The stability of weighted Lebesgue spaces

Sharp Bounds for General Commutators on Weighted Lebesgue Spaces

We show that if an operator T is bounded on weighted Lebesgue space L(w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator [b, T ] with a function b in BMO will obey a quadratic bound with respect to the A2 constant of the weight. We also prove that the kth-order commutator T k b = [b, T k−1 b ] will obey a bound that is a power (k + 1) of the A2 consta...

Decomposition of Lebesgue Spaces

compactifications and function spaces on weighted semigruops

chapter one is devoted to a moderate discussion on preliminaries, according to our requirements. chapter two which is based on our work in (24) is devoted introducting weighted semigroups (s, w), and studying some famous function spaces on them, especially the relations between go (s, w) and other function speces are invesigated. in fact this chapter is a complement to (32). one of the main fea...

The Averaging Integral Operator between Weighted Lebesgue Spaces and Reverse Hölder Inequalities

Let 1 < p ≤ q < +∞ and let v, w be weights on (0,+∞) satisfying: v(x)xρ is equivalent to a non-decreasing function on (0,+∞) for some ρ ≥ 0; [w(x)x] ≈ [v(x)x] for all x ∈ (0,+∞). Let A be the averaging operator given by (Af)(x) := 1 x R x 0 f(t) dt, x ∈ (0,+∞). First, we prove that the operator A : L((0,+∞); v)→ L((0,+∞); v) is bounded if and only if the operator A : L((0,+∞); v)→ L((0,+∞);w) i...

The Sampling Theorem in Variable Lebesgue Spaces

hold. The facts above are well-known as the classical Shannon sampling theorem initially proved by Ogura [10]. Ashino and Mandai [1] generalized the sampling theorem in Lebesgue spaces L0(R) for 1 < p0 < ∞. Their generalized sampling theorem is the following. Theorem 1.1 ([1]). Let r > 0 and 1 < p0 < ∞. Then for all f ∈ L 0(R) with supp f̂ ⊂ [−rπ, rπ], we have the norm inequality C p r ‖f‖Lp0(Rn...