Hardy type inequalities for fractional integral operators.

HARDY TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATORS Vu Kim Tuan1 , Rudolf Gorenflo2 Abstract Hardy type inequalities for the fractional integral operator with sharp constants on finite and infinite intervals are given. Mathematics Subject Classifications: 26A33, 26D10. Key Words and Phrases: Hardy inequality, Fractional integral operator, Singular system.

1. Introduction Let Iaα be the fractional integral operator defined by (Iaα f )(x) =

Z

x

a

(x − t)α−1 f (t) dt. Γ(α)

(1)

For this operator the following inequality, known as the Hardy inequality, (

∞

Z

x

0

−αp

|(I0α f ) (x)|p

dx ≤

Γ(1/p0 ) Γ(α + 1/p0 )

)p Z

∞

|f (x)|p dx,

0

1 < p < ∞,

1 1 + = 1, (2) p p0

is valid ([6]). Here the constant in the right hand side is sharp. Several generalizations of this inequality for Iaα with sharp constants on infinite intervals are given in [6]. Recently, some weighted inequalities for fractional integral operators on finite and infinite intervals are also obtained ([1, 9]), but their method does not allow to get the sharp constants. In this paper we get analogs of the inequality (2) for Iaα on the finite interval (−1, 1) and on the half-line (0, ∞) in L2 space, in which the constants are best possible.

2. Hardy type inequality The folowing lemma is used hereafter to establish Hardy type inequalities. Lemma. Let U, V be Hilbert spaces and assume A : U → V to be an infinite–dimensional linear compact operator. If there exists an orthonomal basis {un }IN 0 of U , an orthonomal system (not neccesary a basis) {vn }IN 0 of V , a sequence of non-increasing positive numbers sn such that Aun = sn vn for n ∈ IN 0 , 1 2

Supported by the Alexander von Humboldt Foundation Supported by NATO Collaborative Research Grant CRG 940843

1

then s0 is the norm of the operator A. Here IN 0 is the set of nonnegative integers. The system {sn , un , vn } is called a singular system of the operator A and plays an important rolle in the theory of ill–posed problems ([2]). The proof of this lemma can be found in [4]. Theorem. For all f ∈ L2 (−1, 1) the inequality Z

1

2 −α

(1 − x )

−1

α |(I−1 f )(x)|2 dx

Γ(1 − α) Z 1 |f (x)|2 dx, ≤ Γ(1 + α) −1

0 < α < 1,

(3)

is valid, with equality attained when f (x) ≡ 1. α

(1+x) α and the equality is easily verified. Proof. When f (x) ≡ 1 we have (I−1 f )(x) = Γ(1+α) 2 −α Let U = L2 (−1, 1) and V = L2 ((−1, 1); (1 − x ) ). Taking now, for n ∈ IN 0 ,

v u u (n + 1/2) n! un (x) = t Pn (x),

Γ(n + 1)

v u u vn (x) = t

(n + 1/2) (1 + x)α Pn(−α,α) (x), 0 < α < 1, Γ(n − α + 1)Γ(n + α + 1)

(4)

where Pn (x) and Pn(α,β) (x) are Legendre and Jacobi polynomials ([3]), and applying the orthogonality relations of Legendre and Jacobi polynomials Pn (x), Pn(−α,α) (x), we obtain Z

1

−1

Pn (x)Pm (x) dx =

Z

1

−1

(1 − x2 )−α Pn−α,α (x)Pm−α,α (x) dx = δmn ,

m, n = 0, 1, . . . ,

that means {un }IN 0 and {vn }IN 0 are orthonomal basis in U and V , respectively. Using now the formula ([7]) Z

1

−1

(x − t)α−1 n! Pn (t)dt = (1 + x)α Pn(−α,α) (x), Γ(α) Γ(n + α + 1)

we have

v u u Γ(n − α + 1) α (I−1 un )(x) = t vn (x).

Γ(n + α + 1)

r

From the Lemma we have that

Γ(1−α) Γ(1+α)

α is the norm of the operator I−1 : U → V . Thus

the Theorem is proved.

3. Some other generalizations of the Hardy inequality

2

1. Let, for n ∈ IN 0 , v u u (2n + β + γ + 1) n! Γ(n + β + γ + 1) un (x) = t (1 + x)β Pn(γ,β) (x), β+γ+1

2

v u u vn (x) = t

sn

Γ(n + β + 1)Γ(n + γ + 1)

(2n + β + γ + 1) n! Γ(n + β + γ + 1) (1 + x)α+β Pn(γ−α,α+β) (x), + α + β + 1)Γ(n + γ − α + 1)

2β+γ+1 Γ(n

v u u Γ(n + β + 1)Γ(n + γ − α + 1) = t ,

Γ(n + γ + 1)Γ(n + α + β + 1)

γ + 1 > α > 0, β > −1.

