Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
Fractional Integration and Fractional
Differentiation for Jacobi Expansions.
Integración y Derivación Fraccionaria para desarrollos de Jacobi.
Cristina Balderrama (cbalde@euler.ciens.ucv.ve)
Departamento de Matemáticas, Facultad de Ciencias, UCV.
Apartado 47009, Los Chaguaramos, Caracas 1041-A Venezuela
Wilfredo Urbina (wurbina@euler.ciens.ucv.ve)
Departamento de Matemáticas, Facultad de Ciencias, UCV.
Apartado 47195, Los Chaguaramos, Caracas 1041-A Venezuela, and
Department of Mathematics and Statistics, University of New Mexico,
Albuquerque, NM 87131, USA.
Abstract
In this article we study the fractional Integral and the fractional
Derivative for Jacobi expansion. In order to do that we obtain an analogous of P. A. Meyer’s Multipliers Theorem for Jacobi expansions. We
also obtain a version of Calderón’s reproduction formula for the Jacobi
measure. Finally, as an application of the fractional differentiation, we
get a characterization for Potential Spaces associated to the Jacobi measure.
Key words and phrases: Fractional Integration, Fractional Differentiation, Jacobi expansions, Multipliers, Potential Spaces.
Resumen
En este trabajo estudiamos la integración y diferenciación fraccionaria para el caso de los desarrollos de Jacobi. Para ello obtenemos
un teorema análogo al teorema de multiplicadores de P.A. Meyer para
desarrollos de Jacobi. También obtenemos una versión de la fórmula de
reproducción de Calderón para la medida de Jacobi. Finalmente, como
una aplicacción de la diferenciación fraccionaria, obtenemos una caracterización de los Espacios Potenciales asociados a la medida de Jacobi.
Palabras y frases clave: Integración fraccionaria, Derivación fraccionaria, Desarrollos de Jacobi, Multiplicadores, Espacios Potenciales.
Received 2006/03/10. Revised 2006/10/18. Accepted 2006/11/06.
MSC (2000): Primary 42C10; Secondary 26A33.
94
1
Cristina Balderrama, Wilfredo Urbina
Introduction.
Let us consider (normalized) Jacobi measure
µα,β (dx) =
1
(1 − x)α (1 + x)β dx,
2α+β+1 B(α + 1, β + 1)
(1)
for x ∈ [−1, 1], where α, β > −1. This normalization gives a probability
measure and it is not usually considered in classical orthogonal polynomial
theory.
The one dimensional Jacobi operator is given by
Lα,β = (1 − x2 )
d2
d
+ (β − α − (α + β + 2)x) .
dx2
dx
(2)
It is easy to see that this is a symmetric operator on L2 ([−1, 1], µα,β ).
Let pα,β
be the normalized Jacobi polynomials of degree n ∈ N. Then the
n
2
family {pα,β
n } is an orthonormal Hilbert basis of L ([−1, 1], µα,β ), that can
be obtained by the Gram–Schmidt orthogonalization process with respect to
the measure µα,β , applied to the monomials. It is well known that Jacobi
polynomials are eigenfunctions of the Jacobi operator Lα,β with eigenvalue
−λn = −n(n + α + β + 1), that is,
Lα,β pα,β
= −n(n + α + β + 1)pα,β
n
n .
(3)
2
Since {pα,β
n } is an orthonormal basis of L ([−1, 1], µα,β ), we have the orthogonal decomposition
L2 ([−1, 1], µα,β ) =
∞
M
Cnα,β ,
(4)
n=0
where, for each n, Cnα,β is the closed subspace generated by pα,β
n . This is
called the Wiener–Jacobi chaos decomposition of L2 ([−1, 1], µα,β ).
Let Jnα,β be the orthogonal projection of L2 ([−1, 1], µα,β ) onto Cnα,β . Then,
for f ∈ L2 ([−1, 1], µα,β ) we have
f=
∞
X
Jnα,β f,
n=0
where Jnα,β f = fˆ(n)pα,β
with
n
Z
fˆ(n) =
1
−1
f (x)pα,β
n (x)µα,β (dx)
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
(5)
Fractional Integration and Fractional Differentiation.
95
the nth-Jacobi–Fourier coefficient of f .
Let us now consider {Ttα,β }t≥0 the Jacobi semigroup. This is the Markov
operator semigroup associated to the Markov probability kernel semigroup
(see [2],[4])
Pt (x, dy) =
∞
X
α,β
α,β
e−λn t pα,β
(t, x, y)µα,β (dy),
n (x)pn (y)µα,β (dy) = p
n=0
that is
Z
Ttα,β f (x) =
Z
1
1
f (y)Pt (x, dy) =
−1
f (y)pα,β (t, x, y)µα,β (dy).
−1
Unfortunately, there is not a reasonable explicit representation for the kernel
pα,β (t, x, y).
The Jacobi semigroup {Ttα,β }t≥0 is a diffusion semigroup, conservative,
symmetric, strongly continuous on Lp ([−1, 1], µα,β ) of positive contractions
on Lp , with infinitesimal generator Lα,β .
Moreover, for α, β > − 12 it can be proved that {Ttα,β }t≥0 is also hypercontractive, that is to say that Ttα,β is not only a contraction on Lp ([−1, 1], µα,β ),
but also for any initial condition 1 < q(0) < ∞ there exists an increasing function q : R+ → [q(0), ∞), such that for every f and all t ≥ 0,
kTtα,β f kq(t) ≤ kf kq(0) .
