Theorem 1. Let 1 < p ≤ q < ∞.

• (i)

  The inequality

holds if and only if

In addition, C ≈ A, where C is the best constant in (5).

• (ii)

  The inequality

holds if and only if

Definition 2. Let K(x, s) be a nonnegative function measurable on the set Ω{(x, s): a < s ≤ x < b} and nonincreasing in the second argument. We say that the function K(x, s) belongs to the class $$ if there exist nonnegative functions w(x) and K_0,1(t, s) measurable on Ω such that

for a < s ≤ t ≤ x < b; moreover, the equivalence coefficients in (9) do not depend on s, t, and x.

Definition 3. Let K(x, s) be a nonnegative function measurable on the set Ω and nonincreasing in the second argument. We say that the function K(x, s) belongs to the class $$ if there exist $$ and nonnegative functions w(x), K_0,2(t, s), and K_1,2(t, s) measurable on Ω such that

for a < s ≤ t ≤ x < b; moreover, the equivalence coefficients in (10) do not depend on s, t, and x.

Definition 4. Let K(x, s) be a nonnegative function measurable on the set Ω and nonincreasing in the second argument. We say that the function K(x, s) belongs to the class $$ if there exist $$, $$, and nonnegative functions w(x), K_0,3(t, s), K_1,3(t, s), and K_2,3(t, s) measurable on Ω such that

for a < s ≤ t ≤ x < b; moreover, the equivalence coefficients in (11) do not depend on s, t, and x.

Let

Theorem 5. Let 1 < p ≤ q < ∞. Let the kernel of operator (2) belong to the class $$. Then, the inequality

holds if and only if B₁(a, b) < ∞. In addition, C ≈ B₁(a, b), where C is the best constant in (15).

Let

Theorem 6. Let 1 < p ≤ q < ∞. Let the kernel of operator (2) belong to the class $$. Then, inequality (15) holds if and only if B₂(a, b) < ∞. In addition, C ≈ B₂(a, b), where C is the best constant in (15).

Let

Theorem 7. Let 1 < p ≤ q < ∞. Let the kernel of operator (2) belong to the class $$. Then, inequality (15) holds if and only if B₃(a, b) < ∞. In addition, C ≈ B₃(a, b), where C is the best constant in (15).

Theorem 8. Let 1 < p < ∞. If the conditions

hold; then, for $$, there exist the finite values f(0), f^′(0), and f^′(∞) such that

Lemma 9. Let $$, where K₁(x, s) ≈ K₁(x, t) + w(x)K_0,1(t, s) for a < s ≤ t ≤ x < b. Then

for a < s ≤ τ ≤ x < b;

Proof.

• (i)

  For a < s ≤ τ ≤ x < b, we have

Therefore, by (10), we get that $$.

• (ii)

  For a < s ≤ τ ≤ x < b, it easily follows that

• (iii)

  Using (28), for a < s ≤ τ ≤ x < b, we have

Then, in view of (11), we obtain that $$. The proof is complete.

Let a = 0 and b = ∞. Assume that

Lemma 10. Let 1 < p ≤ q < ∞ and $$. Then, the inequality

holds if and only if $$. In addition, $$, where $$ is the best constant in (38).

Let

Lemma 11. Let 1 < p ≤ q < ∞ and $$. Then, the inequality

holds if and only if $$. In addition, $$, where $$ is the best constant in (42).

Let

Lemma 12. Let 1 < p ≤ q < ∞. Then, the inequality

holds if and only if A⁻(τ) < ∞. In addition, $$, where $$ is the best constant in (44).

Assume that

Lemma 13. Let 1 < p ≤ q < ∞ and $$. Then, the inequality

holds if and only if $$. In addition, $$, where $$ is the best constant in (49).

Let

Lemma 14. Let 1 < p ≤ q < ∞ and $$. Then, the inequality

holds if and only if $$. In addition, $$, where $$ is the best constant in (54).

Proof. Since $$, by Lemma 9, we have that

Hence, inequality (54) is equivalent to simultaneous fulfilment of the following inequalities:

In addition, $$, where $$ and $$ are the best constants in (56) and (57), respectively. By Theorem 5, inequality (56) holds if and only if $$, and in addition, $$. By part (i) of Theorem 1, inequality (57) holds if and only if A⁺(τ) < ∞, and in addition, $$. Then, inequality (54) holds if and only if $$ and $$. The proof is complete.

Lemma 15. Let 1 < p ≤ q < ∞. Then, the inequality

holds if and only if $$. In addition, $$, where $$ is the best constant in (59).

Theorem 16. Let 1 < p ≤ q < ∞ and $$. Let condition (25) hold. Then, inequality (63) holds if and only if EF < ∞. In addition, C ≈ EF, where C is the best constant in (63).

Proof. Sufficiency. From (25) by Theorem 8, it follows the validity of (26). As in Theorem 2.1 of [5], using (26), for $$, we get

where τ ∈ I. Assuming f^′′ = g in (66), we have that g ∈ L_p,v(I). Moreover, from (26), it follows that $$.

