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Criteria of boundedness and compactness of a class of matrix operators
Journal of Inequalities and Applications volume 2012, Article number: 53 (2012)
Abstract
The aim of this article is to obtain criteria of boundedness and compactness for a wide class of matrix operators from one weighted l p,v space of sequences to another weighted l q,u space, in the case 1 < p ≤ q < ∞. We introduce a general class of matrices. Then we establish necessary and sufficient conditions for the boundedness and compactness of the operators , i ≥ 1 and , j ≥ 1 corresponding to matrices in such classes by using the method of localization. Our classes are more general than those for which corresponding Hardy inequalities are known in the literature.
2010 Mathematics Subject Classification: 26D15; 47B37.
1 Introduction
We consider the problem of boundedness from the weighted l p,v space into the weighted l q,u space of the matrix operators
which is equivalent to the validity of the following Hardy-type inequality
where C is a positive finite constant independent of f and (a i,j ) is a non-negative triangular matrix with entries a i,j ≥ 0 for i ≥ j ≥ 1 and a i,j = 0 for i < j.
Here and further 1 < p, q < ∞, and are positive sequences of real numbers. l p,v is the space of sequences of real numbers such that
For a i,j = 1, i ≥ j ≥ 1, the operators (1), (2) coincide with the discrete Hardy operators of the forms , respectively. References about generalizations of the original forms of the discrete and continuous Hardy inequalities can be found in different books, see e.g., [1–3].
In [4, 5], necessary and sufficient conditions for the validity of (3) have been obtained for 1 < p, q < ∞ under the assumption that there exists d ≥ 1 such that the inequalities
hold.
A sequence is called almost non-decreasing (non-increasing), if there exists c > 0 such that ca i ≥ a k (a k ≤ ca j ) for all i ≥ k ≥ j ≥ 1.
In [6], estimate (3) has been studied under the assumption that there exist d ≥ 1 and a sequence of positive numbers , and a non-negative matrix (b i,j ), where b i,j is almost non-decreasing in i and almost non-increasing in j, such that the inequalities
hold for all i ≥ k ≥ j ≥ 1.
In [7, 8], inequality (3) has been considered under the assumption that there exist d ≥ 1, a sequence of positive numbers , and a non-negative matrix (b i,j ), whose entries b i,j are almost non-decreasing in i and almost non-increasing in j such that the inequalities
hold for all i ≥ k ≥ j ≥ 1.
Conditions (5) and (6) include conditions (4), and complement each other.
In this article, we introduce a general class of matrices. We establish necessary and sufficient conditions for the boundedness and compactness of the operators (1) and (2), where the corresponding matrices belong to such classes. Such classes of matrices are wider than those which have been previously studied in the theory of discrete Hardy-type inequalities.
The content of the article is as follows. In Section 2, we introduce our classes of matrices and their properties. Moreover, in this section we give some auxiliary statements. Section 3 contains the main results. In Section 4, we prove the theorems, which give criteria of boundedness of the operators defined by (1) and (2). In Section 5, we obtain compactness criteria for the operators defined by (1) and (2). Then based on these statements, we prove our main theorems in Section 6. Moreover, in this section we show that one can imply our main results in order to obtain necessary and sufficient conditions for boundedness and compactness of the composition of operators.
Notation: If M and K are real valued functionals of sequences, then we understand that the symbol M ≪ K means that there exists c > 0 such that M ≤ cK, where c is a constant which may depend only on parameters such as p, q, r n and h n . If M ≪ K ≪ M, then we write M ≈ K.
2 Preliminaries and notation
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
for all i ≥ k ≥ j ≥ 1, where and
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
Then for we have
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
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
for all i ≥ k ≥ j ≥ 1, where and
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
From (13) it follows that for all i ≥ k ≥ j ≥ 1
As in (10) every class , m ≥ 0 of matrices is characterized by the following relation
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
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.
3 Main results
We define
We set and
Theorem 3.1. Suppose that 1 < p ≤ q < ∞. Let the matrix (a i,j ) in (1) belong to the class, m ≥ 0. Let A+be the operator defined in (1). Then the following statements hold:
(i) A+ is bounded from l p,v into l q,u if and only if one of the conditionsandholds. Moreoverwhere C is the best constant in (3).
( ii ) A+is compact from l p,v into l q,u if and only if one of the conditionsholds.
Theorem 3.2. Suppose that 1 < p ≤ q < ∞. Let the matrix (a i,j ) in (2) belong to the classLet A- be the operator defined in (2). Then the following statements hold:
(j) A- is bounded from l p,v into l q,u if and only if one of the conditionsandholds. Moreoverwhere C is the best constant in (3).
(jj) A- is compact from l p,v into l q,u if and only if one of the conditionsandholds.
Before proving our main theorems we establish the boundedness and compactness of the operators and from the weighted l p,v space into the weighted l q,u space in particular cases.
4 Boundedness of the matrix operators
Theorem 4.1. Let 1 < p ≤ q < ∞. Let the matrix (a i,j ) in (1) belong to the classThen the estimate (3) for the operator defined by (1) holds if and only if one of the conditionsandholds. Moreoverwhere C is the best constant in (3).
Theorem 4.2. Let 1 < p ≤ q < ∞. Let the matrix (a i,j ) in (2) belong to the classThen the estimate (3) for the operator defined by (2) holds if and only if one of the conditionsandholds. Moreoverwhere C is the best constant in (3).
Here we present only the proof of Theorem 4.2, since the proof of Theorem 4.1 is very similar.
For the proof of Theorem 4.2 we need the following.
