A survey of useful inequalities in fractional calculus

Theorem 2.1

If u, v ∈ C0∞(Rn)and 0 < s < 1, then

If we take u = v in (2.2), multiply by Γ_s(x) = C_s|x|^{2s − n} and integrate, the identity

is valid for 0 < s < 1. Hence,

An other consequence is the following inequality obtained by Cordoba and Cordoba in [15].

Theorem 3.1

Let 0 ≤ α ≤ 2, x ∈ ℝⁿor x ∈ 𝕋ⁿ(the torus) (n = 1, 2, 3 …) andθ∈C02(Rn)or θ ∈ C²(𝕋ⁿ). Then the following inequality holds

Cordoba-Cordoba’s inequality follows from Eilertsen’s identity (2.3) by setting u = v = θ.

This inequality enabled Cordoba and Cordoba to obtain the L^∞-decay estimate for the viscosity solutions of the quasi-geostrophic equation.

This inequality has been generalized by Ju [25] as follows.

Theorem 3.2

Let 0 ≤ α ≤ 2, x ∈ ℝⁿor x ∈ 𝕋ⁿ(the torus) (n = 1, 2, 3 …) andθ∈C02(Rn)or θ ∈ C²(𝕋ⁿ). Then the following inequality holds

Wu [46] proved the following version.

Theorem 3.3

Let 0 ≤ α ≤ 2. Let p₁ = k₁/l₁ ≥ 0 and p₂ = k₂/l₂ ≥ 1 be rational numbers with l₁and l₂being odd, and with k₁l₂ + k₂l₁being even. Then, for any x ∈ ℝⁿand any function θ ∈ C²(ℝⁿ) that decays sufficiently fast at infinity, Then the following inequality holds

Ju’s and Wu’s inequalities have also been used for the quasi-geostrophic equation.

A further generalization has been achieved by Constantin [13].

Theorem 3.4

Letθ∈C02(Rn)or θ ∈ C²(𝕋ⁿ) and Φ be a convex function of one variable. Then

Ju, Caffarelli and Vasseur [12], and Constantin [13] used the “convexity” inequality for the quasi-geostrophic equation too.

Ye and Xu [45] derived an other variant; it reads:

They used this inequality for the 2-D Boussinesq equations.

Recently, Alsaedi, Ahmad and Kirane [5] derived the “convexity” inequality in the Heisenberg group thanks to a result of Ferrari and Franchi [20] concerning an integral representation of the fractional powers of the Laplacian.

Theorem 3.5

Let F ∈ C²(ℝ) be a convex function, 0 ≤ α ≤ 2. Assume thatφ∈C02(R2N+1).Then

Here (−ΔH)α2 is the fractional Laplacian on the Heisenberg group ℍ.

In [5], the convexity inequality is used to prove nonexistence results via the nonlinear capacity method [37] for hyperbolic, parabolic, and hyperbolic equations with linear damping.

Constantin and Vlad [14] derived the following inequality.

Theorem 3.6

Let f ∈ 𝓢(ℝⁿ). For a k ∈ {1, …, n}, let g(x) = ∂_kf(x). Assume thatx ∈ ℝⁿis such thatg(x¯)=maxx∈Rng(x)>0.Then

After the appearance of the “Cordoba-Cordoba inequality”, Diaz, Pieranttozi and Vazquez [16] proved a similar inequality for the Riemann-Liouville fractional time derivative.

Theorem 3.7

Let 0 < α < 1 and u ∈ C([0, T]; ℝ), u′ ∈ L¹(0, T; ℝ) and u be monotone. Then

They conjectured that the inequality (3.9) still holds true without the monotonicity condition imposed on u.

They used inequality (3.9) to obtain finite time extinction for some nonlinear fractional in time equations.

In the same paper, they provided a more general version of Theorem 3.7.

Theorem 3.8

Given the Hilbert space 𝓗 with inner product (, )_𝓗, let 0 < α < 1 and u ∈ L^∞(0, T; 𝓗) such thatD0,tαu ∈ L¹(0, T; 𝓗). Assume that ∥u(.)∥_𝓗is non-increasing (i.e. ∥u(t₂)∥_𝓗 ≤ ∥u(t₁)∥_𝓗for a.e. t₁, t₂ ∈ (0, T) such that t₁ ≤ t₂). Then there exists k(α) > 0 such that for almost every t ∈ (0,T), we have

In [47], Zacher derived the following inequality.

