We introduce two functions
wherefrom, , and .
Theorem 3 If , then we have the following equivalent inequalities:
(13)
(14)
(15)
where the constant B(λ1, λ2) is the best possible in the above inequalities.
Proof. By Beppo Levi's theorem (cf. [19]), there are two expressions for I in (13). In view of (11), for ϖ(x) < B(λ1, λ2), we have (14). By Hälder's inequality, we have
(16)
Then by (14), we have (13). On the other-hand, assuming that (13) is valid, setting
then Jp-1 = ||a||
q
, Ψ. By (11), we find J < ∝. If J = 0, then (14) is valid trivially; if J > 0, then by (13), we have
that is, (14) is equivalent to (13). By (12), since [ϖ(x)]1-q>[B(λ1, λ2)]1-q, we have (15). By Hälder's inequality, we find
(17)
Then by (15), we have (13). On the other-hand, assuming that (13) is valid, setting
then Lq-1 = ║f║
p
, Φ.. By (12), we find L < ∝. If L = 0, then (15) is valid trivially; if L > 0, then by (13), we have
That is, (15) is equivalent to (13). Hence inequalities (13), (14) and (15) are equivalent.
For 0 < ε <pλ1, setting , and
if there exists a positive number k(≤ B(λ1, λ2)), such that (13) is valid as we replace B(λ1, λ2) with k, then in particular, it follows
(18)
(19)
W find
that is, A(ε) = O(1) (ε → 0+). Hence by (18) and (19), it follows
(20)
and B(λ1, λ2) ≤ k(ε → 0+). Hence, k = B(λ1, λ2) is the best value of (13).
Due to the equivalence, the constant factor B(λ1, λ2) in (14) and (15) is the best possible. Otherwise, we can imply a contradiction by (16) and (17) that the constant factor in (13) is not the best possible. □ ▪
Remark 1 (i) Define the first type half-discrete Mulholland's operator as follows: for , we define as
Then by (14), it follows and then T is a bounded operator with ║T║ ≤ B(λ1, λ2). Since by Theorem 1, the constant factor in (14) is the best possible, we have ║T║ = B(λ1, λ2).
(ii) Define the second type half-discrete Mulholland's operator as follows: For a ∈ l
q,
ψ, define as
Then by (15), it follows and then is a bounded operator with . Since by Theorem 1, the constant factor in (15) is the best possible, we have .
Remark 2 We set in (13), (14) and (15). (i) if , then we deduce (7) and the following equivalent inequalities:
(21)
(22)
(ii) if α = 1, then we have the following half-discrete Mulholland's inequality and its equivalent forms:
(23)
(24)
(25)