The proof that we know of this fact is an indirect one, obtained by D. Bakry
in [3], that is based in proving that the Jacobi operator, for the parameters
α, β > −1/2, satisfies a Sobolev inequality, by checking that it satisfies a
curvature-dimension inequality, and therefore a logarithmic Sobolev inequality
and then use the well known equivalency due to L. Gross [8]. A detailed proof
of this can be found in [2], see also [1]. From now on we will consider the
Jacobi semigroup for the parameters α, β > − 12 .
On the other hand, for 0 < δ < 1 we define the generalized Poisson–Jacobi
semigroup of order δ, {Ptα,β,δ }, as
Z ∞
α,β,δ
Pt
f (x) =
Tsα,β f (x)µδt (ds).
0
where {µδt } are the stable measures on [0, ∞) of order δ (∗) . The generalized
(∗) The
stable measures on [0, ∞) of order δ are Borel measures on [0, ∞) such that its
R
δ
Laplace transform verify 0∞ e−λs µδt (ds) = e−λ t . For δ fixed, {µδt } form a semigroup with
respect to the convolution operation in the parameter δ > 0.
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
96
Cristina Balderrama, Wilfredo Urbina
Poisson–Jacobi semigroup of order δ is a strongly continuous semigroup on
Lp ([−1, 1], µα,β ) with infinitesimal generator (−Lα,β )δ .
In the case δ = 1/2, we have the Poisson–Jacobi semigroup, that will be
α,β,1/2
1/2
denoted as Ptα,β = Pt
. In this case we can explicitly compute µt ,
2
t
1/2
µt (ds) = √ e−t /4s s−3/2 ds
2 π
and by Bochner’s formula we have
1
Ptα,β f (x) = √
π
Z
∞
0
e−u α,β
√ Tt2 /4u f (x)du.
u
(6)
Then by (3),
Ttα,β pα,β
= e−λn t pα,β
n
n ,
γ
Ptα,β,δ pnα,β = e−λn t pα,β
n .
Giving a function Φ : N → R the multiplier operator associated to Φ is
defined as
∞
X
TΦ f =
Φ(k)Jkα,β f,
P∞
k=0
α,β
k=0 Jk f,
a polynomial.
for f =
If Φ is a bounded function, then by Parseval’s identity, TΦ is bounded on
L2 ([−1, 1], µα,β ). In the case of Hermite expansions, the P.A. Meyer’s Multiplier Theorem [10] gives conditions over Φ so that the multiplier TΦ can be
extended to a continuous operator on Lp for p 6= 2. In the next section we
will prove an analogous result for the Jacobi expansions.
In section 3 we are going to define the Fractional Integration and Differentiation for Jacobi expansions, as well as Bessel Potentials associated to Jacobi
measure. Using Meyer’s multipliers Theorem we will see the Lp continuity of
the Fractional Integration and of the Bessel Potentials and we give a characterization of the Potential Spaces. We also study the asymptotic behavior of
the Poisson-Jacobi semigroup and we give a version of Calderon’s reproducing
formula.
2
P.A. Meyer’s Multiplier Theorem for Jacobi
expansions.
In order to establish the P.A. Meyer’s Multiplier Theorem for Jacobi expansions we need some previous results. First, let us see that the orthogonal
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
Fractional Integration and Fractional Differentiation.
97
projections Jnα,β can be extended to a continuous function on Lp ([−1, 1], µα,β ).
Lemma 1. If 1 < p < ∞ then for every n ∈ N, Jnα,β can be extended to
a continuous operator to Lp ([−1, 1], µα,β ), that will also be denoted as Jnα,β ,
that is, there exists Cn,p ∈ R+ such that
kJnα,β f kp ≤ Cn,p kf kp ,
for f ∈ Lp ([−1, 1], µα,β ).
Proof. Let us consider p > 2 and for the initial condition q(0) = 2, let
t0 be a positive number such that q(t0 ) = p. Taking g = Jnα,β f, by the
hypercontractive property, Parseval identity and Hölder inequality we obtain,
kTtα,β
gkp = kTtα,β
Jnα,β f kp ≤ kJnα,β f k2 ≤ kf k2 ≤ kf kp .
0
0
Now, as Tt0 Jnα,β f = e−t0 λn Jn f we get
kJnα,β f kp ≤ Cn,p kf kp ,
with Cn,p = et0 λn . For 1 < p < 2 the result follows by duality.
We also need the following technical result,
¤
Lemma 2. Let 1 < p < ∞. Then, for each m ∈ N there exists a constant Cm
such that
α,β
kTtα,β (I − J0α,β − J1α,β − · · · − Jm−1
)f kp ≤ Cm e−tm kf kp .
Proof. Let p > 2 and for the initial condition q(0) = 2, let t0 be a positive
number such that q(t0 ) = p.
If t ≤ t0 , since Ttα,β is a contraction, by the Lp - continuity of the projections
α,β
kTtα,β (I − J0α,β − · · · − Jm−1
)f kp
α,β
≤ k(I − J0α,β − · · · − Jm−1
)f kp
≤ kf kp +
m−1
X
kJkα,β f kp
k=0
≤
(1 +
m−1
X
et0 λk )kf kp .
k=0
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
98
Cristina Balderrama, Wilfredo Urbina
But since et0 λk ≤ et0 λm for all 0 ≤ k ≤ m − 1 and λm ≥ m for all m > 1,
therefore
α,β
kTtα,β (I − J0α,β − · · · − Jm−1
)f kp
≤
≤
(1 + met0 λm )kf kp = Cm e−t0 λm kf kp
Cm e−tm kf kp ,
with Cm = (1 + met0 λm )et0 m .
