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The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications

  • Bing He , Yong Hong EMAIL logo and Qiang Chen
Published/Copyright: May 25, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, we establish equivalent parameter conditions for the validity of multiple integral half-discrete Hilbert-type inequalities with the nonhomogeneous kernel G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) ( λ 1 λ 2 > 0 ) and obtain best constant factors of the inequalities in specific cases. In addition, we also discuss their applications in operator theory.

Keywords: multiple integral half-discrete Hilbert-type inequality; nonhomogeneous kernel; equivalent parameter condition; best constant factor; operator norm; bounded operator
MSC 2010: 26D15; 47A07

1 Introduction and preliminary knowledge

Suppose that 1 p + 1 q = 1 ( p > 1 ) , a ˜ = { a m } ∈ l p , b ˜ = { b n } ∈ L q , the classical Hilbert series inequality was obtained in 1925 [1]:

(1) ∑ n = 1 ∞ ∑ m = 1 ∞ a m b n m + n ≤ π sin ( π / p ) ∥ a ˜ ∥ p ∥ b ˜ ∥ q ,

where the constant factor π sin ( π / p ) is the best.

If f ( x ) ∈ L p ( 0 , + ∞ ) , g ( y ) ∈ L q ( 0 , + ∞ ) , the corresponding Hilbert integral inequality was obtained in 1934 [2]:

(2) ∫ 0 + ∞ ∫ 0 + ∞ f ( x ) g ( y ) x + y d x d y ≤ π sin ( π / p ) ∥ f ∥ p ∥ g ∥ q ,

where the constant factor π sin ( π / p ) is still the best.

Since (1) is of great significance for the study of boundedness and norm of series operator in l p and (2) is of great significance for the study of boundedness and norm of integral operator in L p ( 0 , + ∞ ) , Hilbert inequality has been widely concerned. To study the operator boundedness and operator norm from sequence space l to function space L or from function space L to sequence space l , the half-discrete Hilbert inequality has been paid more attention. In 2011, the following results were obtained [3]: If a ˜ = { a n } ∈ l p , f ( x ) ∈ L q ( 0 , + ∞ ) , then

(3) ∫ 0 + ∞ ∑ n = 1 ∞ a n f ( x ) n + x d x ≤ π sin ( π / p ) ∥ a ˜ ∥ p ∥ f ∥ q ,

the constant factor is also the best. Later on the equivalent conditions for validity of multiple integral half-discrete Hilbert-type inequality with generalized homogeneous kernel were discussed [4]. In [5], the parameter conditions for the optimal constant factor of half-discrete Hilbert-type inequality with homogeneous kernel in one dimension were established. Good results were obtained.

To further discuss the multiple integral half-discrete Hilbert-type inequality, we need to introduce the following notations: Suppose that m ∈ N + , x = ( x 1 , x 2 , … , x m ) , R + m = { x = ( x 1 , x 2 , … , x m ) : x i > 0 , i = 1 , 2 , … , m } . For ρ > 0 , the norm of x is defined by

∥ x ∥ m , ρ = ( x 1 ρ + x 2 ρ + ⋯ + x m ρ ) 1 / ρ .

Spaces l and L are defined by, respectively,

l p α = a ˜ = { a n } : ∥ a ˜ ∥ p , α = ∑ n = 1 ∞ n α ∣ a n ∣ p 1 / p < + ∞ , L q β ( R + m ) = f ( x ) : ∥ f ∥ q , β = ∫ R + m ∥ x ∥ m , ρ β ∣ f ( x ) ∣ q d x 1 / q < + ∞ .

If K ( n , ∥ x ∥ m , ρ ) = G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) ≥ 0 , then K ( n , ∥ x ∥ m , ρ ) is a nonhomogeneous nonnegative function. In this paper, we will discuss the equivalent parameter conditions under which the multiple integral half-discrete Hilbert-type inequality

(4) ∑ n = 1 ∞ ∫ R + m G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) a n f ( x ) d x ≤ M ∥ a ˜ ∥ p , α ∥ f ∥ q , β

can be established when λ 1 λ 2 > 0 . That is, what conditions do the parameters α , β , λ 1 , λ 2 , p , q meet if there is a constant M > 0 such that (4) holds? On the contrary, if there exists a constant M > 0 such that (4) holds, then what conditions do the parameters α , β , λ 1 , λ 2 , p , q satisfy? Such problems are undoubtedly very important theoretical problems, which have not been well solved at present. At the same time, we also discuss the best constant factor of (4) and its application in operator theory. More related literature can be found in [6,7, 8,9,10, 11,12,13, 14,15,16, 17,18,19, 20,21].

2 Some lemmas

By using the Hölder’s inequality of integral and series, we can easily get the following lemma.