Theorem 3.9

Let α ∈ (0, 1), T > 0 and 𝓗 be a Hilbert space with inner product (, )_𝓗. Suppose that v ∈ L²(0, T; 𝓗) and there exists x ∈ 𝓗 such thatv−x∈0H2α([0,T];H):=gα∗w:w∈L2(0,T;H).Then

Zacher used inequality (3.11) to obtain some decay estimate for a nonlinear homogeneous time fractional evolution equation.

Alsaedi, Ahmad, and Kirane [4] looking for stability estimates for various diffusion equations with time-fractional derivatives, derived the following results.

Theorem 3.10

Let one of the following conditions be satisfied:

• u ∈ C([0, T]), v ∈ C^β([0,T]), α < β ≤ 1;

• v ∈ C([0, T]), u ∈ C^β([0,T]), α < β ≤ 1;

• u ∈ C^β([0,T]), v ∈ C^δ([0,T]), α < β + δ, 0 < β < 1, 0 < δ < 1.

Then we have:

The immediate consequences are:

1. If u and v have the same sign and are both increasing or both decreasing, then

   D0+α(uv)(t)≤u(t)D0+αv(t)+v(t)D0+αu(t).(3.13)

   By setting u = v in inequality (3.13) and taking only u ∈ C^β([0,T]), α < 2 β, β ≤ 1 we obtain the inequality conjectured by J. I. Diaz, T. Pierantozi and L. Vázquez [16]

   2u(t)D0+αu(t)≥D0+αu2(t).(3.14)

   In the case β < 1, our requirement on u is weaker than the one of [16] as u is not differentiable. However, in the case β < 1, by Rademacher’s theorem, u is almost everywhere differentiable [19].

2. By induction, one can show that, for any integer p ≥ 2,

   pu(p−1)(t)D0+αu(t)≥D0+αup(t),(3.15)

   for p even, or p odd whenever u ≥ 0.

Remark 3.1

For the Caputo derivative, inequality (3.13) reads

Alikhanov [1], looking for some stability estimates in L² for diffusion equations with time-fractional derivative, derived the following equality.

Theorem 3.11

Let 0 < α < 1 and u absolutely continuous on [0, T]. Then

The vectorial case:

In [17], the vectorial version of the “Leibniz’ inequality” is presented.

Theorem 3.12

Let X: [t₀, +∞) → ℝ^ℕbe a vectorial differentiable function. Then, for any time instant t ≥ t₀,

This inequality is used in [17] to prove Lyapunov uniform stability for fractional order systems.

Theorem 4.1

(Analogue of Fermat’s theorem [39], p. 56)

Let the function u(t) ∈ L¹([A, B]) attain at x ∈ (A, B) its extremum and let there exists δ > 0 such that u(t) in the one-sided neighborhood ω_δ of the point x satisfies Hölder’s condition with exponent h > α. Then for any α ∈ [0, 1] and a ∈ (A, B), a ≠ x, we have

Here ω_δ = [x − δ, x] when x ≥ a and ω_δ = [x, x + δ] when x ≤ a, δ > 0.

Corollary 4.1

If x is a point of extremum of the function u(t) defined in some ε–neighborhoodSεx = [x − ε, x + ε] of x, then eitherDa,xαu(t), 0 < α < 1, does not exist, or it satisfies one of the inequalities(4.1), (4.2), where a is any point ofSεx. In particular,

Nakhushev [39] also derived the following result that may be useful for various fractional differential equations.

Theorem 4.2

Let:

1. the function u(t) ∈ L¹([A, B]) and it attains a maximum value at a point x ∈ (A, B) where it is differentiable;

2. there exists δ > 0 such that u′(t) on the segment ω_δ satisfies the Hölder condition with exponent h > α − 1.

