Weighted inequalities for a class of matrix operators: the case p ≤ q

Abstract

We prove a new discrete Hardy-type inequality A f q,u a i, j f j , a i, j 0 . Moreover, we study the problem of compactness of the operator A , and the dual result is stated. (2000): 26D10, 26D15, 47B37. Mathematics subject classification

Hardy-type inequalities for matrix operators

“…The inequalities (1) and (2) have been investigated for the case 0 < p, q < ∞ with a triangular matrix (a i,j ≥ 0 for 1 ≤ j ≤ i and a i,j = 0 for i < j) in [2][3][4][5] and the references given therein. However, these inequalities have not been studied for the case 0 < p ≤ q < ∞ and p ≤ 1.…”

“…However, these inequalities have not been studied for the case 0 < p ≤ q < ∞ and p ≤ 1. Only, in 1991 G. Bennett [3] studied the inequality (1) for the this case with the identity matrix. He proved that the inequality (1) holds if and only if…”

“…The main aim of this paper is to investigate that the problems necessary and sufficient conditions the validity of inequalities (1) and (2) with the case 0 < p ≤ q < ∞, 0 < p < 1 and under weaker conditions on the matrices (a i,j ) in operators defined by (3) and (4) for all sequences f ≥ 0 (see theorems 2.1-2.2). Moreover, we study these problems on the cone of monotone sequences (see theorems 2.3-2.6).…”

Estimates for norms of two‐weighted summation operators on a tree under some restrictions on weights

“…The similar problem for two-weighted integration operators on a semiaxis was solved by Bradley [5], Mazya and Rozin [21]. Later, these results were generalized for matrix operators and integration operators with different kernels (see, e.g., papers of Heinig and Andersen [1,14], Stepanov [30,31], Oinarov [23], Prokhorov and Stepanov [27], Stepanov and Ushakova [32], Rautian [28], Farsani [11], Oinarov, Persson and Temirkhanova [24], Okpoti, Persson and Wedestig [25,26], and the books [13,15,16]). In the case p = q = 2 Naimark and Solomyak [22] showed that the problem of estimating the norm of weighted integration operator on a regular tree with weights depending only on distance from the root can be reduced to a problem on estimating the norm of some weighted Hardy-type operator on a half-axis.…”

Criteria of boundedness and compactness of a class of matrix operators

“…In [6], estimate (3) has been studied under the assumption that there exist d ≥ 1 and a sequence of positive numbers {ω k } ∞ k=1 , 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…”