Lemma 2.1

Assume that 1 p + 1 q = 1 ( p > 1 ) , a n ( x ) ≥ 0 , b n ( x ) ≥ 0 , Ω is measurable. Then, the mixed Hölder’s inequality can be obtained

∫ Ω ∑ n = 1 ∞ a n ( x ) b n ( x ) d x = ∑ n = 1 ∞ ∫ Ω a n ( x ) b n ( x ) d x ≤ ∫ Ω ∑ n = 1 ∞ a n p ( x ) d x 1 / p ∫ Ω ∑ n = 1 ∞ b n q ( x ) d x 1 / q .

Lemma 2.2

[22] If m ∈ N + , ρ > 0 , r > 0 , ψ ( u ) is measurable, then

∫ ∥ x ∥ m , ρ ≤ r ψ ( ∥ x ∥ m , ρ ) d x = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∫ 0 r ψ ( u ) u m − 1 d u , ∫ ∥ x ∥ m , ρ ≥ r ψ ( ∥ x ∥ m , ρ ) d x = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∫ r + ∞ ψ ( u ) u m − 1 d u ,

where Γ ( t ) is the Gamma function, and ∥ x ∥ m , ρ ≤ r represents the region Ω r = { x = ( x 1 , x 2 , … , x m ) : x i ≥ 0 , ∥ x ∥ m , ρ ≤ r } .

According to Lemma 2.2, one gets

∫ R + m ψ ( ∥ x ∥ m , ρ ) d x = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∫ 0 + ∞ ψ ( u ) u m − 1 d u .

Lemma 2.3

Assume that m ∈ N + , ρ > 0 , 1 p + 1 q = 1 ( p > 1 ) , λ 1 λ 2 > 0 , K ( n , ∥ x ∥ m , ρ ) = G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) is nonnegative measurable, 1 λ 2 m λ 1 + α λ 2 p − λ 2 + β λ 1 q = c , and K ( t , 1 ) t − α + 1 p + c is monotonically decreasing in ( 0 , + ∞ ) . Denote that

W 1 = ∫ 0 + ∞ K ( 1 , t ) t − β + m q + m − 1 d t , W 2 = ∫ 0 + ∞ K ( t , 1 ) t − α + 1 p + c d t .

Then,

λ 1 W 2 = λ 2 W 1

and

ω 1 ( n ) = ∫ R + m K ( n , ∥ x ∥ m , ρ ) ∥ x ∥ m , ρ − β + m q d x = n λ 1 λ 2 β + m q − m Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) W 1 , ω 2 ( x ) = ∑ n = 1 ∞ K ( n , ∥ x ∥ m , ρ ) n − α + 1 p + c ≤ ∥ x ∥ m , ρ λ 2 λ 1 α + 1 p − 1 − c W 2 .

Proof

It follows from m λ 1 + α λ 2 p − λ 2 + β λ 1 q = λ 2 c that 1 λ 1 − α λ 2 p + λ 2 q + λ 2 c − 1 = − β + m q + m − 1 , then

W 2 = ∫ 0 + ∞ K 1 , t λ 1 λ 2 t − α + 1 p + c d t = λ 2 λ 1 ∫ 0 + ∞ K ( 1 , u ) u λ 2 λ 1 − α + 1 p + c + λ 2 λ 1 − 1 d u = λ 2 λ 1 ∫ 0 + ∞ K ( 1 , u ) u 1 λ 1 − α λ 2 p + λ 2 q + λ 2 c − 1 d u = λ 2 λ 1 ∫ 0 + ∞ K ( 1 , u ) u − β + m q + m − 1 d u = λ 2 λ 1 W 1 .

Hence, λ 1 W 2 = λ 2 W 1 .

By Lemma 2.2, one gets

ω 1 ( n ) = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∫ 0 + ∞ K ( n , t ) t − β + m q + m − 1 d t = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∫ 0 + ∞ K 1 , t ⋅ n λ 1 λ 2 t − β + m q + m − 1 d t = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) n − λ 1 λ 2 − β + m q + m − 1 − λ 1 λ 2 ∫ 0 + ∞ K ( 1 , u ) u − β + m q + m − 1 d u = n λ 1 λ 2 β + m q − m Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) W 1 .