Assume that $$. Then, in (66), the condition $$ is equivalent to the condition $$. Replacing (66) into the left-hand side of (63), we find that

Therefore, inequality (63) has the form

In the left-hand side of (68), using the Minkowski’s inequality for sums, then, applying Lemmas 10, 11, 12, 13, and 14 to each term, we get

Since the left-hand side of inequality (63) does not depend on τ ∈ I, then, taking in the right-hand side of (69) infimum with respect to τ ∈ I, we can conclude that

where C is the best constant in (63).

Necessity. By the conditions of Theorem 16, we have that v⁻¹ ∈ L_p′(I). Then, for any τ ∈ I, there exists k_τ such that

in addition, k_τ increases in τ, $$ and $$.

Let us use the ideas in the paper [5]. For τ ∈ I, we consider two sets $$ and $$. For each $$ and $$, we, respectively, construct the functions $$ and $$ so that g(t) = g₁(t) for 0 < t ≤ τ and g(t) = g₂(t) for t > τ belongs to the set $$.

We define a strictly increasing function ρ : (0, τ)⟶(τ, ∞) from the relation

where ρ⁻¹ is inverse to ρ. From (73), it follows that the functions ρ and ρ⁻¹ are locally absolutely continuous, ρ(τ) = τ and $$.Differentiating both relations in (73), we have

Then, for $$, we construct

while for $$, we construct

Changing the variables ρ⁻¹(t) = s and using the first equality in (74), we find that

Similarly, using the second equality in (74), we get

From (77) and (78), assuming that g(t) = g₁(t) for 0 < t ≤ τ and g(t) = g₂(t) for t > τ, we have

i.e., g ∈ L_p,v(I). For any τ ∈ I integrating both sides of (75) from τ to ∞ and (76) from 0 to τ, we find that

Hence, constructed from the functions $$ and $$, the function g belongs to $$. Replacing it into (68), we get

where all terms in the left-hand side are nonnegative.

Let the function $$ constructed from the function $$. Then, from (81) and (79), we have

Due to arbitrariness of g₁ ∈ L_p,v(0, τ), by Lemmas 10, 11, and 12, the latter gives that

Similarly, due to (81) and (79), for the function $$ constructed from the function $$, we obtain

From (83) and (84), we find that

Therefore, EF ≪ C, which, together with (70), yields that EF ≈ C, where C is the best constant in (63). The proof is complete.

Theorem 17. Let 1 < p ≤ q < ∞ and $$. Let conditions in (25) hold. Then, inequality (3) holds if and only if max{G, EF} < ∞. In addition, C ≈ max{G, EF}, where C is the best constant in (3).

Proof. Due to (60), we consider inequality (3) on the set

The function f ∈ H has the form

Hence, f^′′(t) = 0 almost everywhere on t ∈ I. Therefore, on the basis of (61), we have

Let f₀ ∈ H be such that

Then, from (89), we obtain

which implies that C ≫ G.

Let max{|c₀| + |c₁|, |c₂|} = L. Then, |f(t)| ≤ L(χ_{(0, 1)}(t) + χ_(1,∞)(t)t). Replacing the function f into the left-hand side of (89), we get

The latter, together with C ≫ G, gives that C ≈ G. Then, by Theorem 16, it follows that C ≈ max{G, EF}, where C is the best constant in (3). The proof is complete.

Corollary 18. Let 1 < p ≤ q < ∞ and the kernel of (2) satisfy condition (93). Let conditions (25) hold.

• (i)

  Inequality (63) holds if and only if $$. In addition, $$, where C is the best constant in (63)

• (ii)

  Inequality (3) holds if and only if $$. In addition, $$, where C is the best constant in (3)

Let n ≥ 3. Instead of operator (2), we consider the operator of Riemann-Liouville I_n−2, defined by

The kernel (x − t)ⁿ⁻³ of the operator I_n−2 satisfies condition (93), and therefore, it belongs to the class $$. In this case, we replace K(x, t) and K_0,1(x, t) by (x − t)ⁿ⁻³ and assume that w(x) ≡ 1. For the kernel (x − t)ⁿ⁻³ inequality, (38) has the form

Then, according to Theorem 5, we have $$, where

For the sum of kernels of the operators in (42) and (44), we deduce that

Then instead of inequalities (42) and (44), we, respectively, have

By part (i) of Theorem 1, this yields that

Assume that $$. Now, for the sum of kernels of the operators in (49) and (54), we get

Then instead of inequalities (49) and (54), we obtain

Hence, by part (i) of Theorem 1 we have

For the kernel (x − t)ⁿ⁻³ inequality (59) can be written as follows:

Therefore, by using part (ii) of Theorem 1, we find

Assume that

Thus, for inequality (63) with operator (99)

we can conclude the following statement.

Corollary 19. Let 1 < p ≤ q < ∞ and conditions (25) hold. Then, inequality (115) holds if and only if $$. In addition, $$, where C is the best constant in (115).

Corollary 20. Let 1 < p ≤ q < ∞ and conditions (25) hold. Then, inequality (116) with conditions (117) holds if and only if $$. In addition, $$, where C is the best constant in (116).