(5)

Then {un }n∈IN 0 is an orthonomal basis in the space L2 ((−1, 1); (1−x)γ (1+x)−β ), {vn }n∈IN 0 is an orthonomal basis in the space L2 ((−1, 1); (1 − x)γ−α (1 + x)−α−β ), and {sn }n∈IN 0 is a sequence of positive decreasing numbers approaching 0. The integral ([7]) Z

x

−1

Γ(n + β + 1) (x − t)α−1 (1 + t)β Pn(γ,β) (t) dt = (1 + x)α+β Pn(γ−α,α+β) (x) Γ(α) Γ(n + α + β + 1)

(6)

can be rewritten in the form α (I−1 un )(x) = sn vn (x),

where {sn , un , vn } are as in (5). Therefore, applying the Lemma, we have Z

1

−1

α (1 − x)γ−α (1 + x)−α−β |(I−1 f )(x)|2 dx

Γ(β + 1)Γ(γ − α + 1) Z 1 ≤ (1 − x)γ (1 + x)−β |f (x)|2 dx, γ + 1 > α > 0, β > −1, Γ(γ + 1)Γ(α + β + 1) −1

(7)

and equality is attained with f (x) = (1 + x)β . 2. Taking in formula (7) β = γ = 0 we obtain the Theorem. Putting β = −γ = −α we get Z 1 Γ(1 − α) Z 1 α 2 (1 − x2 )α |f (x)|2 dx, 0 < α < 1. (8) |(I−1 )f )(x)| dx ≤ Γ(1 + α) −1 −1 3. Let, for n ∈ IN 0 , s

n! xγ L(γ) n (x), Γ(n + γ + 1)

s

n! xα+γ Ln(α+γ) (x), Γ(n + α + γ + 1)

un (x) = vn (x) = sn

v u u = t

Γ(n + γ + 1) , Γ(n + α + γ + 1)

γ > −1.

(9)

Here the functions Lγn (x) are generalized Laguerre polynomials ([3]). From properties of generalized Laguerre polynomials it follows that {un }n∈IN 0 is an orthonomal basis 3

in the space L2 ((0, ∞); x−γ exp (−x)), {vn }n∈IN 0 is an orthonomal basis in the space L2 ((0, ∞); x−α−γ exp (−x)). It is easy to see that {sn }n∈IN 0 is a sequence of positive decreasing numbers approaching 0. The integral (see [7]) Z 0

x

(x − t)α−1 γ (γ) Γ(n + γ + 1) t Ln (t) dt = xα+γ Ln(α+γ) (x) Γ(α) Γ(n + α + γ + 1)

(10)

can be rewritten in the form (I0α un )(x) = sn vn (x), where {sn , un , vn } are as in (10). Therefore, applying the Lemma, we have Z

∞

x

−α−γ

0

exp (−x)|(I0α )f )(x)|2

Γ(γ + 1) Z ∞ −γ dx ≤ x exp (−x)|f (x)|2 dx, Γ(α + γ + 1) 0 α, γ + 1 > 0, (11)

and equality is attained with f (x) = xγ . 4. An analogous inequality can be obtained for the Weyl fractional integral operator I−α , defined by Z ∞ (t − x)α−1 α (I− f )(x) = f (t) dt. (12) Γ(α) x Let, for n ∈ IN 0 , s

n! exp (−x)L(γ) n (x) , Γ(n + γ + 1)

s

n! exp (−x)L(γ−α) (x) , n Γ(n + γ − α + 1)

un (x) = vn (x) = sn

v u u Γ(n + γ − α + 1) = t ,

Γ(n + γ + 1)

γ + 1 > α > 0.

(13)

Then {un }n∈IN 0 and {vn }n∈IN 0 are orthonormal bases in the spaces L2 (IR+ ; xγ exp (x)) and L2 (IR+ ; xγ−α exp (x)) respectively, and {sn }n∈IN 0 is a sequence of decreasing positive numbers. Applying the formula Z

∞

x

(t − x)α−1 (γ−α) exp (−t)L(γ) (x) , n (t)dt = exp (−x)Ln Γ(α)

(14)

(see [7], p. 463, formula (7) ), we obtain (I−α un )(x) = sn vn (x),

(15)

where {sn , un , vn } are as in (13). Therefore, from the Lemma we get Z 0

∞

x

γ−α

exp (x)|(I−α f )(x)|2 dx

≤

Z

∞

xγ exp (x)|f (x)| dx, γ + 1 > α > 0,

0

4

(16)

with equality attained when f (x) = exp (−x). Remark. The method used here is applied in [5, 10] to get singular values of fractional integral operators (when it is possible) and their asymptotics.

References [1] Andersen K.F. and Heinig H.P. Weighted norm inequalities for certain integral operators. SIAM J. Math. Anal. 14(1983), no. 4, p. 834–844. [2] Baumeister J. Stable Solution of Inverse Problems. Vieweg und Sohn, Braunschweig/Wiesbaden, 1987. [3] Erdelyi A. et al. Higher Transcendental Functions, vol. 2, McGraw–Hill, New York– Toronto–London, 1953. [4] Gohberg I.C. and Krein M.G. Introduction to the Theory of Linear Nonselfadjoint Operators. Amer. Math. Soc., Providence, 1969. [5] Gorenflo R. and Vu Kim Tuan. Singular value decomposition of fractional integration operators in L2 –spaces with weights. Journal of Inverse and Ill-posed Problems (to appear). [6] Hardy G.H., Littlewood J.E. and Polya G. Inequalities. Cambridge Univ. Press, Cambridge, 1959. [7] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev. Integrals and Series, Vol. 2: Special Functions. Gordon and Breach Science Publishers, New York–London–Paris– Montreux–Tokyo–Melbourne, 1990. [8] Samko S.G., Kilbas A.A. and Marichev O.I. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science Publishers, Switzerland–Australia–Belgium–France–Germany–Great Britain–India– Japan–Malaysia–Netherlands–Russia–Singapore–USA, 1993. [9] Stepanov V.D. Two–weighted estimates of Riemann–Liouville integrals. Math. USSR Izvestiya 36(1991), no. 3, p. 669–681. Translated from the Russian. [10] Vu Kim Tuan and Gorenflo R. Asymptotics of singular values of fractional integral operators. Inverse Problems 10(1994), p. 949–955.

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