   Then for each number α ∈ (1, 2) and any a ∈ [A, B], a ≠ x the following inequality holds true:

   Daxαu(t)≤u(x)|x−a|−αΓ(1−α).(4.4)

   If u′(t) ∈ Lip^h([A, B]) (the space of functions satisfying Hölder’s condition with exponent h ∈ (0, 1]) and h > α − 1, then in a neighborhoodSεxof the point x there exists a point a distinct from x such that for all α ∈ (1, 2) the following equality holds true:

   Daxαu(t)=u(x)|x−a|−αΓ(1−α).(4.5)

   Similar results also appeared in the papers of Al-Refai and Luchko [2], [3].

Theorem 5.1

Let 1 < p < ∞, and let I^α, ℜ α > 0 (ℜ for real part), with domain D(I^α) = {f ∈ L^p(0, ∞): I^αf ∈ L^p(0, ∞)}. If f ∈ D(I^γ), ℜ γ > 0, then f ∈ D(I^α) whenever 0 < ℜ α < ℜ γand that, if γ is real and 0 < α < β < γ < L, then

An inequality similar to (5.1) for the Weyl fractional inetgral was first derived by Hardy, Landau and Littlewood [24].

Based on Theorem 5.1, Hughes deduced the following theorem.

Theorem 5.2

Let D^β denote the β-th Riemann-Liouville fractional derivative acting in the weighted spaceLωp (0, ∞), 0 < β < α, and let the weight w satisfy the Muckenhoupt (A_p) condition[27]. Then the following Hardy-Landau-Littlewood inequality is valid:

These two theorems are useful in intermediary estimates, for example, for equations with forcing terms containing fractional derivative of order less than the leading derivative in the equation.

Geisberg [23] proved the following inequality for the Marchaud fractional derivative

Theorem 5.3

Let 0 < α < 1. The Marchaud fractional derivativeD+αenjoys the following inequality

Other inequalities have been proved by Arestov [7], Babenko and col. [9], [10].

Theorem 6.1

Let 1/p + 1/q = 1 withp, q > 1, letγ ≥ 0, ν > γ + 1 −1/p, and let f ∈ L(0, x) have an integrable fractional derivative D^νf ∈ L^∞(0, x), and letD^ν−jf(0) = 0 for j = 1, …, [ν]+1. Then

Theorem 6.2

Letν > γ ≤ 0, and let f ∈ L(0, x) have an integrable fractional derivative D^νf ∈ L^∞(0, x), and letD^ν−jf(0) = 0 for j = 1, …, [ν]+1. Then

Theorem 6.3

Let 1/p + 1/q = 1 withp, q > 1, letγ ≥ 0, ν > γ + 1 −1/p, and let f ∈ L(0, x) have an integrable fractional derivative D^νf ∈ L^∞(0, x), and letD^ν−jf(0) = 0 forj = 1, …, [ν] + 1. Then for anym > 0,

Definition 7.1

Let A be a closed operator densely defined in the complex Banach space 𝓧. The operator A is said to be of type (ω, M) if there exist 0 < ω < π and M ≥ 1 such that ρ(A) ⊃ {λ: |arg(λ) > ω |} and ∥λ(A − λ)⁻¹ ∥ ≤ M for λ < 0, and if there exists a number M_ε such that ∥λ(A − λ)⁻¹ ∥ ≤ M_ε holds in | arg(λ) | > ω + ε for all > 0.

Theorem 7.1

LetAbe of type (ω, M) and suppose that 0 ∈ ρ(A). For 0 ≤ α < β ≤ 1, there exists a constant C_α,βdepending only onM, α, andβ, such that, for all u ∈ D(A^β),

Remark 7.1

([43], p. 39) A more general form of the moment inequality can be described as follows. For any α < β < γ and for any u ∈ D(A^γ),

For more details, we refer to Krein [33], p. 115.