P∞
Now suppose t > t0 . For f = k=0 Jkα,β f , by the hypercontractive property,
α,β
kTtα,β
Ttα,β (I − J0α,β − · · · Jm−1
)f k2p
0
≤
=
α,β
kTtα,β (I − J0α,β − · · · Jm−1
)f k22
∞
X α,β
Jk f )k22
kTtα,β (
k=m
=
=
≤
k
∞
X
e−tλk Jkα,β f k22
k=m
∞
X
−2tλk
e
k=m
∞
X
kJkα,β f k22
e−2tk kJkα,β f k22 ,
k=m
as λm ≥ m for all m ≥ 1. Then
∞
X
e−2tk kJkα,β f k22
∞
X
α,β 2
kJk+m
k2 ≤ e−2tm
≤
e−2tm
=
e−2tm kf k22 ≤ e−2tm kf k2p .
k=m
k=0
∞
X
kJkα,β k22
k=0
Thus
α,β
kTtα,β
Ttα,β (I − J0α,β − J1α,β − · · · − Jm−1
)f kp
0
≤
e−tm kf kp .
In particular,
α,β
α,β
α,β
kTtα,β (I −J0α,β −· · ·−Jm−1
)f kp = kTtα,β
Tt−t
(I −J0α,β −· · ·−Jm−1
)f kp
0
0
≤ e−(t−t0 )m kf kp = Cm e−tm kf kp ,
with Cm = et0 m . For 1 < p < 2 the result follows by duality.
¤
Now, by the Minkowski integral inequality, we have an analogous result
for the generalized Poisson–Jacobi semigroup.
Lemma 3. Let 1 < p < ∞. Then for each m ∈ N, there exists Cm such that
γ
α,β
kPtα,β,γ (I − J0α,β − J1α,β − · · · − Jm−1
)f kp ≤ Cm e−tm kf kp .
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
Fractional Integration and Fractional Differentiation.
99
From the generalized Poisson–Jacobi semigroup let us define a new family
α,β
of operators {Pk,γ,m
}k∈N by the formula
α,β
Pk,γ,m
f=
1
(k − 1)!
Z
0
∞
α,β
tk−1 Ptα,β,γ (I − J0α,β − J1α,β − · · · − Jm−1
)f dt.
By the preceding lemma and the Minkowski integral inequality we have,
Proposition 4. If 1 < p < ∞, then for every m ∈ N there is a constant Cm
such that
α,β
kPk,γ,m
f kp ≤
Cm
kf kp .
mγk
Observe that in particular if f = pα,β
n , n≥m
α,β
Pk,γ,m
pα,β
=
n
1
λγk
n
pα,β
n .
(7)
Now we are ready to present P.A. Meyer’s Multipliers Theorem for Jacobi
expansions.
Theorem 5. If for some n0 ∈ N and 0 < γ < 1
µ ¶
1
Φ(k) = h
,
k ≥ n0 ,
λγk
with h an analytic function in a neighborhood of zero, then TΦ , the multiplier
operator associated to Φ, admits a continuous extension to Lp ([−1, 1], µα,β ).
Proof. Let
TΦ f = Tφ1 f + TΦ2 f =
nX
0 −1
Φ(k)Jkα,β f +
k=0
∞
X
Φ(k)Jkα,β f.
k=n0
By the lemma 1 we have that
kTΦ1 f kp
≤
nX
0 −1
|Φ(k)|kJkα,β f kp
k=0
that is, TΦ1 is Lp continuous.
continuous.
≤
Ãn −1
0
X
!
|Φ(k)|Ck
kf kp ,
k=0
It remains to be seen that TΦ2 is also Lp
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
100
Cristina Balderrama, Wilfredo Urbina
P∞By then hypothesis let us assume that h can be written as h(x) =
n=0 an x , for x in a neighborhood of zero, then
TΦ2 f
=
∞
X
Φ(k)Jkα,β f
=
k=n0
µ
h
k=n0
but since (7), for k ≥ n0 ,
TΦ2 f
∞
X
=
=
1
Jkα,β f
λγn
k
∞ X
∞
X
1
λγk
¶
Jkα,β f
∞ X
∞
X
=
an
k=n0 n=0
1 α,β
Jk f,
λγn
k
α,β
= Pn,γ,n
J α,β f , we have
0 k
α,β
an Pn,γ,n
J α,β f =
0 k
k=n0 n=0
∞
∞
X
X
α,β
Jkα,β f
an Pn,γ,n
0
n=0
k=0
∞
X
an
n=0
=
∞
X
∞
X
α,β
Pn,γ,n
J α,β f
0 k
k=0
α,β
an Pn,γ,n
f.
0
n=0
α,β
Since Pn,γ,n
is Lp continuous, by proposition 4, we obtain,
0
kTΦ2 f kp ≤
≤
∞
X
α,β
|an |kPn,γ,n
f kp
0
n=0
̰
X
1
|an |Cn0 γn
n
0
n=0
!
Ã
kf kp = Cn0
∞
X
1
|an | γn
n
0
n=0
!
kf kp .
Therefore, TΦ is continuous in Lp ([−1, 1], µα,β ).
3
¤
Fractional Integration and Differentiation.
As in the classical case, for γ > 0 we define the Fractional Integral of order γ,
Iγα,β , with respect to Jacobi measure, as
Iγα,β = (−Lα,β )−γ/2 .
(8)
Iγα,β is also called Riesz Potential of order γ.
Observe that, since zero is an eigenvalue of Lα,β , then Iγα,β is not defined over all L2 ([−1, 1], µα,β ). Let Π0 = I − J0α,β and denote also by Iγα,β
the operator (−Lα,β )−γ/2 Π0 . Then, this operator is well defined over all
L2 ([−1, 1], µα,β ). In particular, for Jacobi polynomials we have
Iγα,β pα,β
=
k
1 α,β
p .
γ/2 k
λk
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
(9)
Fractional Integration and Fractional Differentiation.
