Journal of Function Spaces

Journal of Function Spaces / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 7414861 | https://doi.org/10.1155/2020/7414861

Qiang Chen, Bing He, Yong Hong, Zhen Li, "Equivalent Parameter Conditions for the Validity of Half-Discrete Hilbert-Type Multiple Integral Inequality with Generalized Homogeneous Kernel", Journal of Function Spaces, vol. 2020, Article ID 7414861, 6 pages, 2020. https://doi.org/10.1155/2020/7414861

Equivalent Parameter Conditions for the Validity of Half-Discrete Hilbert-Type Multiple Integral Inequality with Generalized Homogeneous Kernel

Academic Editor: Konstantin M. Dyakonov
Received10 Jun 2020
Accepted23 Jul 2020
Published11 Aug 2020

Abstract

Let be a homogeneous nonnegative function of order . By using the weight coefficient method, the equivalent parameter conditions and best constant factors for the validity of the following half-discrete Hilbert-type multiple integral inequality are discussed. Finally, its applications in operator theory are discussed.

1. Preliminaries

Suppose that . In this paper, it is always assumed that , For any given , we write

Let be a homogeneous nonnegative function of order . We call a generalized homogeneous function. Obviously, satisfies

We will discuss the equivalent parameter conditions for the validity of the half-discrete Hilbert-type multiple integral inequality with generalized homogeneous kernel and the optimal constant factors of the inequality under certain special conditions. Finally, the boundedness and norm of the corresponding series operator and integral operator are discussed. The relevant literature can be found in [116].

Lemma 1 (see [17]). Let be measurable. represents the region . Then,where is the Gamma function.
In view of Lemma 1, one has

Lemma 2. Assume that , is a homogeneous nonnegative measurable function of order , , , is monotonically decreasing on . DenoteThen,

Proof. According to , after a simple calculation, we have . Then,Hence, .
It follows from Lemma 1 thatNotice that is monotonically decreasing on ;we have

2. Main Results

Theorem 3. Suppose that , , is a homogeneous nonnegative measurable function of order , , and are monotonically decreasing on , andis convergent. Then,(i)The necessary and sufficient condition for the validity of inequalityfor some constant is , where (ii)When , that is, , the best constant factor of (11) is.

Proof. Denote . (i) Suppose that (11) holds. We prove that by using reduction to absurdity. If , for , takeIt follows from Lemma 1 thatSince is monotonically decreasing on , we haveIt follows from (11), (14), and (15) thatSince , then , which contradicts (16). Therefore, .
Suppose that . It follows from Hölder’s inequality and Lemma 2 thatTake arbitrarily; one can get (11).(ii)When , if (12) does not hold, then, there exists constant , such thatFor and are sufficiently small, takeThen,It follows from (19), (21), and (22) thatLet ; then,In addition, let ; we haveThis contradicts (18). Hence, (12) holds.

3. Applications

Define series operator and singular integral operator with kernel by, respectively,

According to the basic theory of Hilbert-type inequality (11) can be equivalently written as the following two expressions:

Thus, Theorem 3 is equivalent to the following theorem.

Theorem 4. Assume that , is a homogeneous nonnegative measurable function of order , and are monotonically decreasing on , operators and are as defined in (26), andis convergent. Then,(i) and are bounded operators if and only if (ii)When , the operator norms of and are.

Corollary 5. Suppose that , , . DenoteDefine operators and by, respectively,Then,(i) is a bounded operator from to , and is a bounded operator from to if and only if (ii)When , the operator norms of and are

Proof. DenoteThen, is a homogeneous nonnegative function of order.
It follows from that and . Hence, is convergent andIn view of