Theorem 8.1

Let m, q, θ ∈ ℝ\{0} with q ≠ mθ > 0, 0 < s < n, 1 < p < n/s and 1 < r/(q_mθ). Then the inequality

Corollary 8.1

(Fractional Sobolev inequality) For 0 < s < n, 1 < p<nsandq=npn−ps,we have

Theorem 8.2

Let v ∈ Cc∞ (ℝ^d), u ∈ L²∩ L¹(ℝ^d), α > 0, αnot in ℕ, and k ∈ ℕ such thatα − k > 0 andα − k − 1 < 0. Assume thatDxiα−m ∈ L² ∩ L¹(ℝ^d) for every i ∈ {1, …, d} and m = 0, 1, …, k + 1. Then there exists a positive constant C such that for every M > 0,

Theorem 9.1

Theorem 9.2

Theorem 9.3

Theorem 10.1

Let f and g be defined on [0, ∞) such that for allτ ≥ 0, ρ ≥ 0, (f(τ) − f(ρ)) ((g(τ) − g(ρ)) ≥ 0 (in this case f and g are said synchronous), then

Remark 10.1

The inequality (10.1) is reversed if the functions f and g are asynchronous.

Theorem 10.2

Let f andgbe defined on [0, ∞) such that f is increasing, gis differentiable with bounded derivative, m := min_t≥0g′(t), M := max_t≥0g′(t), then

Theorem 10.3

Let p be a positive function and let f and g be two differentiable functions on [1, ∞). If f′ ∈ L^r([1, ∞)), g′ ∈ L^s([1, ∞)), r > 1, r + s = rs, then for all t > 1 andα > 0, β > 0

Theorem 11.1

Let a(t)≥ 0 be a nondecreasing C¹-function on [0, T] (0 < T < ∞), let F(t) ≥ 0 be continuous on [0, T], 0 < β < 1, r ≥ 1, and letω : ℝ₊ → ℝ₊be a continuous, nondecreasing, positive function. Assume thatu(t) ≥ 0 is a continuous function on [0, T] satisfying the inequality(11.1). Then

2^q−1a(0)^q ≥ v₀ > 0, Gqr−1is the inverse of 𝓖_qr, a = a(t)

Theorem 11.2

Letu(t), a(t), b(t) andf(t) be nonnegative continuous functions fort > 0. Let p and q be constants with p ≥ q ≥ 0. Ifu(t) satisfies

1. ifα ∈ (0, 1], β ∈ (1/2, 1), γ ≥ 3/2 − β,

   u(t)≤{a(t)+M1βt(α+1)(β−1)+γb(t)[A11−β(t)+Kq−ppM1β[1−(1−V1(t))1−β]−1×(∫0ts(α+1)(β−1)+γ1−βf11−β(s)b11−β(s)A1(s)V1(s)ds)1−β]}1p,(11.5)

   where

   M1=1αBβ+γ−1αβ,2β−1β,A(t)=qpKq−ppa(t)+p−qpKqp,A1(t)=∫0tf11−β(s)A11−β(s)ds,V1(t)=exp−Kp−qp(1−β)M1β1−β∫0ts(α+1)(β−1)+γ1−βf11−β(s)b11−β(s)ds,

   and

   B[σ;η]=∫01sσ−1(1−s)η−1ds.

2. ifα ∈ (0, 1], β ∈ (0, 1/2], γ > (1 − 2β²)/(1 − β²), then

   u(t)≤{a(t)+M21+3β1+4βt[α(β−1)+γ](1+4β)−β1+4βb(t)[A2β1+4β(t)+Kq−ppM21+3β1+4β×[1−(1−V2)β1+4β]−1∫0ts[α(β−1)+γ](1+4β)−ββf1+4ββ(s)b1+4ββ(s)A2V2dsβ1+4β]}1p,(11.6)

   where

   M2=1αBγ(1+4β)−βα(1+3β),4β21+3β,A2(t)=∫0tf1+4ββ(s)A1+4ββ(s)ds

   and

   V2(t)=exp−K(q−p)(1+4β)pβM21+3ββ∫0ts[α(β−1)+γ](1+4β)−ββf1+4ββ(s)b1+4ββ(s)ds.

Theorem 11.3

Suppose that the following conditions are satisfied:

(H₁) The functions p and r ∈ C([t₀, T), ℝ₊).

(H₂) The functionϕ ∈ C([βt₀, t₀], ℝ₊) with max_{s∈[βt0, t0]}ϕ(s) > 0, where 0 < β < 1.

(H₃) The function u ∈ C([βt₀, T), ℝ₊) with

Then the following assertions hold:

(R₁) Suppose α > 12 , then

(R₂) Suppose 0 < α ≤ 12 , then