;(;

161. )
=
1X
k=1
j
hf ; P;

162. k i
^ hn
(;

163. )
j2 (k + +

164. + 1)2
4k(k + +

165. + 1)
p
1 x2P(+1;

166. +1)
k1
2
2;(;

167. )
=
1X
k=1
j
hf ; P;

168. k i
^ hn
(;

169. )
j2 (k + +

170. + 1)2
4k(k + +

171. + 1)
4k
(k + +

172. + 1)
P(;

173. )
k
2
2;(;

174. )
=
1X
k=1
k ij2
jhf ; P;

175. (2k + +

176. + 1)( + 1)(

177. + 1)(k + 1)(k + +

178. ( +

179. + 2)(k + + 1)(k +

180. + 1)

181. Main Results
Rf
2
2; =
1X
k=1
k ij2 (2k + 2) [( + 1=2)]2 (k + 1)(k + 2(jhf ; P(1=2;1=2)
2 + 1) [(k + + 1=2)]2
=
1X
k=1
k ij2 (2k + 2) [( + 1=2)]2 (k + 1)(k + 2)
jhf ;C
(2 + 1) [(k + + 1=2)]2
[(2)]2 [(k + + 1=2)]2
[( + 1=2)]2 [(k + 2)]2
1X
k=1
k ij2 k!
jhf ; k=2C
p
(2 + 1=)(2 + 2=) : : : (2 + (k 1)=)
2

182. Main Results
On the other hand, again using Parseval’s identity, we have that the L2-norm
of the Gaussian Riesz transform for f 2 L2 (R;
) is given by
kR
f k22
;
=
1X
k=1
jhf ;Hkij2
22k(k!)2 2k kHk1k22
;
=
1X
k=1
jhf ;Hkij2
p
k!2k
:
Then, taking f(x) = f (
p
x)1[1;1](x) and using the asymptotic relation
and the previous proposition, we get
kR
f k22
;
=
1X
k=1
1
k!2k
p
jhf ;Hkij2 =
1X
k=1
1
(k!)2
k!
p
2(k1)
2
jhf ;Hkij2
= lim
!1
1X
k=1
k!
2
p
(2 + 1
)(2 + 2
) : : : (2 + (k1)
)
jhf; k=2C
k ij2
lim
!1
Rf
2
2;
C2 lim
!1
kfk22
; = C2 kf k22
;
:

183. Main Results.
The Laguerre case is essentially analogous.
For the general case p6= 2 we will follow the argument given by Betancour
et al.
Take 2 C1
p
x); for 0 and x 2 R . If big
0 (R); and (x) = (
enough sop [1; 1]; then
kRkLp([1;1]) CkkLp([1;1])
i.e.
1X
n=1
c(n)r()
n
p
1 x2C+1
n1
Lp([1;1])
CkkLp([1;1])

184. Main Results.
Taking the change of variable x = y p
, we get
(Z p
p

189. 1X
n=1
c(n)r()
n
r
1
y2
C+1
n1 (
y
p

194. )
p
Z()(1
y2
)1=2dy
)1=p
CkkLp([1;1])
where Z() = 1=2[()]222
2(2)
(Z p
p

199. 1X
n=1
c(n)r()
n
1
y2
=p1=2p+1=2
e
y2
p C+1
n1 (
y
p

204. )
p
ey2
p
)1=p
dy
C(Z())1=pkkLp([1;1]);

205. Main Results.
and also we have
8
:
Z p
p

210. 1X
n=1
c(n)r()
n
1
y2
=21=4+1=2
e y2
2 C+1
n1 (
y
p

215. )
2
ey2
p
dy
9=
1=2
;
C(Z())1=2kkL2([1;1])
Let k 2 N and 0 such that
p
k define
F;k(y) =
8
:
1P
n=1
c(n)r()
n1 ( y p
n C+1
)
1 y2
=21=4+1=2
e y2
2 if jyj k
0 si jyj k
and
f;k(y) =
8
: 1P
n=1
c(n)r()
n1 ( y p
n C+1
)
1 y2
=p1=2p+1=2
e
y2
p if jyj k
0 si jyj k

216. Main Results.
p
;
for y 2 [
p
], both series converges and F;k = f;k
; where
(y) = e
y2
y2
2 p
1
y2
=2=p1=4+1=2p
:
is bounded in [k; k].