Note that K ( t , 1 ) t − α + 1 p + c is monotonically decreasing in ( 0 , + ∞ ) , thus

ω 2 ( x ) = ∑ n = 1 ∞ K ( n ⋅ ∥ x ∥ m , ρ λ 2 / λ 1 , 1 ) n − α + 1 p + c = ∥ x ∥ m , ρ λ 2 λ 1 α + 1 p − c ∑ n = 1 ∞ K ( n ⋅ ∥ x ∥ m , ρ λ 2 / λ 1 , 1 ) ( n ⋅ ∥ x ∥ m , ρ λ 2 / λ 1 ) − α + 1 p + c ≤ ∥ x ∥ m , ρ λ 2 λ 1 α + 1 p − c ∫ 0 + ∞ K ( t ⋅ ∥ x ∥ m , ρ λ 2 / λ 1 , 1 ) ( t ⋅ ∥ x ∥ m , ρ λ 2 / λ 1 ) − α + 1 p + c d t = ∥ x ∥ m , ρ λ 2 λ 1 α + 1 p − 1 − c ∫ 0 + ∞ K ( u , 1 ) u − α + 1 p + c d u = ∥ x ∥ m , ρ λ 2 λ 1 α + 1 p − 1 − c W 2 . □

3 Main results

Theorem 3.1

Suppose that m ∈ N + , ρ > 0 , λ 1 λ 2 > 0 , 1 p + 1 q = 1 ( p > 1 ) , K ( n , ∥ x ∥ m , ρ ) = G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) is nonnegative measurable, 1 λ 2 m λ 1 + α λ 2 p − λ 2 + β λ 1 q = c , K ( t , 1 ) t − α + 1 p and K ( t , 1 ) t − α + 1 p + c are monotonically decreasing in ( 0 , + ∞ ) , and

W 0 = ∣ λ 1 ∣ ∫ 0 + ∞ K ( t , 1 ) t − α + 1 p d t < + ∞ .

Then,

  1. The necessary and sufficient conditions for the validity of inequality

    (5) ∫ R + m ∑ n = 1 ∞ K ( n , ∥ x ∥ m , ρ ) a n f ( x ) d x ≤ M ∥ a ˜ ∥ p , α ∥ f ∥ q , β

    with some constant M > 0 is c ≥ 0 , where a ˜ = { a n } ∈ l p α , f ( x ) ∈ L q β ( R + m ) .

  2. For c = 0 , the best constant factor of (5) is

    inf M = W 0 ∣ λ 1 ∣ 1 / q ∣ λ 2 ∣ 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p .

Proof

(i) Suppose that (5) holds. If c < 0 , for ε = − c 2 ∣ λ 1 ∣ > 0 , taking

a n = n ( − α − 1 − ∣ λ 1 ∣ ε ) / p , n = 1 , 2 , …

and

f ( x ) = ∥ x ∥ m , ρ ( − β − m + ∣ λ 2 ∣ ε ) / q , 0 < ∥ x ∥ m , ρ ≤ 1 , 0 , ∥ x ∥ m , ρ > 1 .

It follows from Lemma 2.2 that

M ∥ a ˜ ∥ p , α ∥ f ∥ q , β = M ∑ n = 1 ∞ n − 1 − ∣ λ 1 ∣ ε 1 / p ∫ 0 < ∥ x ∥ m , ρ ≤ 1 ∥ x ∥ m , ρ − m + ∣ λ 2 ∣ ε d x 1 / q ≤ M 1 + ∫ 1 + ∞ t − 1 − ∣ λ 1 ∣ ε d t 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∫ 0 1 t − 1 + ∣ λ 2 ∣ ε d t 1 / q = M ε ∣ λ 1 ∣ 1 / p ∣ λ 1 ∣ 1 / q ( 1 + ∣ λ 1 ∣ ε ) 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p = 2 M − c λ 1 λ 2 1 / q 1 − c 2 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p < + ∞ .

Since K ( t , 1 ) t − α + 1 p is monotonically decreasing in ( 0 , + ∞ ) , then

∫ R + m ∑ n = 1 ∞ K ( n , ∥ x ∥ m , ρ ) a n f ( x ) d x = ∫ 0 < ∥ x ∥ m , ρ ≤ 1 ∥ x ∥ m , ρ ( − β − m + ∣ λ 2 ∣ ε ) / q ∑ n = 1 ∞ K ( n , ∥ x ∥ m , ρ ) n − ( α + 1 + ∣ λ 1 ∣ ε ) / p d x = ∫ 0 < ∥ x ∥ m , ρ ≤ 1 ∥ x ∥ m , ρ − β − m + ∣ λ 2 ∣ ε q − λ 2 λ 1 − α − 1 − ∣ λ 1 ∣ ε p ∑ n = 1 ∞ K n ⋅ ∥ x ∥ m , ρ λ 2 λ 1 , 1 n ⋅ ∥ x ∥ m , ρ λ 2 λ 1 − α + 1 + ∣ λ 1 ∣ ε p d x ≥ ∫ 0 < ∥ x ∥ m , ρ ≤ 1 ∥ x ∥ m , ρ − β + m − ∣ λ 2 ∣ ε q + λ 2 λ 1 α + 1 + ∣ λ 1 ∣ ε p ∫ 1 ∞ K u ⋅ ∥ x ∥ m , ρ λ 2 λ 1 , 1 u ⋅ ∥ x ∥ m , ρ λ 2 λ 1 − α + 1 + ∣ λ 1 ∣ ε p d u d x = ∫ 0 < ∥ x ∥ m , ρ ≤ 1 ∥ x ∥ m , ρ − β + m − ∣ λ 2 ∣ ε q + λ 2 λ 1 α + 1 + ∣ λ 1 ∣ ε p − λ 2 λ 1 ∫ ∥ x ∥ m , ρ λ 2 λ 1 ∞ K ( t , 1 ) t − α + 1 + ∣ λ 1 ∣ ε p d t d x ≥ ∫ 0 < ∥ x ∥ m , ρ ≤ 1 ∥ x ∥ m , ρ − m + λ 2 λ 1 c + ∣ λ 2 ∣ ε ∫ 1 + ∞ K ( t , 1 ) t − α + 1 + ∣ λ 1 ∣ ε p d t d x = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∫ 0 1 t − 1 + λ 2 λ 1 c + ∣ λ 2 ∣ ε d t ∫ 1 + ∞ K ( t , 1 ) t − α + 1 + ∣ λ 1 ∣ ε p d t = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∫ 0 1 t − 1 + λ 2 2 λ 1 c d t ∫ 1 + ∞ K ( t , 1 ) t − α + 1 + ∣ λ 1 ∣ ε p d t .

