Integral operators and weighted amalgams

Boundedness of positive operators on weighted amalgams

“…In this sense, our result includes, as particular cases, most of the results cited above from the papers [3][4][5][6][7], as well as other corresponding to operators whose behavior on weighted amalgams has not been studied yet. Thus, if M -is the one-sided Hardy-Littlewood maximal operator defined by…”

“…Carton-Lebrun, Heinig, and Hoffmann studied in [3] holds for all f, with a constant C independent of f, whenever 1 <q ≤ q < ∞. The characterization of the pairs (u, v) for (1.1) to hold in the case 1 < q <q < ∞ has been recently completed by Ortega and Ramírez ( [4]), who have also characterized the weak type inequality where ||g|| p,∞;u,q = n∈Z ||gX (n,n+1) || q p,∞,u 1 q .…”

“…Specifically, Carton-Lebrun, Heinig, and Hoffmann studied in [3] weighted inequalities in amalgams for the Hardy-Littlewood maximal operator as well as for some integral operators with kernel K(x, y) increasing in the second variable and decreasing in the first one. On the other hand, Rakotondratsimba ( [5]) characterized some weighted inequalities in amalgams (corresponding to the cases 1 <p ≤ p < ∞ and 1 <q ≤ q < ∞) for the fractional maximal operators and the fractional integrals.…”

Integral Operator in Amalgam

“…We omit the detail. When X u is weighted Lebesgue space, φ 1 ≡ ψ 1 ≡ 1 and f ≥ 0, Lemma A has been proved in [3].…”

“…Boundedness of the Hardy operator (T f )(x) = x −∞ f (t)dt and it's adjoint operator (T * f )(x) = ∞ x f (t)dt between ℓ q (X u ) and ℓ b (L r , v) has been considered in [6], while ℓ q (L p , u) − ℓq(Lp, v) boundedness of T and T * has been proved in [3]. The notion of BFS was introduced by Luxemburg [12].…”