Lemma 4.1. Let the matrix of (2) belongs to the classThen we have the following equivalence
where
By (18) it follows that
Indeed, this equivalence easily follows from (16).
PROOF OF THEOREM 4.2. Necessity. Suppose that the matrix of the operator (2) belongs to the class and (3) holds.
For k > 1 we assume that
It is known that inequality (3) holds if and only if the following dual inequality
holds for the conjugate operator A*, which coincides with operator defined by (1). Moreover, the best constants in (3) and (20) coincide (see e.g., [3]).
Hence applying in (20) and using (15) we obtain
Therefore
Now for 1≤ r < M < ∞, we assume that where
Applying f to inequality (3) and using (15) we find that
which implies
Since inequality (21) holds for all γ = 0,1,..., m and r ≥ 1 is arbitrary, passing to the limit as M → ∞ we have
By using Lemma 4.1 we obtain
The proof of the necessity is complete.
Sufficiency. Let the matrix (a i,j ) of the operator (2) belong to the class Let and at least one of the conditions and hold. Assume that m = 0. By the definition of the matrix of the operator (2) has the form Then the estimate (3) coincides with the estimate (17) and the operator (2) is the matrix operator Hence from Theorem A it follows that
Based on Lemma 4.1 it follows that the inequality (3) holds for m = 0 and for the best constant in (3) the following estimate is valid
Now we assume that the inequality (3) holds for m = 0,1,..., n - 1, n ≥ 1 and for the best constant in (3) the estimate (24) is valid. We consider the inequality
where is given by (2) with the matrix
Now our aim is to show that the inequality (25) holds for m = n with the estimate (24).
Let where h n is the constant in (12) with m = n. For all j ≥ 1we define the following set:
where ℤ is the set of integers. We assume that k j = inf T j , if T j ≠ ∅ and k j = ∞, if T j = ∅. In order to avoid trivial cases we directly suppose that Since is non-increasing in j, we have k j ≤ k j +1. If k j < ∞, then
Let m1 = 0, and M1 = {j ∈ ℕ: k j = k1 = , where ℕ is the set of natural numbers. Suppose that m2 is such that sup M1 = m2. Obviously m2> m1 and if the set M1 is bounded from above, then m2< ∞ and m2 = max M1. We now define the numbers 0 = m1< m2m s < ∞, s ≥ 1 by induction. To define m s +1 we assume that m s +1 = sup M s , where M s = {j ∊ ℕ: k j =
Let N0 = {s ∊ ℕ: m s <∞}. Further, we assume that = n s +1, s ∊ N0. From the definition of m s and from (26) it follows that, for s ∊ N0,
and
Therefore for 0 ≤ f ∊ l p,v the left-hand side of (3) has the following form
We assume that
There are two possible cases: N0 = ℕ and N0 ≠ ℕ.
1. If N0 = ℕ, then we estimate (28) in the following way.
Clearly inequalities n s +1< n s +2< n s +3 imply that -n s +3 + 1 ≤ -n s +1 - 1 for all s ∊ ℕ. Hence, (27), (16) imply that
Now, by using (27) and (29), we can estimate (28) in the following way.
where
and
To estimate I n we apply Hölder's and Jensen's inequalities and find that
We introduce the sequence defined by , j = m
s
+3 and , s ∊ N. Hence, we can rewrite I
γ
, γ = 0,..., n - 1 in the following form:
By the assumptions on we have the validity of (25). Therefore,
where
Using (14) and taking into account that is non-decreasing in i and non-increasing in j, we find that
By combining (33), (34), and (35), we obtain that
Thus, from (30), (31), and (36) it follows that
i.e., inequality (3) is valid and by Lemma 4.1, we obtain that
2. If N0 ≠ N, i.e., max N0< ∞ and N0 = {1, 2,..., s0}, s0 ≥1. Therefore, < ∞ and We assume that and We have two possible cases: < ∞ and = ∞. We consider these cases separately:
-
1)
If < ∞, then from (28) it follows that
(39)
If J1 ≠ 0 then for s0> 3, we estimate J1 using (29) and the previous proof for the case N0 = ℕ as in estimate I γ . Hence we get
If J2 ≠ 0 then by using (27) and applying Hölder's and Jensen's inequalities, we obtain the following estimate
Using (27) and applying Hölder's inequality we estimate J3 in the following way.
By (39), (40), (41), and (42) we obtain (37) and, consequently (38).
-
2)
If = ∞, which means that then by the definition of we have and By the assumption that it follows that Therefore, m 2 < ∞ and s 0 ≥ 2. Thus by (28) we have
(43)
By estimating and as J1 and J2, respectively, from (43) we obtain (37) and, consequently (38). Therefore, we see that inequality (25) holds for m = n and the estimate (24) is valid. This means that inequality (25) holds for all m ≥ 0 with the estimate (24), which together with (23) gives The proof is complete.
5 Compactness of the matrix operators
Theorem 5.1. Let 1 < p ≤ q < ∞. Let the matrix (a i,j ) of (1) belong to the classThen the operator defined by (1) is compact from l p,v into l q,u if and only if one of the following conditions holds
Theorem 5.2. Let 1 < p ≤ q < ∞. Let the matrix (a i,j ) of (2) belong to the classThen the operator defined by (2) is compact from l p,v into l q,u if and only if one of the following conditions holds
Now we give the proof of compactness for the class
PROOF OF THEOREM 5.1. For the proof of Theorem 5.1, we need the following equivalence
where
The equivalence directly follows from (10).
Necessity. Suppose that the matrix of operator (1) belongs to the class Let the operator (1) be compact from l p,v into l q,u .
For r ≥ 1, we introduce the following sequence:
where