Hardy-Leindler Type Inequalities for Multiple Integrals on Time Scales

Definition 1 (Time scale)

A non empty as well as closed subset of ℝ is called a time scale, denoted by 𝕋 [5, 6]. The sets ℝ, ℤ, and ℕ are some examples of time scales.

Definition 2 (Classification of points, [5: Definition 1.1])

The operator σ: 𝕋 → 𝕋 defined by σ(γ₁) := inf {z₁ ∈ 𝕋 ; z₁ > γ₁} is called forward jump operator and the operator ρ: 𝕋 → 𝕋 defined by ρ(γ₁) := sup {z₁ ∈ 𝕋 ; z₁ < γ₁} is called backward jump operator, where γ₁ ∈ 𝕋. A function μ: 𝕋 → [0, ∞) is called the graininess function if μ(γ₁) := σ(γ₁) − γ₁ for γ₁ ∈ 𝕋.

The point γ₁ satisfying σ(γ₁) > γ₁ is right-scattered and called left-scattered if ρ(γ₁) < γ₁. The points are called isolated if they are left-scattered and right-scattered simultaneously. Also the point γ₁ ∈ 𝕋 is called right-dense if γ₁ < sup 𝕋 and σ(γ₁) = γ₁ and is called left-dense if γ₁ > inf 𝕋 and ρ(γ₁) = γ₁. The points that are both left-dense and right-dense are dense points.

Definition 3 (rd-Continuous function, [5: Definition 1.58])

Suppose a function h₁: 𝕋 → ℝ which satisfies:

1. h₁ is continuous at right dense points of 𝕋,

2. the left hand limits exist and are finite at left dense points in 𝕋, then h₁ is called rd-continuous on 𝕋.

Notation ([5])

We write the notation h1σ(⋅)=h1(σ(⋅)) for any function h₁: 𝕋 → ℝ.

Definition 4 (Delta differentiable function, [5: Definition 1.10])

Consider a function h₁: 𝕋 → ℝ and fix γ₁ ∈ 𝕋 such that h1Δ(γ1) exists and satisfies the property that, for given ε > 0, γ₁ has a neighborhood O with the condition,

for all ς ∈ O, then h₁ is delta differentiable at γ₁ or h^Δ(γ₁) is the (delta) derivative of h₁ at γ₁.

Proposition 2.1 (Properties of Delta Derivative)

Assume h₁, h₂: 𝕋 → ℝ are delta differentiable at γ₁ ∈ 𝕋, then the following properties hold:

• Product Rule [5: Theorem 1.20 (iii)]: The product h₁, h₂: 𝕋 → ℝ is also differentiable at γ₁ with

  2.1 (h1h2)Δ(γ1)=h1Δ(γ1)h2(γ1)+h1σ(γ1)h2Δ(γ1)=h1(γ1)h2Δ(γ1)+h1Δ(γ1)h2σ(γ1).

• Quotient Rule [5: Theorem 1.20 (iv)]: If h2(γ1)h2σ(γ1)≠0, then h1h2 is differentiable at γ₁ and

  2.2 (h1h2)Δ(γ1)=h1Δ(γ1)h2(γ1)−h1(γ1)h2Δ(γ1)h2(γ1)h2σ(γ1).

• Chain Rule [5: Theorem 1.87]: Assume that ϑ: 𝕋 → ℝ is delta differentiable on 𝕋, ϑ: ℝ → ℝ is continuous and θ: ℝ → ℝ is continuous as well as differentiable such that the following holds

  2.3 [θ(ϑ(γ1))]Δ=θ′(ϑ(c))ϑΔ(γ1),c∈[γ1,σ(γ1)],γ1∈T.

Proposition 2.2 (Delta Integrable Function, [5: Theorem 1.74])

Every rd-continuous function h₁: 𝕋 → ℝ has an anti-derivative. In particular if t₀ ∈ 𝕋 then H₁(·) defined by

is antiderivative of h₁.

Proposition 2.3 (Properties of Integration)

• Scaler Multiplication, [5: Theorem 1.77 (ii)]: For c > 0

  2.4 ∫abcθ(γ1)Δγ1=c∫abθ(γ1)Δγ1,γ1∈T.