101
Thus, for f a polynomial in L2 ([−1, 1], µα,β ) with Jacobi expansion
P∞
α,β
k=0 Jk f, we have
Iγα,β f =
∞
X
1
γ/2
k=1 λk
Jkα,β f.
For the Fractional Integral of order γ > 0 we have the following integral
representation,
Z ∞
1
α,β
tγ−1 Ptα,β f dt,
Iγ f =
(10)
Γ(γ) 0
for f polynomial, where Ptα,β is the Poisson–Jacobi semigroup. In order to
prove that observe that for the Jacobi polynomials, we have, by the change
1/2
of variables s = λk t,
Z ∞
Z ∞
1/2
1
1
tγ−1 Ptα,β pα,β
tγ−1 e−λk t dt pα,β
dt
=
k
k
Γ(γ) 0
Γ(γ) 0
Z ∞
1
1
1
sγ−1 e−s ds pα,β
=
= γ/2 pα,β
k
k .
Γ(γ) λγ/2 0
λ
k
k
The Meyer’s multiplier theorem allows us to extend Iγα,β as a bounded
operator on Lp ([−1, 1], µα,β ), as next theorem shows.
Theorem 6. The the Fractional Integral of order γ, Iγα,β admits a continuous
extension, that it will also be denoted as denote Iγα,β , to Lp ([−1, 1], µα,β ).
Proof. If γ/2 < 1, then Iγα,β is a multiplier with associated function
Φ(k) =
1
γ/2
λk
Ã
=h
1
!
γ/2
λk
where h(z) = z, which is analytic in a neighborhood of zero. Then the results
follows immediately by Meyer’s theorem.
γ
Now, if γ/2 ≥ 1, let us consider β ∈ R, 0 < β < 1 and δ = 2β
. Then
γ
δ
δβ = 2 . Let h(z) = z , which is analytic in a neighborhood of zero. Then we
have
à !
1
1
1
= δβ = γ/2 = Φ(k).
h
λβk
λk
λk
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
102
Cristina Balderrama, Wilfredo Urbina
Again the results follows applying Meyer’s theorem.
¤
Now the Bessel Potential of order γ > 0, Jγα,β , associated to the Jacobi
measure is defined as
Jγα,β = (I − Lα,β )−γ/2 .
(11)
Observe that for the Jacobi polynomials we have
Jγα,β pα,β
=
k
1
pα,β ,
(1 + λk )γ/2 k
and, therefore if f ∈ L2 ([−1, 1], µα,β ) polynomial with expansion
Jγα,β f =
∞
X
1
J α,β f.
γ/2 k
(1
+
λ
)
k
k=0
P∞
k=0
Jkα,β f
(12)
Again Meyer’s theorem allows us to extend Bessel Potentials to a
continuous operator on Lp ([−1, 1], µα,β ),
Theorem 7. The operator Jγα,β admits a continuous extension, that it will
also be denoted as Jγα,β , to Lp ([−1, 1], µα,β ).
Proof. Bessel Potential of order γ is a multiplier associated to the function
³
´γ/2
³ β ´γ/2
1
z
Φ(k) = 1+λ
.
Let
β
∈
R,
β
>
1
and
h(z)
=
. Then h is an
z β +1
k
analytic function on a neighborhood of zero and
Ã
! µ
¶γ/2
1
1
h
=
= Φ(k).
1/β
1 + λk
λ
k
The results follows applying Meyer’s theorem.
¤
Finally as in the classical case, we define the Fractional Derivative of order
γ > 0, Dγα,β , with respect to Jacobi measure as
Dγα,β = (−Lα,β )γ/2 .
(13)
Observe that, for the Jacobi polynomials we have,
γ/2
Dγα,β pα,β
= λk pα,β
k
k ,
(14)
and therefore, by the density of the polynomials in Lp ([−1, 1], µα,β ), 1 < p <
∞, Dγα,β can be extended to Lp ([−1, 1], µα,β ).
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
Fractional Integration and Fractional Differentiation.
103
For the Fractional Derivative of order 0 < γ < 1 we also have a integral
representation,
Dγα,β f =
1
cγ
Z
∞
0
t−γ−1 (Ptα,β f − f )dt,
(15)
R∞
for f polynomial, where cγ = 0 s−γ−1 (e−s − 1)ds, since, for the Jacobi
1/2
polynomials, we have, by the change of variables s = λk t,
Z
∞
0
Z
t−γ−1 (Ptα,β pα,β
− pα,β
k
k ) dt =
∞
1/2
t−γ−1 (e−λk t − 1) dt pα,β
k
0
Z ∞
γ/2
= λk
s−γ−1 (e−s − 1) ds pα,β
k
=
0
γ/2
λk cγ pα,β
k .
Now, if f is a polynomial, by (9) and (14) we have,
Iγα,β (Dγα,β f ) = Dγα,β (Iγα,β f ) = Π0 f.
(16)
In [5] H. Bavinck has defined Fractional Integration and Differentiation for
Jacobi expansions. Nevertheless the motivation, the methods and techniques
use in his paper are totally different from ours.
Now we are going to give an alternative representation of Dγα,β and Iγα,β
that are very useful in what follows. Before that, we need the following
technical result of the asymptotic behavior of {Ptα,β }t≥0 at infinity.
R1
Lemma 8. If −1 f (y)µα,β (dy) = 0 and f has continuos derivatives up to the
second order, then
¯
¯
¯
¯ ∂ α,β
¯ Pt f (x)¯ ≤ Cf,α,β (1 + |x|)e−(α+β+2)1/2 t .
(17)
¯
¯ ∂t
As a consequence we have that the Poisson-Jacobi semigroup {Ptα,β }t≥0 , has
L∞
α,β
exponential decay on (C0α,β )⊥ =
n=1 Cn . More precisely, if we have
R1
f (y)µα,β (dy) = 0,
−1
|Ptα,β f (x)| ≤ Cf,α,β (1 + |x|)e−(α+β+2)
1/2
t
.