Consequently,

(6) ∫ 0 1 t − 1 + λ 2 2 λ 1 c d t ∫ 1 + ∞ K ( t , 1 ) t − α + 1 + ∣ λ 1 ∣ ε p d t ≤ 2 M − c λ 1 λ 2 1 / q 1 − c 2 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) − 1 / q < + ∞ .

Note that λ 2 2 λ 1 c < 0 , hence ∫ 0 1 t − 1 + λ 2 2 λ 1 c d t = + ∞ , which contradicts (6). Therefore, c ≥ 0 .

On the contrary, suppose that c ≥ 0 . It follows from the mixed Hölder’s inequality and Lemma 2.3 that

∫ R + m ∑ n = 1 ∞ K ( n , ∥ x ∥ m , ρ ) a n f ( x ) d x = ∫ R + m ∑ n = 1 ∞ n ( α + 1 − p c ) / ( p q ) ∥ x ∥ m , ρ ( β + m ) / ( p q ) ⋅ a n ∥ x ∥ m , ρ ( β + m ) / ( p q ) n ( α + 1 − p c ) / ( p q ) ⋅ f ( x ) K ( n , ∥ x ∥ m , ρ ) d x ≤ ∫ R + m ∑ n = 1 ∞ n ( α + 1 − p c ) / q ∥ x ∥ m , ρ ( β + m ) / q ⋅ ∣ a n ∣ p ⋅ K ( n , ∥ x ∥ m , ρ ) d x 1 / p × ∫ R + m ∑ n = 1 ∞ ∥ x ∥ m , ρ ( β + m ) / p n ( α + 1 − p c ) / p ⋅ ∣ f ( x ) ∣ q ⋅ K ( n , ∥ x ∥ m , ρ ) d x 1 / q = ∑ n = 1 ∞ n ( α + 1 − p c ) / q ∣ a n ∣ p ω 1 ( n ) 1 / p ∫ R + m ∥ x ∥ m , ρ ( β + m ) / p ∣ f ( x ) ∣ q ω 2 ( x ) d x 1 / q ≤ W 1 1 / p W 2 1 / q Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p ∑ n = 1 ∞ n α + 1 − p c q + λ 1 λ 2 β + m q − m ∣ a n ∣ p 1 / p × ∫ R + m ∥ x ∥ m , ρ β + m p + λ 2 λ 1 α + 1 p − 1 − c ∣ f ( x ) ∣ q d x 1 / q = W 0 ∣ λ 1 ∣ 1 / q ∣ λ 2 ∣ 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p ∑ n = 1 ∞ n α − p c ∣ a n ∣ p 1 / p ∫ R + m ∥ x ∥ m , ρ β ∣ f ( x ) ∣ q d x 1 / q ≤ W 0 ∣ λ 1 ∣ 1 / q ∣ λ 2 ∣ 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p ∑ n = 1 ∞ n α ∣ a n ∣ p 1 / p ∫ R + m ∥ x ∥ m , ρ β ∣ f ( x ) ∣ q d x 1 / q = W 0 ∣ λ 1 ∣ 1 / q ∣ λ 2 ∣ 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p ∥ a ˜ ∥ p , α ∥ f ∥ q , β .

Taking M ≥ W 0 ∣ λ 1 ∣ 1 / q ∣ λ 2 ∣ 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p arbitrarily, one can get (5).

