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MODIFIED FRACTIONAL INTEGRAL OPERATORS IN Lp SPACE WITH POWER–LOGARITHMIC WEIGHT Vu Kim Tuan∗ Department of Mathematics and Computer Science Faculty of Science, Kuwait University P.O. Box 5969, Safat 13060, Kuwait Megumi Saigo Department of Applied Mathematics Faculty of Science, Fukuoka University Fukuoka 814-80, Japan
Abstract The modified fractional integral operators, introduced in [4], are proved to be bounded from some Lp – space into some Lq – space with power-logarithmic weights.
1. Introduction In [4] the multidimensional modified fractional integrals of order α > 0 are defined as follows α Z 1 ∂n x1 xn = min ,..., − 1 f (t)dt, Γ(α + 1) ∂x1 . . . ∂xn R+n t1 tn + α Z n n x ∂ x (−1) 1 n α ,..., f (t)dt. X−,n f (x) = 1 − max Γ(α + 1) ∂x1 . . . ∂xn R+n t1 tn + α X+,n f (x)
(1) (2)
n Here R+ = {(t1 , . . . , tn ) | t1 , . . . , tn > 0} and ϕ+ (x) is constructed from a real-valued function ϕ(x) by ( ϕ(x) , ϕ(x) > 0 ϕ+ (x) = . (3) 0 , ϕ(x) ≤ 0 α α Operators X+,n , X−,n generalize the one-dimensional Riemann-Liouville and Weyl fractional integral operators [2], respectively. Some of their properties (product rules, index laws, mapping properties, composition structures, inverses, . . . ) are established in the ∗
Supported by Kuwait University research grant SM 112 and by the Fukuoka University
1
n ) [5]. space of functions Mγ (R+ α α n In this paper the modified fractional integral operators X+,n , X−,n on Mγ (R+ ) are proved to satisfy some Lp – Lq weighted inequalities. Hence, they can be continuously extended to some bounded operators in the classical space Lp with power-logarithmic n α α ), therefore, established in [4] only on Mγ (R+ and X−,n weight. Many properties of X+,n are still valid in the space Lp with weight.
2. Pitt’s Inequality for Mellin Transform Let fˆ(ξ) be the Fourier transform of f (x) : Rn → C fˆ(ξ) =
Z Rn
f (x)e−ixξ dx.
(4)
The following Pitt’s inequality holds k|ξ|−b fˆ(ξ)kr ≤ Ck|x|a f (x)kp if
(5)
!
n 1 1 0≤b< , 0≤a=b+n 1− − , 1 < p ≤ r < ∞, r p r
(6)
(see [3]). Inequality (5) under condition (6) is still valid after a linear transformation k|ξ − τ |−b fˆ(ξ)kr ≤ Ck|x − t|a f (x)kp ,
τ, t ∈ Rn .
(7)
Consider now the Mellin transform ∗
f (s) =
Z n R+
f (t)ts−1 dt,
and its inversion f (t) =