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
(18)
104
Cristina Balderrama, Wilfredo Urbina
¯
¯
¯ ∂ α,β
¯
Proof. First, let us prove that ¯ ∂t
Tt f (x)¯ ≤ Cf,α,β (1 + |x|)e−(α+β+2)t .
Since
∂ α,β
f
∂t Tt
= Lα,β Ttα,β f and
∂ α,β
T f
∂x t
∂ 2 α,β
T f
∂x2 t
¶
∂f
∂x
µ 2 ¶
∂ f
= e−2(α+β+3)t Ttα+2,β+2
∂x2
µ
= e−(α+β+2)t Ttα+1,β+1
we have,
¯
¯
¯ ∂ α,β
¯
¯ Tt f (x)¯
¯ ∂t
¯
2
−2(α+β+3)t
Ttα+2,β+2
µ¯ 2 ¯¶
¯∂ f ¯
¯
¯
¯ ∂x2 ¯ (x) +
≤
|1 − x |e
+
µ¯ ¯¶
¯ ∂f ¯
(|β − α + 1| + (α + β + 2)|x|)e−(α+β+2)t Ttα+1,β+1 ¯¯ ¯¯ (x).
∂x
Also
e−2(α+β+3)t ≤ e−(α+β+2)t , |1−x2 | ≤ 1+|x|, |β −α+1|+(α+β +2)|x| ≤ Cα,β (1+|x|)
and as f has continue
derivatives
¯ ¯
¯ to the second order, there exist a constant
¯ 2 up
¯ ∂f ¯
¯∂ f ¯
Cf such that ¯ ∂x ¯ ≤ Cf and ¯ ∂x2 ¯ ≤ Cf , therefore,
¯
¯
¯ ¯¯ ∂
¯
¯ α,β α,β
¯ ¯
α,β
¯L Tt f (x)¯ = ¯ Tt f (x)¯¯ ≤ Cf,α,β (1 + |x|)e−(α+β+2)t .
∂t
Now,
∂ α,β
1
P f=√
∂t t
π
Z
∞
0
e−u t α,β α,β
√
L Tt2 /4u f du
u 2u
hence, by the change of variables u = (α + β + 2)s we have
¯
¯
Z
¯ ∂ α,β
¯
|x|) ∞ −u −3/2 −(α+β+2)t2 /4u
¯ Pt f (x)¯ ≤ Cf,α,β (1 +
√
e u
te
du
¯ ∂t
¯
2 π
Z 0∞
2
t
= Cf,α,β (1 + |x|)
e−(α+β+2)s √ s−3/2 e−t /4s ds
2
π
Z0 ∞
1/2
= Cf,α,β (1 + |x|)
e−(α+β+2)s µt (ds)
0
=
Cf,α,β (1 + |x|)e
−(α+β+2)1/2 t
By hypothesis, since we are assuming that
R1
−1
.
f (y)µα,β (dy) = 0,
lim Ptα,β f (x) = 0.
t→∞
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
(19)
Fractional Integration and Fractional Differentiation.
Then
¯
¯
¯ α,β
¯
¯Pt f (x)¯
Z
≤
t
=
∞
¯
¯
Z
¯
¯ ∂ α,β
¯ Ps f (x)¯ ds ≤ Cf,α,β
¯
¯ ∂s
∞
105
(1 + |x|)e−(α+β+2)
1/2
s
ds
t
Cf,α,β (1 + |x|)e−(α+β+2)
1/2
t
.
¤
Remember that, since {Ptα,β }t≥0 is an strongly continuos semigroup, we
have
lim+ Ptα,β f (x) = f (x)
(20)
t→0
Now we are ready to give the alternate representation of Dγα,β and Iγα,β ,
R1
Proposition 9. Suppose f ∈ C 2 ([−1, 1]) such that −1 f (y)µα,β (dy) = 0,
then
Z ∞
1
∂
α,β
Dγ f =
t−γ Ptα,β f dt, 0 < γ < 1,
(21)
γcγ 0
∂t
Z ∞
1
∂
Iγα,β f = −
tγ Ptα,β f dt, γ > 0.
(22)
γΓ(γ) 0
∂t
Proof. Let us start proving (21). Integrating by parts in (15) we get
Z b
³
´
1
Dγα,β f (x) =
lim+
t−γ−1 Ptα,β f (x) − f (x) dt
cγ a→0 a
b→∞
½ −γ ³
¾
Z
´¯
1
t
1 b −γ ∂ α,β
b
α,β
¯
lim
Pt f (x) − f (x) a +
t
P f (x)dt
=
cγ a→0+ −γ
γ a
∂t t
b→∞
Z ∞
1
∂
t−γ Ptα,β f (x)dt
=
γcγ 0
∂t
since, by (19), (20) and the previous lemma, we have
Ã
!
Pbα,β f (x) − f (x)
lim
=0
b→∞
bγ
and
¯
¯ α,β
¯ Pa f (x) − f (x) ¯
¯
¯
lim ¯
¯
aγ
a→0+
≤
≤
1
lim+ γ
a→0 a
Z
a
0
¯
¯
¯
¯ ∂ α,β
¯ Ps f (x)¯ ds
¯
¯ ∂s
Cf,α,β (1 + |x|) lim+
a→0
1 − e−(α+β+2)
aγ
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
1/2
a
= 0.