(ii) For c = 0 , assuming the best constant factor of (5) is M 0 , then we can see from the previous proof that

M 0 ≤ W 0 ∣ λ 1 ∣ 1 / q ∣ λ 2 ∣ 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p ,

∫ R + m ∑ n = 1 ∞ K ( n , ∥ x ∥ m , ρ ) a n f ( x ) d x ≤ M 0 ∥ a ˜ ∥ p , α ∥ f ∥ q , β .

For ε > 0 , sufficiently small and N sufficiently large, taking

a n = n ( − α − 1 − ∣ λ 1 ∣ ε ) / p , n = 1 , 2 , … f ( x ) = ∥ x ∥ m , ρ ( − β − m + ∣ λ 2 ∣ ε ) / q , 0 < ∥ x ∥ m , ρ ≤ N , 0 , ∥ x ∥ m , ρ > N .

Then,

M 0 ∥ a ˜ ∥ p , α ∥ f ∥ q , β = M 0 ∑ k = 1 ∞ n − 1 − ∣ λ 1 ∣ ε 1 / p ∫ 0 < ∥ x ∥ m , ρ ≤ N ∥ x ∥ m , ρ − m + ∣ λ 2 ∣ ε d x 1 / q ≤ M 0 1 + ∫ 1 + ∞ t − 1 − ∣ λ 1 ∣ ε d t 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∫ 0 N t − 1 + ∣ λ 2 ∣ ε d t 1 / q = M 0 ε ∣ λ 1 ∣ 1 / p ∣ λ 2 ∣ 1 / q Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / q N ∣ λ 2 ∣ ε / q ( 1 + ∣ λ 1 ∣ ε ) 1 / p ,

∫ R + m ∑ n = 1 ∞ K ( n , ∥ x ∥ m , ρ ) a n f ( x ) d x = ∑ n = 1 ∞ n − ( α + 1 + ∣ λ 1 ∣ ε ) / p ∫ 0 < ∥ x ∥ m , ρ ≤ N K ( n , ∥ x ∥ m , ρ ) ∥ x ∥ m , ρ ( − β − m + ∣ λ 2 ∣ ε ) / q d x = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∑ n = 1 ∞ n − ( α + 1 + ∣ λ 1 ∣ ε ) / p ∫ 0 N K ( n , t ) t − β + m − ∣ λ 2 ∣ ε q + m − 1 d t = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∑ n = 1 ∞ n − ( α + 1 + ∣ λ 1 ∣ ε ) / p ∫ 0 N K ( 1 , t ⋅ n λ 1 / λ 2 ) t − β + m − ∣ λ 2 ∣ ε q + m − 1 d t = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∑ n = 1 ∞ n − α + 1 + ∣ λ 1 ∣ ε p + λ 1 λ 2 β + m − ∣ λ 2 ∣ ε q − m + 1 − λ 1 λ 2 ∫ 0 N ⋅ n λ 1 / λ 2 K ( 1 , u ) u − β + m − ∣ λ 2 ∣ ε q + m − 1 d u = Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∑ n = 1 ∞ n − 1 − ∣ λ 1 ∣ ε ∫ 0 N ⋅ n λ 1 / λ 2 K ( 1 , u ) u − β + m − ∣ λ 2 ∣ ε q + m − 1 d u ≥ Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∑ n = 1 ∞ n − 1 − ∣ λ 1 ∣ ε ∫ 0 N K ( 1 , u ) u − β + m − ∣ λ 2 ∣ ε q + m − 1 d u ≥ Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) ∫ 1 + ∞ t − 1 − ∣ λ 1 ∣ ε d t ∫ 0 N K ( 1 , u ) u − β + m − ∣ λ 2 ∣ ε q + m − 1 d u = Γ m ( 1 / ρ ) ∣ λ 1 ∣ ε ρ m − 1 Γ ( m / ρ ) ∫ N + ∞ K ( 1 , u ) u − β + m − ∣ λ 2 ∣ ε q + m − 1 d u .

Consequently,

Γ m ( 1 / ρ ) ∣ λ 1 ∣ ε ρ m − 1 Γ ( m / ρ ) ∫ 0 N K ( 1 , u ) u − β + m − ∣ λ 2 ∣ ε q + m − 1 d u ≤ M 0 ε ∣ λ 1 ∣ 1 / p ∣ λ 2 ∣ 1 / q Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / q N ∣ λ 2 ∣ ε / q ( 1 + ∣ λ 1 ∣ ε ) 1 / p , ∣ λ 1 ∣ 1 / p ∣ λ 2 ∣ 1 / q ∣ λ 1 ∣ Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p ∫ 0 N K ( 1 , u ) u − β + m − ∣ λ 2 ∣ ε q + m − 1 d u ≤ M 0 ⋅ N ∣ λ 2 ∣ ε / q ( 1 + ∣ λ 1 ∣ ε ) 1 / p .

