Assume that a(m, n) is decreasing in the first variable.
If. for (m, n) ∈ Ω, then (2) Assume that a(m, n) is decreasing in the first variable.
Suppose u, a, b, f, g, h, w ∈ ℘+, and b, f, g, h, w are decreasing in the first variable, while nondecreasing in the second variable.
Specifically, Carton-Lebrun, Heinig, and Hoffmann studied in [3] weighted inequalities in amalgams for the Hardy-Littlewood maximal operator as well as for some integral operators with kernel K x, y) increasing in the second variable and decreasing in the first one.
Suppose u, a, b, w ∈ ℘+ with a(m, n) not equivalent to zero, and w is decreasing both in the first variable and the second variable, a is decreasing in the first variable, and b is decreasing in the second variable, α, β are defined as in Theorem 2.2, and L is defined as in Theorem 2.8.
Suppose u, a, b, f, g, h, w ∈ ℘+ with a(m, n) not equivalent to zero, and f, g, h, w are decreasing both in the first variable and the second variable, a is decreasing in the first variable, and b is decreasing in the second variable, α, β are defined as in Theorem 2.2, and p, q, r l are defined as in Theorem 2.1.
Suppose u, a, b, f, g, h, w ∈ ℘+, and b, f, g, h, w are nondecreasing in the first variable, while decreasing in the second variable.
provided that, and is nondecreasing in the first variable and decreasing in the second variable, where K > 0 is a constant, and (21).
Remark 7. If we take Ω = ℕ0 × ℕ0, w(m, n) ≡ 0, α(m) = m, β(n) = n, and omit the conditions "w is nondecreasing in the first variable, while decreasing in the second variable", " is nondecreasing in the first variable and decreasing in the second variable", and "b is decreasing in the second variable" in Theorem 2.8, then Theorem 2.8 reduces to [[13], Theorem 5].
If we take Ω = ℕ0 × ℕ0, w(m, n) ≡ 0, α(m) = m, β(n) = n, and omit the conditions "w is decreasing both in the first variable and the second variable", " is decreasing both in the first variable and the second variable" and "b is decreasing in the second variable" in Theorem 2.9, then Theorem 2.9 reduces to [[13], Theorem 6].