106
Cristina Balderrama, Wilfredo Urbina
Let us prove now (22). Again, by integrating by parts, we have
Z b
1
Iγα,β f (x) =
lim+
tγ−1 Ptα,β f (x)dt
Γ(γ) a→0 a
b→∞
½ γ
¾
Z
¯b
t α,β
1 b γ ∂ α,β
1
¯
lim
P f (x) a −
t
P f (x)dt
=
Γ(γ) a→0+ γ t
γ a
∂t t
b→∞
Z ∞
1
∂
= −
tγ Ptα,β f (x)dt,
γΓ(γ) 0
∂t
since, by the previous result
¯
¯
−1/2
¯
¯
b
lim ¯Pbα,β f (x)bγ ¯ ≤ Cd,f (1 + |x|) lim bγ e−(α+β+2)
=0
b→∞
b→∞
and
¯
¯
lim+ ¯Paα,β f (x)aγ ¯ = 0.
a→0
¤
Observe that the previous proposition is also true for the Jacobi polynomials of order
R 1 n > 0, and therefore is true for any nonconstant polynomial f
such that −1 f (y)µα,β (dy) = 0.
Now let us write
Z ∞
1/2
Ptα,β f (x) =
Tsα,β f (x)µt (ds)
0
Z
Z
1
=
∞
[
−1
Z 1
=
0
1/2
pα,β (s, x, y)µt (ds)]f (y)µα,β (dy)
k α,β (t, x, y)f (y)µα,β (dy),
−1
where
Z
k
α,β
(t, x, y) =
0
∞
1/2
pα,β (s, x, y)µt (ds)
(23)
and define the operator Qα,β
as
t
Qα,β
t f (x) = −t
∂ α,β
P f (x) =
∂t t
Z
1
q α,β (t, x, y)f (y)µα,β (dy),
(24)
−1
∂ α,β
k (t, x, y).
with q α,β (t, x, y) = −t ∂t
Following [6] we get immediately from (21) and (22), the following formulas
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
Fractional Integration and Fractional Differentiation.
107
Corollary 10. SupposeR f differentiable with continuos derivatives up to the
1
second order such that −1 f (y)µα,β (dy) = 0, then we have
Z ∞
1
−γDγα,β f =
t−γ−1 Qα,β
(25)
t f dt, 0 < γ < 1,
cγ 0
Z ∞
1
α,β
tγ−1 Qα,β
(26)
γIγ f =
t f dt, γ > 0.
Γ(γ) 0
An interesting use of the family {Qα,β
t } is that it allows to give a version
of Calderón’s reproduction formula for the Jacobi measure,
R1
Theorem 11. i) Suppose f ∈ L1 ([−1, 1], µα,β ) such that −1 f (y)µα,β (dy) =
0, then we have
Z ∞
dt
f=
Qα,β
.
(27)
t f
t
0
R1
ii) Suppose f a polynomial such that −1 f (y)µα,β (dy) = 0, then we have
Z ∞Z ∞
¡ α,β ¢ ds dt
f = Cγ
t−γ sγ Qα,β
Qs f
, 0 < γ < 1.
(28)
t
s t
0
0
Also,
Z
0
∞
Z
0
∞
¡ α,β ¢ ds dt
t−γ sγ Qα,β
Qs f
=
t
s t
Z
∞
u
0
∂ 2 α,β
P f du.
∂u2 u
(29)
Formula (28) is the version of Calderón’s reproduction formula for the
Jacobi measure.
Proof. i) Using (20) and (19) we have,
Z ∞
Z b
¯b
∂ α,β
α,β dt
Qt f
= lim+ (−
Pt f dt) = lim+ (−Ptα,β f )¯a = f.
t
a→0
a→0
0
a ∂t
b→∞
b→∞
R1
Let us prove (28), given f a polynomial such that −1 f (y)µα,β (dy) = 0, by
Corollary 10, we have
Z ∞
¡ α,β ¢
¡
¢
1
t−γ−1 Qα,β
Dγα,β Iγα,β f = −
Iγ f dt.
(30)
t
γcγ 0
Now, by (26) and the linearity of Qα,β
t , we have
Z ∞
¡ α,β ¢
1
α,β
sγ−1 Qα,β
Qα,β
I
f
=
t (Qs f )(y)ds.
t
γ
γΓ(γ) 0
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
108
Cristina Balderrama, Wilfredo Urbina
Substituting in (30)
¡
¢
f = Dγα,β Iγα,β f = Cγ
Z
∞
0
Z
∞
0
¡ α,β ¢
t−γ−1 sγ−1 Qα,β
Qs f dsdt,
t
with Cγ = − γ 2 cγ1Γ(γ) .
To show (29), let us integrate by parts, and by Lemma 8 we have
¯∞ Z ∞
Z ∞
¯
∂2
∂
∂ α,β
u 2 Puα,β f (x)du = u Puα,β f (x)¯¯ −
Pu f (x)du
∂u
∂u
∂u
0
0
0
Z ∞
¯∞
∂ α,β
= −
Pu f (x)du = −Puα,β f (x)¯0
∂u
0
=
P0α,β f (x) = f (x).
¤
Now let us consider the Jacobi Potential Spaces. The Jacobi Potential
space of order γ > 0, Lpγ ([−1, 1], µα,β ), for 1 < p < ∞, is defined as the
completion of the polynomials with respect to the norm
kf kp,γ := k(I − Lα,β )γ/2 f kp .
That is to say f ∈ Lpγ ([−1, 1], µα,β ) if, and only if, there is a sequence of
polynomials {fn } such that limn→∞ kfn − f kp,γ = 0.
As in the classical case, the Jacobi Potential space Lpγ ([−1, 1], µα,β ) can
also be defined as the image of Lp ([−1, 1], µα,β ) under the Bessel Potential
Jγα,β , that is,
Lpγ ([−1, 1], µα,β ) = Jγα,β Lp ([−1, 1], µα,β ).