Let ε → 0 + , then

∣ λ 1 ∣ 1 / p ∣ λ 2 ∣ 1 / q ∣ λ 1 ∣ Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p ∫ 0 N K ( 1 , u ) u − β + m q + m − 1 d u ≤ M 0 .

In addition, let N → + ∞ , we have

1 ∣ λ 1 ∣ 1 / q ∣ λ 2 ∣ 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p ∣ λ 2 ∣ ∫ 0 + ∞ K ( 1 , u ) u − β + m q + m − 1 d u ≤ M 0 .

It follows from Lemma 2.3 that

W 0 ∣ λ 1 ∣ 1 / q ∣ λ 2 ∣ 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p ≤ M 0 .

Hence, the best constant factor of (5) is

M 0 = W 0 ∣ λ 1 ∣ 1 / q ∣ λ 2 ∣ 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p . □

4 Applications in operator theory

The series operator T 1 and singular integral operator T 2 are defined by, respectively,

(7) T 1 ( a ˜ ) ( x ) = ∑ n = 1 ∞ K ( n , ∥ x ∥ m , ρ ) a n , T 2 ( f ) n = ∫ R + m K ( n , ∥ x ∥ m , ρ ) f ( x ) d x .

According to the basic theory of Hilbert-type inequality, (5) can be equivalently written as the following two expressions:

∥ T 1 ( a ˜ ) ∥ p , β ( 1 − p ) ≤ M ∥ a ˜ ∥ p , α and ∥ T 2 ( f ) ∥ q , α ( 1 − q ) ≤ M ∥ f ∥ q , β .

Thus, by Theorem 3.1, one has

Theorem 4.1

Assume that m ∈ N + , ρ > 0 , λ 1 λ 2 > 0 , 1 p + 1 q = 1 ( p > 1 ) , K ( n , ∥ x ∥ m , ρ ) = G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) is nonnegative measurable, 1 λ 2 m λ 1 + α λ 2 p − λ 2 + β λ 1 q = c , K ( t , 1 ) t − α + 1 p and K ( t , 1 ) t − α + 1 p + c are monotonically decreasing in ( 0 , + ∞ ) , and

W 0 = ∣ λ 1 ∣ ∫ 0 + ∞ K ( t , 1 ) t − α + 1 p d t

is convergent. Then,

  1. T 1 : l p α → L p β ( 1 − p ) ( R + m ) and T 2 : L q β ( R + m ) → l q α ( 1 − q ) are bounded operators if and only if c ≥ 0 ;

  2. For c = 0 , i.e., m λ 1 + α λ 2 p = λ 2 + β λ 1 q , the operator norms of T 1 and T 2 are as follows:

    ∥ T 1 ∥ = ∥ T 2 ∥ = W 0 ∣ λ 1 ∣ 1 / q ∣ λ 2 ∣ 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p .

Corollary 4.1

Suppose that m ∈ N + , ρ > 0 , 1 p + 1 q = 1 ( p > 1 ) , a > 0 , λ 1 > 0 , λ 2 > 0 , 1 λ 2 m λ 1 + α λ 2 p − λ 2 + β λ 1 q = l , p q + max { p λ 1 ( b − a ) , p ( λ 1 b − 1 ) , p ( λ 1 c − 1 ) , p ( λ 1 b + l − 1 ) , p ( λ 1 c + l − 1 ) } < α < p q + p λ 1 c , and

W 0 = ∫ 0 1 1 ( 1 + t ) a t c + 1 λ 1 ( 1 − α + 1 p ) − 1 + t a − b − 1 λ 1 ( 1 − α + 1 p ) − 1 d t .

The operators T 1 and T 2 are defined by, respectively,

T 1 ( a ˜ ) ( x ) = ∑ n = 1 ∞ ( max { 1 , n λ 1 ∥ x ∥ m , ρ λ 2 } ) b ( min { 1 , n λ 1 ∥ x ∥ m , ρ λ 2 } ) c ( 1 + n λ 1 ∥ x ∥ m , ρ λ 2 ) a ⋅ a n , T 2 ( f ) n = ∫ R + m ( max { 1 , n λ 1 ∥ x ∥ m , ρ λ 2 } ) b ( min { 1 , n λ 1 ∥ x ∥ m , ρ λ 2 } ) c ( 1 + n λ 1 ∥ x ∥ m , ρ λ 2 ) a ⋅ f ( x ) d x .

Then,

  1. T 1 is a bounded operator from l p α to L p β ( 1 − p ) ( R + m ) and T 2 is a bounded operator from L q β ( R + m ) to l q α ( 1 − q ) if and only if l ≥ 0 ;

  2. For l = 0 , i.e., m λ 1 + α λ 2 p = λ 2 + β λ 1 q , the operator norms of T 1 and T 2 are as follows:

    ∥ T 1 ∥ = ∥ T 2 ∥ = W 0 λ 1 1 / q λ 2 1 / p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p .

