Content uploaded by Nermin Okicic
Author content
All content in this area was uploaded by Nermin Okicic on Oct 14, 2015
Content may be subject to copyright.
ADV MATH
SCI JOURNAL
ON WEIGHTED BANACH SEQUENCE SPACES
Abstract
We consider Banach sequence spaces lp;� with a weighted sequence �,
which are generalizations of standard sequence spaces. We investigate the relationships
between these spaces for a xed p (1 � p � +1) and for di�erent weighted
functions, as well as for xed � and various p; q (1 � p < q � +1). We also present
the representation of bounded linear functionals on these spaces.
ResearchGate has not been able to resolve any citations for this publication.
Let A=(a n,k ) n,k≥1 and B=(b n,k ) n,k≥1 be two non-negative matrices. Denote by L v,p,q,B (A), the supremum of those L, satisfying the following inequality: ∥Ax∥ v,B(q) ≥L∥x∥ v,B(p) , where x≥0 and x∈l p (v,B) and also v=(v n ) n=1 ∞ is an increasing, non-negative sequence of real numbers. In this paper, we obtain a Hardy-type formula for L v,p,q,B (H μ ), where H μ is the Hausdorff matrix and 0<q≤p≤1. Also for the case p=1, we obtain ∥A∥ w,B(1) , and for the case p≥1, we obtain L w,B(p) (A).
In this article, we study the boundedness of matrix operators acting on weighted sequence Besov spaces ˙ b α,q p,w . First we obtain the necessary and sufficient condition for the boundedness of diagonal matrices acting on weighted sequence Besov space ˙ b α,q p,w , and investigate the duals of ˙ b α,q p,w , where the weight is non-negative and locally integrable. In particular, when 0 < p < 1, we find a type of new sequence sapces which characterize the dual space of ˙ b α,q p,w . We also use the duals of ˙ b α,q p,w to characterize an algebra of matrix operators acting on weighted sequence Besov spaces ˙ b α,q p,w and find the necessary and suffi-cient conditions to such a characterization. Note that we do not require that the given weight satisfies the doubling condition in this situation. Using these results, we give some applications to characterize the boundedness of Fourier-Haar multipliers and paraproduct operators. In this situation, we need to require that the weight w is an A p weight.
We give the exact formulas for the so-called moduli of near convexity of the Baernstein space. This space is frequently used in the geometric theory of Banach spaces and in other branches of nonlinear functional analysis.
In a recent paper [8], the author has discoverd the norm for the Cesaro, Copson and Hilbert operators on Lorentz sequence space d(w,1). The purpose of this note is to establish analogous norms for arbitrary weighted mean matrices(with non-negative entries) acting on arbitrary ‘1(w)(d(w,1)) spaces. Key Words:Norm, Weighted Mean Matrix,Weighted Sequence Space.
The problem considered is thedetermination of of matrix operators on the spaces\ell_p(w) or d(w,p). Under fairly general conditions, thesolution is the same for both spaces and is given by the infimum of a certain sequence. Specific casesare considered, with the weighting sequence defined by w_n = 1/n^\alpha . The exactsolution is found for the Hilbert operator. For the averaging operator, two different upper bounds aregiven, and for certain values of p and \alpha it is shown thatthe smaller of these two bounds is the exact solution.
The problem addressed is the exact determination of the norms of the classical Hilbert, Copson and averaging operators on weighted 'p spaces and the corresponding Lorentz sequence spaces d(w,p), with the power weighting sequence wn = n or the variant defined by w1+···+wn = n1 . Exact values are found in each case except for the averaging operator with wn = n , for which estimates deriving from various different methods are obtained and compared.
Workable necessary and sufficient conditions for a non-negative matrix to be a bounded operator from l p to l q when 1 < q ≤ p < ∞ are discussed. Alternative proofs are given for some known results, thereby filling a gap in the proof of the case p = q of a result of Koskela's. The case 1 < q < p < ∞ of Koskela's result is refined, and a weakened form of the Vere-Jones conjecture concerning matrix operators on l p is shown to be false.
The sequence space ℓ(p) was introduced and studied by Maddox [I.J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford (2) 18 (1967) 345–355]. In the present paper, the sequence spaces ℓ(u,v;p) of non-absolute type which is derived by the generalized weighted mean is defined and proved that the spaces ℓ(u,v;p) and ℓ(p) are linearly isomorphic. Besides this, the β- and γ-duals of the space ℓ(u,v;p) are computed and the basis of that space is constructed. Further, it is established that the sequence space ℓp(u,v) has AD property and given f-dual of the space ℓp(u,v). Finally, the matrix mappings from the sequence spaces ℓ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence space ℓ(u,v;p) are characterized.
We find a new expression for the norm of a function in the weighted Loreritz space, with respect to the distribution funetion, and obtain as a simple consequence a generalization of the classical embeddings Lp, 1 ⊂ · · · ⊂ Lp ⊂ · · · ⊂ Lp, ∞ and a new definition of the weak space Λp, ∞u (w). We also give some applications to the boundedness of the Hardy operator Sƒ = ∫x0ƒ from Λp0u0(w0) into Λp1u1 (w1) with 0 < p0 ≤ p1.