For the details of the proof of this equivalence, we refer to [2].
Now, let us consider some inclusion properties among Jacobi Potential
Spaces,
Proposition 12.
i) If p < q, then Lqγ ([−1, 1], µα,β ) ⊆ Lpγ ([−1, 1], µα,β )
for each γ > 0.
ii) If 0 < γ < δ, then Lpδ ([−1, 1], µα,β ) ⊆ Lpγ ([−1, 1], µα,β ) for each 0 < p <
∞.
Proof. i) For γ fixed, it follows immediately by Hölder’s inequality.
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
Fractional Integration and Fractional Differentiation.
109
ii) Let f be a polynomial and consider
φ = (I − Lα,β )δ/2 f =
∞
X
(1 + λk )δ/2 Jkα,β f,
k=0
which is also a polynomial. Then φ ∈ Lpδ ([−1, 1], µα,β ), kφkp = kf kp,δ and
α,β
J(γ−δ)
φ = (I − Lα,β )(γ−δ)/2 φ = (I − Lα,β )γ/2 f. By the Lp continuity of Bessel
Potentials,
kf kp,γ = k(I − Lα,β )γ/2 f kp = kJ(γ−δ) φkp ≤ Cp kf kp,δ .
Now let f ∈ Lpδ ([−1, 1], µα,β ). Then there exists g ∈ Lp ([−1, 1], µα,β ) such
that f = Jδα,β g and a sequence of polynomials {gn } in Lp ([−1, 1], µα,β ) such
that limn→∞ kgn − gkp = 0. Set fn = Jδα,β gn . Then limn→∞ kfn − f kp,δ = 0,
and
kfn − f kp,γ = k(I − Lα,β )γ/2 (fn − f )kp = k(I − Lα,β )γ/2 (I − Lα,β )−δ/2 (gn − g)kp
= k(I − Lα,β )(γ−δ)/2 (gn − g)kp = kJ(γ−δ) (gn − g)kp .
By the Lp continuity of Bessel Potentials limn→∞ kfn − f kp,γ = 0.
Therefore,
kf kp,γ
≤ kfn − f kp,γ + kfn kp,γ
≤ kfn − f kp,γ + kfn kp,δ ,
taking limit as n goes to infinity, we obtain the result.
Let us consider the space
[
Lγ ([−1, 1], µα,β ) =
Lpγ ([−1, 1], µα,β ).
¤
p>1
Lγ ([−1, 1], µα,β ) is the natural domain of Dγα,β . We define it in this space as
follows.
Let f ∈ Lγ ([−1, 1], µα,β ), then there is p > 1 such that f ∈
Lpγ ([−1, 1], µα,β ) and a sequence {fn } polynomials such that limn→∞ fn = f
in Lpγ ([−1, 1], µα,β ). We define for f ∈ Lγ ([−1, 1], µα,β )
Dγα,β f = lim Dγα,β fn
n→∞
in Lp ([−1, 1], µα,β ). The next theorem shows that Dγα,β is well defined and
also inequality (31) gives us a characterization of the Potential Spaces,
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
110
Cristina Balderrama, Wilfredo Urbina
Theorem 13. Let γ > 0 and 1 < p, q < ∞.
i) If {fn } is a sequence of polynomials such that limn→∞ fn = f in
Lpγ ([−1, 1], µα,β ), then limn→∞ Dγα,β fn ∈ Lp ([−1, 1], µα,β ) and the limit
does not depend on the choice of the sequence {fn }.
T
If f ∈ Lpγ ([−1, 1], µα,β ) Lqγ ([−1, 1], µα,β ), then the limit does not depend on the choice of p or q.
Thus Dγα,β is well defined on Lγ ([−1, 1], µα,β ).
ii) f ∈ Lpγ ([−1, 1], µα,β ) if, and only if, Dγα,β f ∈ Lp ([−1, 1], µα,β ). Moreover, there exists positive constants Ap,γ and Bp,γ such that
Bp,γ kf kp,γ ≤ kDγα,β f kp ≤ Ap,γ kf kp,γ .
(31)
Proof.
P∞
ii) First, let us note that for p = n=0 Jnα,β p polynomial,
Dγα,β Jγα,β p
¶γ/2
∞ µ
X
λn
=
Jnα,β p,
1
+
λ
n
n=0
that is, the operator Dγα,β Jγα,β is a multiplier with associated function
³
´γ/2
³
´γ/2
λn
1
1
Φ(n) = 1+λ
=
h(
)
where
h(z)
=
, and therefore by
λn
z+1
n
Meyer’s theorem it is Lp -continuos.
Let f be a polynomial and let φ be a polynomial such that f = Jγα,β φ.
We have that kf kp,γ = kφkp and by the continuity of the operator Dγα,β Jγα,β
kDγα,β f kp = kDγα,β Jγα,β φkp ≤ Ap,γ kφkp = Ap,γ kf kp,γ .
To prove the converse, let us suppose that f polynomial, then Dγα,β f is
also a polynomial, and therefore Dγα,β f ∈ Lp ([−1, 1], µα,β ). Consider
φ = (I − Lα,β )γ/2 f =
∞
X
(1 + λk )γ/2 Jkα,β f =
k=0
k=0
The mapping
g=
∞
X
k=0
¶γ/2
∞ µ
X
1 + λk
Jkα,β g 7→
¶γ/2
∞ µ
X
1 + λk
k=0
λk
λk
Jkα,β (Dγα,β f ).
Jkα,β g
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
Fractional Integration and Fractional Differentiation.
³
is a multiplier with associated function Φ(k) =
1+λk
λk
111
´γ/2
= h( λ1k ) where
h(z) = (z + 1)γ/2 , so by Meyer’s theorem, taking g = Dγα,β f we have
kf kp,γ = kφkp ≤ Bp,γ kDγα,β f kp .