Proof

It follows from p q + p λ 1 ( b − a ) < α < p q + p λ 1 c that c + 1 λ 1 1 − α + 1 p > 0 , a − b − 1 λ 1 1 − α + 1 p > 0 , and the integral in W 0 is convergent. Denote that

K ( n , ∥ x ∥ m , ρ ) = ( max { 1 , n λ 1 ∥ x ∥ m , ρ λ 2 } ) b ( min { 1 , n λ 1 ∥ x ∥ m , ρ λ 2 } ) c ( 1 + n λ 1 ∥ x ∥ m , ρ λ 2 ) a .

Then,

W 0 = λ 1 ∫ 0 + ∞ K ( t , 1 ) t − α + 1 p d t = λ 1 ∫ 0 + ∞ ( max { 1 , t λ 1 } ) b ( min { 1 , t λ 1 } ) c ( 1 + t λ 1 ) a ⋅ t − α + 1 p d t = ∫ 0 + ∞ ( max { 1 , u } ) b ( min { 1 , u } ) c ( 1 + u ) a ⋅ u 1 λ 1 ( 1 − α + 1 p ) − 1 d u = ∫ 0 1 u c + 1 λ 1 ( 1 − α + 1 p ) − 1 ( 1 + u ) a d u + ∫ 1 + ∞ u b + 1 λ 1 ( 1 − α + 1 p ) − 1 ( 1 + u ) a d u = ∫ 0 1 1 ( 1 + t ) a t c + 1 λ 1 1 − α + 1 p − 1 + t a − b − 1 λ 1 1 − α + 1 p − 1 d t .

According to α > p q + p ( λ 1 b − 1 ) and α > p q + p ( λ 1 c − 1 ) , one has λ 1 b − α + 1 p < 0 , λ 1 c − α + 1 p < 0 , and

K ( t , 1 ) t − α + 1 p = ( max { 1 , t λ 1 } ) b ( min { 1 , t λ 1 } ) c ( 1 + t λ 1 ) a ⋅ t − α + 1 p = 1 ( 1 + t λ 1 ) a ⋅ t λ 1 c − α + 1 p , 0 < t ≤ 1 , 1 ( 1 + t λ 1 ) a ⋅ t λ 1 b − α + 1 p , t > 1 .

Thus, K ( t , 1 ) t − α + 1 p is monotonically decreasing in ( 0 , + ∞ ) .

Note that α > p q + p ( λ 1 b + l − 1 ) and α > p q + p ( λ 1 c + l − 1 ) , we get λ 1 b − α + 1 p + l < 0 , λ 1 c − α + 1 p + l < 0 , and

K ( t , 1 ) t − α + 1 p + l = 1 ( 1 + t λ 1 ) a ⋅ t λ 1 c − α + 1 p + l , 0 < t ≤ 1 , 1 ( 1 + t λ 1 ) a ⋅ t λ 1 b − α + 1 p + l , t > 1 .

Therefore, K ( t , 1 ) t − α + 1 p + l is monotonically decreasing in ( 0 , + ∞ ) .

Finally, it follows from Theorem 4.1 that Corollary 4.1 holds.□

Taking c = b in Corollary 4.1, according to the properties of Beta function, one gets

W 0 = ∫ 0 1 1 ( 1 + t ) a t b + 1 λ 1 ( 1 − α + 1 p ) − 1 + t a − b + 1 λ 1 ( 1 − α + 1 p ) − 1 d t = B b + 1 λ 1 1 − α + 1 p , a − b + 1 λ 1 ( 1 − α + 1 p ) .

Hence, we have

Corollary 4.2

Assume that m ∈ N + , ρ > 0 , 1 p + 1 q = 1 ( p > 1 ) , a > 0 , λ 1 > 0 , λ 2 > 0 , 1 λ 2 m λ 1 + α λ 2 p − λ 2 + β λ 1 q = l , p q + max { p λ 1 ( b − a ) , p ( λ 1 b − 1 ) , p ( λ 1 b + l − 1 ) } < α < p q + p λ 1 b , operators T 1 and T 2 are defined by, respectively,

T 1 ( a ˜ ) ( x ) = ∑ n = 1 ∞ ( n λ 1 ∥ x ∥ m , ρ λ 2 ) b ( 1 + n λ 1 ∥ x ∥ m , ρ λ 2 ) a ⋅ a n , T 2 ( f ) n = ∫ R + m ( n λ 1 ∥ x ∥ m , ρ λ 2 ) b ( 1 + n λ 1 ∥ x ∥ m , ρ λ 2 ) a ⋅ f ( x ) d x .