217. Main Results.
On the other hand, it can be proved that
lim
!1
(Z())1=pkkLp([1;1]) lim
!1
C
(Z p
p
j(y)jp ey2
p
)1=p
dy
= C
(Z 1
1
j(y)jp ey2
p
dy
)1=p
= CkkLp(R;
);
and moreover,
(Z())1=pkkLp([1;1]) CkkLp(R;
): (29)
Then,
kF;kkL2(R;
) CkkL2(R;
):
and
kF;kkLp(R;
) CkkLp(R;
)
for all
p
k:

218. Main Results.
Thus, fF;kg is a bounded sequence in L2 (R;
) and in Lp (R;
) : By
Bourbaki-Alaoglu’s theorem,there exists a subsequence (j)j2N such that
limj!1 j = 1and functions Fk 2 L2 (R;
) and fk 2 Lp (R;
) satisfying
I Fj;k ! Fk; as j ! 1; in the weak topology of L2 (R;
)
I Fj;k ! fk; as j ! 1; in the weak topology of Lp (R;
):
Moreover, sopFk [ sopfk [k; k]; and
kFkkL2(R;
) lim
j!1
kFj;kkL2(R;
) CkkL2(R;
): (30)
Analogously one gets,
kfkkLp(R;
) CkkLp(R;
): (31)
and moreover Fk = fk a:e (k; k); and as k is arbitrary we have Fk = fk
a.e. so we get
kFkkLp(R;
) CkkLp(R;
): (32)

219. Main Results.
Then, there exists a monotone increasing sequence (j)j2N (0;1) such
that limj!1 j = 1; and a function F 2 Lp (R;
) L2 (R;
) ; satisfying
I For each k 2 N; Fj;k ! F; as j ! 1; in the weak topology of
L2 (R;
) and in the weak topology of Lp (R;
)
I kFkLp(R;
) CkkLp(R;
):
Then, to finish the proof we have to prove F is the Gaussian-Riesz transform
of ,
F(y) = R
(y); a:e:;
and then
kR
kLp(R;
) = kFkLp(R;
) CkkLp(R;
):
The Laguerre case is essentially analogous.

220. Main Results
We can also obtain the Lp-continuity of the Gaussian-Littlewood-Paley g
function and the Lp-continuity of the Laguerre-Littlewood-Paley g function
from the Lp-continuity of the Jacobi–Littlewood-Paley g function using the
asymptotic relations (but that is another talk!).
Theorem
Let ;

221. 1 and 1 p 1, then the Lp(;

222. ) boundedness for the
Jacobi-Littlewood-Paley g function
kg(;

223. )f kp;(;

224. ) Cpkf kp;(;

225. ) (33)
implies
i) the Lp(
) boundedness for the Gaussian-Littlewood-Paley g function
kg
f kp;
Cpkf kp;
: (34)
ii) the Lp() boundedness for the Laguerre-Littlewood-Paley g function
kgf kp; Cpkf k2;: (35)

226. References
References
I Betancor, J., Fari˜na, J., Rodriguez, L., Sanabria, A. Transferring
boundedness from conjugate operators associated whit Jacobi,
Laguerre and Fourier-Bessel expansions to conjugate operators in the
Hankel setting. J. Fourier Anal. Appl. 14 (2008), no. 4, 493–513.
I Cafarelli, L. Sobre conjugaci´on y sumabilidad de series de Jacobi,
Univ. Buenos Aires, 1971
I Guti´errez, C. On the Riesz transforms for the Gaussian measure. J.
Func. Anal. 120 (1) (1994) 107-134.
I Li, Z. Conjugate Jacobi series and conjugate functions. J. Approx.
Theory 86 (1996), no. 2, 179196; MR1400789 (98g:42044)
I Meyer, P. A. Transformations de Riesz pour les lois Gaussiennes.
Lectures Notes in Math 1059 (1984) Springer-Verlag 179-193.
I Muckenhoupt, B. Hermite conjugate expansions. Trans. Amer. Math.
Soc. 139 (1969), 243260; MR0249918 (40 3159)
I Muckenhoupt, B. Conjugate functions for Laguerre expansions. Trans.
Amer. Math. Soc. 147 (1970), 403418; MR0252945 (40 6160)
I Muckenhoupt, B. Stein, E.M. Classical Expansions. Trans. Amer.
Math. Soc. 147 (1965) 17-92.

227. References
I Navas, E Urbina, W. A transference result of the Lp continuity of the
Jacobi Riesz transform to the Gaussian and Laguerre Riesz transforms.
J. Fourier Anal. Appl. 19 (2013), no. 5, 910942. arXiv: 0441166.
I Navas, E Urbina, W. A transference result of the Lp continuity of the
Jacobi Littlewood-Paley g function to the Gaussian and Laguerre
Littlewood-Paley g function. (2014) In preparation.
I Nowak, A. On Riesz transforms for Laguerre expansions. J. Funct.
Anal. 215 (2004), no. 1, 217–240.
I Nowak, A. Stempak, K. L2-theory of Riesz transforms for orthogonal
expansions. J. Fourier Anal. Appl. 12 (2006), no. 6, 675–711.
I Novak, A.; S¨ogren, P. Riesz Transforms for Jacobi expansions J. Anal.
Math. 104 (2008), 341369.
I Stein, E.M. Topics in Harmonic Analysis related to the
Littlewood-Paley Theory. Princeton Univ. Press. Princeton (1970).
I Szeg¨o, G. Orthogonal polynomials. Colloq. Publ. 23. Amer. Math.
Soc. Providence (1959).
I Urbina, W. On singular integrals with respect to the Gaussian measure.
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 4, 531567.