Thus we get (31) for polynomials. For the general case consider f ∈
Lpγ ([−1, 1], µα,β ), then there exists g ∈ Lp ([−1, 1], µα,β ) such that f = Jγα,β g
and a sequence {gn } of polynomials such that limn→∞ kgn − gkp = 0. Let
fn = Jγα,β gn , then limn→∞ kfn − f kp,γ = 0. Then, by the continuity of the
operator Dγα,β Jγα,β and as limn→∞ kgn − gkp = 0,
lim kDγα,β (fn − f )kp = lim kDγα,β Jγα,β (gn − g)kp = 0.
n→∞
Then, as
n→∞
Bp,γ kfn kp,γ ≤ kDγα,β fn kp ≤ Ap,γ kfn kp,γ ,
the results follows by taking the limit as n goes to infinity in this inequality.
i) Let {fn } be a sequence of polynomials such that limn→∞ fn = f in
Lpγ ([−1, 1], µα,β ). Then, by (31)
lim kDγα,β fn kp ≤ Bp,γ lim kfn kp,γ = Bp,γ kf kp,γ ,
n→∞
n→∞
hence, limn→∞ Dγα,β fn ∈ Lp ([−1, 1], µα,β ).
Now suppose that {qn } is another sequence of polynomials such that
limn→∞ qn = f in Lpγ ([−1, 1], µα,β ). By (31)
Bp,γ kfn kp,γ ≤ kDγα,β fn kp ≤ Ap,γ kfn kp,γ
and
Bp,γ kqn kp,γ ≤ kDγα,β qn kp ≤ Ap,γ kqn kp,γ .
Taking the limit as n goes to infinity
Bp,γ kf kp,γ ≤ lim kDγα,β fn kp ≤ Ap,γ kf kp,γ
n→∞
Bp,γ kf kp,γ ≤ lim kDγα,β qn kp ≤ Ap,γ kf kp,γ ,
n→∞
limn→∞ Dγα,β fn
therefore
= limn→∞ Dγα,β qn in Lp ([−1, 1], µα,β ) and the limit
does not depends on the choice of the sequence.
T
Finally, let us suppose that f ∈ Lpγ ([−1, 1], µα,β ) Lqγ ([−1, 1], µα,β )
and, without loss of generality, that p ≤ q. By Proposition 12, i),
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
112
Cristina Balderrama, Wilfredo Urbina
Lqγ ([−1, 1], µα,β ) ⊆ Lpγ ([−1, 1], µα,β ) and therefore f ∈ Lqγ ([−1, 1], µα,β ).
Then, if {fn } is a sequence of polynomials such that limn→∞ fn = f in
Lqγ ([−1, 1], µα,β ) (hence in Lpγ ([−1, 1], µα,β )), we have
lim Dγα,β fn ∈ Lq ([−1, 1], µα,β ) = Lp ([−1, 1], µα,β )
n→∞
\
Lq ([−1, 1], µα,β ).
Therefore the limit does not depends on the choice of p or q.
¤
References
[1] Ané C., Blacheré, D., Chaifaı̈ D. , Fougères P., Gentil, I. Malrieu F. ,
Roberto C., G. Sheffer G. Sur les inégalités de Sobolev logarithmiques.
Panoramas et Synthèses 10, Société Mathématique de France. Paris,
(2002).
[2] Balderrama, C. Sobre el semigrupo de Jacobi. Tesis de Maestrı́a. UCV.
February 2006.
[3] Bakry, D. Remarques sur les semi-groupes de Jacobi. In Hommage a P.A.
Meyer et J. Neveau. 236, Asterique, 1996, 23–40.
[4] Bakry, D. Functional inequalities for Markov semigroups. Notes of the
CIMPA course in Tata Institute, Bombay, November 2002.
[5] Bavinck, H. A Special Class of Jacobi Series and Some Applications. J.
Math. Anal. and Applications. 37, 1972, 767–797.
[6] Gatto A. E, Segovia C, Vági S. On Fractional Differentiation and Integration on Spaces of Homogeneous Type, Rev. Mat. Iberoamericana 12
(1996) 111-145.
[7] Graczyk P., Loeb J.J., López I.A., Nowak A., Urbina W. Higher order
Riesz Transforms, fractional derivatives and Sobolev spaces for Laguerre
expansions. Math. Pures Appl. (9), 84 2005, no. 3, 375–405.
[8] L. Gross, Logarithmic Sobolev inequalities and contractivity properties of
semigroups, Dirichlet forms (Varenna, 1992), Springer, Berlin, 1993, p.
54–88.
[9] López I.A., Urbina W. Fractional Differentiation for the Gaussian Measure and applications. Bull. Sci. math. 128, 2004, 587–603.
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113
Fractional Integration and Fractional Differentiation.
113
[10] Meyer, P. A. Transformations de Riesz pour les lois Gaussiennes. Lectures Notes in Math 1059, 1984 Springer-Verlag. 179–193.
[11] Szegö, G. Orthogonal polynomials. Colloq. Publ. 23. Amer. Math. Soc.
Providence 1959.
[12] Urbina, W. Análisis Armónico Gaussiano. Trabajo de ascenso, Facultad
de Ciencias, UCV. 1998.
[13] Urbina, W. Semigrupos de Polinomios Clásicos y Desigualdades Funcionales. Notas de la Escuela CIMPA–Unesco–Venezuela. Mérida (2006).
[14] Zygmund, A. Trigonometric Series. 2nd. ed. Cambridge Univ. Press.
Cambridge (1959).
Divulgaciones Matemáticas Vol. 15 No. 2(2007), pp. 93–113