Abstract
We obtain an atomic decomposition of weighted Lorentz spaces for a class of weights satisfying the Δ2condition. Consequently, we study operators such as the multiplication and composition operators and also provide Hölder’s-type and duality-Riesz type inequalities on these weighted Lorentz spaces.
Highlights
Weighted spaces are studied in most cases as a generalization of a special case
We prove that the weighted Lorentz spaces have an atomic decomposition and in the third section, we utilize this atomic decomposition to show the boundedness of some operators on theses weighted spaces
The special atoms spaces AΦ(μ) originally introduced by de Souza in [12] for Φ(t) = t1/p seem to have an interesting role in analysis with its connection to Lipschitz spaces through Holder’s inequality and duality
Summary
Weighted spaces are studied in most cases as a generalization of a special case. The Lorentz spaces, introduced by Lorentz in [1, 2], are no exception to this. (∫0∞ (f∗(x))pw(x)dx)1/p < ∞}, where f∗ is the decreasing rearrangement of f and w is a weight function He proved that, for p ≥ 1, ‖ ⋅ ‖Λp(w) is a norm if and only if the weight w is decreasing. We continue the ideas in [8] and show that the weighted Lorentz spaces admit an atomic decomposition, for a certain class of weights. We introduce the necessary notions needed; namely, we define the conditions on our weight functions, and provide some preliminary definitions and results. We prove that the weighted Lorentz spaces have an atomic decomposition and, we utilize this atomic decomposition to show the boundedness of some operators on theses weighted spaces. The last section opens up a discussion about the relevance of this line of research
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