Then,

  1. T 1 : l p α → L p β ( 1 − p ) ( R + m ) and T 2 : L q β ( R + m ) → l q α ( 1 − q ) are bounded operators if and only if l ≥ 0 .

  2. For l = 0 , i.e., m λ 1 + α λ 2 p = λ 2 + β λ 1 q , the operator norms of T 1 and T 2 are

    ∥ T 1 ∥ = ∥ T 2 ∥ = 1 λ 1 1 / q λ 2 1 / p B b + 1 λ 1 1 − α + 1 p , a − b + 1 λ 1 1 − α + 1 p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p .

We can get the following results by taking b = 0 in Corollary 4.2.

Corollary 4.3

Assume that m ∈ N + , ρ > 0 , 1 p + 1 q = 1 ( p > 1 ) , a > 0 , λ 1 > 0 , λ 2 > 0 , 1 λ 2 m λ 1 + α λ 2 p − λ 2 + β λ 1 q = l , max { p ( 1 − λ 1 a ) − 1 , − 1 , p l − 1 } < α < p − 1 , operators T 1 and T 2 are defined by, respectively,

T 1 ( a ˜ ) ( x ) = ∑ n = 1 ∞ a n ( 1 + n λ 1 ∥ x ∥ m , ρ λ 2 ) a and T 2 ( f ) n = ∫ R + m f ( x ) d x ( 1 + n λ 1 ∥ x ∥ m , ρ λ 2 ) a .

Then,

  1. T 1 : l p α → L p β ( 1 − p ) ( R + m ) and T 2 : L q β ( R + m ) → l q α ( 1 − q ) are bounded operators if and only if l ≥ 0 .

  2. For l = 0 , i.e., m λ 1 + α λ 2 p = λ 2 + β λ 1 q , the operator norms of T 1 and T 2 are expressed as follows:

    ∥ T 1 ∥ = ∥ T 2 ∥ = 1 λ 1 1 / q λ 2 1 / p B 1 λ 1 1 − α + 1 p , a − 1 λ 1 1 − α + 1 p Γ m ( 1 / ρ ) ρ m − 1 Γ ( m / ρ ) 1 / p .

In Corollary 4.1, let m = 1 , α = 1 λ 2 ( p σ − λ 1 ) and β = 1 λ 1 ( q σ − λ 2 ) , then m λ 1 + α λ 2 p = σ = λ 2 + β λ 1 q . The following results can be obtained.

Corollary 4.4

Assume that 1 p + 1 q = 1 ( p > 1 ) , a > 0 , λ 1 > 0 , λ 2 > 0 , λ 1 p + λ 2 q + max { λ 1 λ 2 ( b − a ) , λ 1 λ 2 b − λ 2 , λ 1 λ 2 c − λ 2 } < σ < λ 1 p + λ 2 q + λ 1 λ 2 c , α = 1 λ 2 ( p σ − λ 1 ) , β = 1 λ 1 ( q σ − λ 2 ) . Denote that

W 0 = ∫ 0 1 1 ( 1 + t ) a t 1 λ 1 λ 2 λ 1 p + λ 2 q + λ 1 λ 2 c − σ − 1 + t a − 1 λ 1 λ 2 λ 1 p + λ 2 q + λ 1 λ 2 b − σ − 1 d t .

Operators T 1 and T 2 are defined by, respectively,

T 1 ( a ˜ ) ( x ) = ∑ n = 1 ∞ ( max { 1 , n λ 1 x λ 2 } ) b ( min { 1 , n λ 1 x λ 2 } ) c ( 1 + n λ 1 x λ 2 ) a ⋅ a n , T 2 ( f ) n = ∫ 0 + ∞ ( max { 1 , n λ 1 x λ 2 } ) b ( min { 1 , n λ 1 x λ 2 } ) c ( 1 + n λ 1 x λ 2 ) a ⋅ f ( x ) d x .

Then, T 1 : l p α → L p β ( 1 − p ) ( 0 , + ∞ ) and T 2 : L q β ( 0 , + ∞ ) → l q α ( 1 − q ) are bounded operators, and the operator norms of T 1 and T 2 are expressed as follows:

∥ T 1 ∥ = ∥ T 2 ∥ = W 0 λ 1 1 / q λ 2 1 / p .

  1. Funding information: This work was supported by the National Natural Science Foundation of China (No. 12071491) and the Innovation Team Construction Project of Guangdong Province (No. 2018KCXTD020).

  2. Conflict of interest: Authors state no conflict of interest.

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Received: 2020-09-02
Revised: 2021-02-02
Accepted: 2021-02-04
Published Online: 2021-05-25

© 2021 Bing He et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2021-0023
Keywords for this article
multiple integral half-discrete Hilbert-type inequality; nonhomogeneous kernel; equivalent parameter condition; best constant factor; operator norm; bounded operator
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Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
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  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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