Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights, such as $A_p$-weights of Muckenhoupt and $B_p$-weights of Ariño and Muckenhoupt. Our purpose is to show that different classes of weights are related by means of composition with classical transforms. A typical example is the family $M_p$ of weights w for which the Hardy transform is $L_p(w)$-bounded. A $B_p$-weight is precisely one for which its Hardy transform is in $M_p$, and also a weight whose indefinite integral is in $A_{p+1}$
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If C is a capacity on a measurable space, we prove that the restriction of the K-functional $K(t,f;L^p(C),L^∞(C))$ to quasicontinuous functions f ∈ QC is equivalent to $K(t,f;L^p(C) ∩ QC, L^∞(C) ∩ QC)$. We apply this result to identify the interpolation space $(L^{p₀,q₀}(C) ∩ QC,L^{p₁,q₁}(C) ∩ QC)_{θ,q}$.
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