On a k -Fold Beta Integral Formula
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On a k-Fold Beta Integral Formula Di Wu1 · Zuoshunhua Shi2 · Xudong Nie3 · Dunyan Yan4 Received: 29 January 2019 © Mathematica Josephina, Inc. 2019
Abstract In this paper, we investigate some necessary and sufficient conditions which ensure validity of a k-fold Beta integral formula
k
Rn i=1
|x i − t|−di dt = Cd1 ,...,dk
−αi j i x − x j
(0.1)
1≤i< j≤k
for any x i ∈ Rn and some nonzero real numbers di with i = 1, 2, . . . , k. We establish that Eq. (0.1) holds if and only if max{d1 , d2 } < n < d1 + d2 when k = 2 ,
(0.2)
max{d1 , d2 , d3 } < n, d1 + d2 + d3 = 2n when k = 3 .
(0.3)
and
This yields a complete answer to the question raised by Grafakos and Morpurgo in [4]. In addition, it turns out that formula (0.1) does not hold if k ≥ 4. For those k, d1 , d2 , . . . , dk not satisfying (0.2) or (0.3), we prove that the real-valued integral k |x i − t|−di dt can be represented as a function of distances of consecutive Rn i=1
differences of the sequence x 1 , x 2 , . . . , x k . Keywords Multilinear fractional integral · Beta integral formula · Gamma Function AMS Subject Classification (2010) 42B20 · 42B35
Extended author information available on the last page of the article
123
D. Wu et al.
1 Introduction Multi-fold Beta integral is a generalization of Euler’s Beta integral and has a lot of important applications in number theory and analysis. Multilinear fractional integral inequalities can be used to investigate the endpoint estimates for restriction theorems of the Fourier transform [2] and are related with Beta integral [1]. Estimates on Beta integral formula can lead to the endpoint estimates for multilinear fractional integral. Furthermore, precise calculations on Beta integral formula imply the sharp constant of some important inequalities such as Hardy–Littlewood–Sobolev inequality [4] and [9]. It is well known that the following formula Rn
n−z 1 −z 2 −z 1 −z 2 1 dt = C z 1 ,z 2 ,n x 1 − x 2 x − t x 2 − t
(1.1)
holds for Re z i < n with i = 1, 2 and Re z 1 + Re z 2 > n, where x 1 , x 2 ∈ Rn , and n
C z 1 ,z 2 ,n = π 2
n−z 2 z 1 +z 2 −n 1 ( n−z ) 2 )( 2 )( 2
( z21 )( z22 )(n −
z 1 +z 2 2 )
.
In [4], Grafakos and Morpurgo obtained the following equations
3 −di n−d1 −d2 i 1 x − t dt = Cd1 ,d2 ,d3 ,n x 1 − x 2 x
Rn i=1
n−d1 −d3 n−d2 −d3 2 −x 3 , x − x 3
(1.2)
and
3 n−d1 −d2 −di i 1 ξ − τ dτ = Cd1 ,d2 ,d3 ,n ξ 1 − ξ 2 ξ
Sn i=1
n−d1 −d3 n−d2 −d3 2 −ξ 3 , ξ − ξ 3
(1.3)
for di < n with i = 1, 2, 3, and d1 + d2 + d3 = 2n, where x 1 , x 2 , x 3 ∈ Rn , ξ 1 , ξ 2 , ξ 3 ∈ Sn , and n
Cd1 ,d2 ,d3 ,n = π 2
123
n−d2 n−d3 1 ( n−d 2 )( 2 )( 2 )
( d21 )( d22 )( d23 )
.
On a k-Fold Beta Integral Formula
They utilized Eq. (1.2) to compute a threefold Selberg’s integral. Specifically, they proved the following equation
3 n−d2 n−d3 n−di −d j 1 ( n−d i 2 )( 2 )( 2 ) dξ 1 dξ 2 dξ 3 = |Sn |(2π )n ξ − ξ j ( d21 )( d22 )(
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