For n ≥ 1, we introduce the classes and of matrices (a
i,j
). We assume that if or .
We define the classes , n ≥ 0 by induction. Let (a
i,j
) be a matrix which is non-negative and non-decreasing in the first index for all i ≥ j ≥ 1. By definition matrices of the type , ∀i. ≥ j ≥ 1 belong to the class . Let the classes , γ = 0, 1,..., n - 1, n ≥ 1 be defined. By definition, the matrix belongs to the class if and only if there exist matrices , γ = 0, 1,..., n - 1 and a number r
n
> 0 such that
(7)
for all i ≥ k ≥ j ≥ 1, where and
(8)
From (8) it follows that entries of the matrices do not decrease in the first index and do not increase in the second index. And (8) provides the validity of the following inequality
(9)
Then for we have
(10)
for all i ≥ k ≥ j ≥ 1.
REMARK 1. It is easy to show that if for the matrix, n ≥ 0 there exist matrices, γ = 0, 1,..., n - 1, and matrices, γ = 0, 1,..., n such that the equivalence (10) is valid for all i ≥ k ≥ j ≥ 1, thenand. Hence we may assume that the matricesare arbitrary non-negative matrices which satisfy (10).
For the proof of our main results we also need the following inequality. Let n ≥ l ≥ γ. Then we have
(11)
Indeed, using (9), for i ≥ s ≥ k ≥ 1, n ≥ l ≥ γ we obtain
As above, we introduce the classes , m ≥ 0. Let (a
i,j
) be a matrix which is non-negative and non-increasing in the second index for all i ≥ j ≥ 1. By definition a matrix belongs to the class if and only if it has the form for all i ≥ j ≥ 1. Let the classes , γ = 0, 1,..., m - 1, m ≥ 1 be defined. A matrix belongs to the class if and only if there exist matrices , γ = 0, 1,..., m - 1 and a number h
m
> 0 such that
(12)
for all i ≥ k ≥ j ≥ 1, where and
(13)
From the definition of the matrix , γ = 0, 1,..., m - 1, m = 0, 1,..., it is obvious that the entries of the matrix do not decrease in the first index and do not increase in the second index and for m ≥ l ≥ γ, k ≥ s ≥ j satisfy the following inequality
(14)
From (13) it follows that for all i ≥ k ≥ j ≥ 1
(15)
As in (10) every class , m ≥ 0 of matrices is characterized by the following relation
(16)
for all i ≥ k ≥ j, where , γ = 0, 1,..., m are defined by the formula (13).
REMARK 2. As mentioned before we may assume that the matrices, γ = 0, 1,..., m, m ≥ 0 are arbitrary non-negative matrices which satisfy (16).
REMARK 3. By the definitions of the classes, n ≥ 0 we have
In particular, the matrices of the classes and are characterized by the following relations, respectively,
It is easy to see that the class include the matrices, whose entries satisfy conditions (4) and (5). Also it should be noted that the matrices with conditions (4) and (6) belong to the class . This implies that the classes , n ≥ 1 and , m ≥ 1 are wider than the classes of matrices which have been used in this connection before.
The matrices of the classes and are described by the following relations, respectively,
Next, we show properties of the classes of matrices and , n ≥ 0.
We set
Then we have the following
Lemma 2.1. Let. Then.
PROOF OF LEMMA 2.1. Since , there exist matrices , γ = 0, 1,..., n-1, and matrices such that
for all i ≥ l ≥ k ≥ 1.
Since , there exist matrices , μ = 0, 1,..., m - 1, and matrices such that
for all j ≥ l ≥ k ≥ 1.
We set
First, we consider the case when m ≥ 0, n = 0. In this case , ∀i ≥ j ≥ 1. For ∀i ≥ l ≥ k we obtain
where , μ = 0, 1,..., m. Suppose that . Since , μ = 0, 1,..., m, by definition we easily see that . By induction, we assume that for n = 0, 1,..., r - 1, belongs to the class . For i ≥ l ≥ k we have
where , γ = 0,..., r - 1 and μ = 0,..., m. We denote γ + m + 1 by μ. Then we have
where
and
Since , μ = 0, 1,..., r + m we obtain that . The proof is complete.
Now we set
Then we have the following lemma.
Lemma 2.2. Let, . Then.
Lemma 2.2 can be proved in the same way as Lemma 2.1.
For the proof of our main theorem we will need the following well-known result for the discrete weighted Hardy inequality (see [1, 9]) and the criteria of precompactness of sets in l
p
(see [10, p. 32]).
Theorem A. Let 1 < p ≤ q < ∞. Then the inequality
(17)
holds if and only if
Moreover, H ≈ C, where C is the best constant in (17).
Theorem B. Let T be a set in l
p
, 1 ≤ p < ∞. The set T is compact if and only if T is bounded and for all ε > 0 there exists N = N(ε) such that for allthe inequality
holds.