• Integration by Parts Formula, [5: Theorem 1.77 (v)]: For a, b ∈ 𝕋 and θ, ϑ ∈ C_rd(𝕋, ℝ), the formula for integration by parts is given by

  2.5 ∫abθ(γ1)ϑΔ(γ1)Δγ1≤|θ(γ1)ϑ(γ1)|ab−∫abθΔ(γ1)ϑσ(γ1)Δγ1,γ1∈T.

Theorem 2.4.

Suppose ℓ₁, ℓ₂ ∈ 𝕋 and θ, ϑ: [ℓ₁, ℓ₂] → ℝ are rd-continuous functions, then

where p > 1 and q = p/(p − 1).

Theorem 2.5

Let 𝕋₁ and 𝕋₂ be two time scales, h₁: 𝕋₁ × 𝕋₂ → ℝ is integrable with respect to both time scales and if the function φ(y1)=∫T1h1(x1,y1)Δx1 exists for almost every y₁ ∈ 𝕋₂ and ψ(x1)=∫T2h1(x1,y1)Δy1 exists for almost every x₁ ∈ 𝕋₁, then

Notation

The following notation is used in the paper for partial delta derivative

Theorem 3.1

Consider ι ∈ {1,…,n} and time scales 𝕋_ι for a_ι ∈ [0, ∞)_𝕋ι. Take ξ_ι: 𝕋_ι → ℝ₊, such that

exists, where A_ι(∞) = 0 for all ι.

Consider g: 𝕋₁ × … × 𝕋_n → ℝ₊ such that

Then for p > 1,

where n ≥ 1.

Proof

Mathematical induction technique is used to prove this result. The statement is true for n = 1, see [17: Theorem 2.1].

Next assume (3.3) holds for 1 ≤ n ≤ k. Now for n = k + 1, right hand side of (3.3) takes the form

Denote Ik+1=∫ak+1∞ξk+1(ℏk+1)ϕk+1p(σ1(ℏ1),…,σk+1(ℏk+1))Δℏk+1.

Apply integration by parts (2.5) with

θk+1Δk+1(ℏk+1)=ξk+1(ℏk+1), ϑk+1σk+1(ℏk+1)=ϕk+1p(σ1(ℏ1),…,σk+1(ℏk+1)) and use the facts that φ_{k + 1}(ℏ₁,…,ℏ_k, a_{k + 1}) = 0 and A_{k + 1}(∞) = 0 to get

where −θ_{k + 1}(ℏ_{k + 1}) = A_{k + 1}(ℏ_{k + 1}). Therefore,

Use chain rule formula (2.3) to get

where d_{k + 1} ∈ [ℏ_{k + 1}, σ_{k + 1}(ℏ_{k + 1})]. Since

Therefore, (3.7) implies

where

Use (3.8) and d_{k + 1} ≤ σ_{k + 1}(ℏ_{k + 1}) in (3.6) to get

Substitute (3.9) in (3.5) to obtain,

Using Hölder’s inequality (2.6) to (3.10), one’s get

Simplification in (3.11) gives

Put (3.12) in (3.4) to have

Exchange the integrals k times by using Fubini’s Theorem on right hand side of (3.13)

Use induction hypothesis for φ_k(σ₁(ℏ₁),…,σ_k(ℏ_k), ℏ_{k + 1}) instead of φ_k(σ₁(ℏ₁),…,σ_k(ℏ_k)) for fixed ℏ_{k + 1} ∈ 𝕋_{k + 1} on right hand side of (3.14) and once again apply the Fubini’s Theorem k times to obtain

Hence by the principle of mathematical induction, the result is true for all positive integers n. □

Remark 1

As a special case of Theorem 3.1, when 𝕋_i = ℝ, (note that in this case, we have σ_i(ℏ_i) = ℏ_i and ℏ_i ∈ T_i, for all i ∈ {1,…, n}). We have the Wirtinger type inequality,

where p > 1 and ∂n∂(ℏ1)⋯∂(ℏn)ϕ(ℏ1,…,ℏn)=g(ℏ1,…,ℏn). Moreover φ(ℏ₁,…,ℏ_n) is differentiable function with φ(a₁,…,a_n) = 0.

