Abstract
Some character of the symmetric homogenous kernel of 1-order in Hilbert-type operator is obtained. Two equivalent inequalities with the symmetric homogenous kernel of -order are given. As applications, some new Hilbert-type inequalities with the best constant factors and the equivalent forms as the particular cases are established.
Highlights
If the real function k(x, y) is measurable in (0,∞) × (0,∞), satisfying k(y,x) = k(x, y), for x, y ∈ (0,∞), one calls k(x, y) the symmetric function
Where kp is a positive constant independent of x, T ∈ B(lr→lr), T is called the Hilberttype operator and T r ≤ kp (r = p, q); (ii) if for fixed x > 0, ε ≥ 0 and r = p, q, the functions k(x, t)(x/t)(1+ε)/r are decreasing in t ∈ (0, ∞); kr(ε, x) = kp(ε) (r = p, q; ε ≥ 0) is independent of x, satisfying kp(ε) = kp + o(1) (ε→0+), and
The following inequalities are equivalent: ambn n=1 m=1 max mλ, nλ
Summary
If the real function k(x, y) is measurable in (0,∞) × (0,∞), satisfying k(y,x) = k(x, y), for x, y ∈ (0,∞), one calls k(x, y) the symmetric function. Suppose that p > 1, 1/ p + 1/q = 1, lr (r = p, q) are two real normal spaces, and k(x, y) is a nonnegative symmetric function in (0, ∞) × (0, ∞). Define the operator T as follows: for a = {am}∞m=1 ∈ lp,. The function k(x, y) is said to be the symmetric kernel of T
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