As a particular case if ξi(ℏi)=1ℏi and p = 2, then replace upper limit of the integral by 1 to get the following famous inequality due to Hardy (in several variables),

where φ(0,…,0) = 0, with the constant 14n as best possible.

Remark 2

Assume that 𝕋_i = ℕ, p > 1 and a_i = 1 for all i = 1,…, n in Theorem 3.1. Furthermore assume that,

is convergent, then (3.3) becomes the following discrete Leindler’s inequality for p > 1.

where φ_n(m₁,…,m_n) is a positive sequences with φ_n(1,…,1) = 0 and Δ_ι represents forward difference operator with respect to m_ι.

Theorem 3.2

Consider ι = {1,…,n} and 𝕋_i be a time scales with a_ι ∈ [0,∞)_𝕋ι. Take ξ_ι: 𝕋_i → ℝ₊ such that

exists and B_ι(∞) = 0 for all ι. Furthermore let

for any ℏι∈[aι,∞)Tι and g:T1×⋯×Tn→R+. Then for p > 1,

Proof

We use mathematical induction method to prove the required result. For n = 1, the result is obvious, see [17: Theorem 2.2]. Let (3.17) holds for 1 ≤ n ≤ k, then for n = k + 1 left hand side of (3.17) takes the form,

Denote Ik+1=∫ak+1∞ξk+1(ℏk+1)φk+1p(ℏ1,…,ℏk+1)Δℏk+1.

Apply integration by parts (2.5) and use φk+1(ℏ1,…,ℏk,∞)=0, Bk+1(ak+1)=0 to get

Apply chain rule (2.3) to find

where dk+1∈[ℏk+1,σk+1(ℏk+1)]. Since

and dk+1≥ℏk+1. Therefore (3.20) implies,

where φk(ℏ1,…,ℏk+1)=∫ℏ1∞⋯∫ℏk∞g(ς1,…,ςk,ℏk+1)Δςk⋯Δς1.

Use (3.21) in (3.19) to get

Apply Hölder’s inequality to (3.22) to get

which results in

Put (3.23) in (3.18) to get

Exchange the integrals k times by using Fubini’s Theorem on right hand side of (3.24) to get

Use induction hypothesis for φk(ℏ1,…,ℏk,ℏk+1) instead of φk(ℏ1,…,ℏk) for some fixed ℏk+1∈Tk+1 on right hand side of (3.26) and again apply Fubini’s Theorem k times to obtain

Hence statement is true for all integers n. □

Remark 3

As a special case of Theorem 3.2, when 𝕋_i = ℝ_i and p > 1, we have the following integral inequality of Leindler type (for functions of several variables), note that in this case we have

Therefore,

Remark 4

Assume that T1×T2×⋯×Tn=N×N×⋯×N in Theorem 3.2, p > 1 and a_i = 1. In this case (3.17) becomes the following discrete Leindler’s inequality

where Bι(mι)=∑kι=1mι−1ξι(kι).

Theorem 4.1

Consider ι = {1,…,n} and 𝕋_ι are time scales with aι∈[0,∞)Tι. Take ξι:Tι→Ri such that

where C_ι(∞) = 0. Furthermore assume

where g:T1×⋯×Tn→R+. Then for 0 < p ≤ 1,

Proof

For n = 1, the statement is true by [17: Theorem 2.3]. Next suppose, for 1 ≤ n ≤ k, the statement holds. To observe the statement for n = k + 1, the left hand side of the inequality (4.1) takes the form

Denote Ikx+1=∫ak+1∞ξk+1(ℏk+1)ψk+1p(σ1(ℏ1),…,σk+1(ℏk+1))Δℏk+1. Apply integration by parts (2.5) with θΔk+1(ℏk+1)=ξk+1(ℏk+1) and ϑ(σk+1(ℏk+1))=ψk+1(σ1(ℏ1),…,σk+1(ℏk+1)) and use the fact ψk+1(σ1(ℏ1),…,σk(ℏk),ak+1)=0=θk+1(∞) to get