### Interpolation of Operators on Decreasing Functions.

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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 ${L}_{p}\left(w\right)$-bounded. A ${B}_{p}$-weight is precisely one for which its Hardy transform is in ${M}_{p}$, and also a weight whose indefinite...

We develop an abstract extrapolation theory for the real interpolation method that covers and improves the most recent versions of the celebrated theorems of Yano and Zygmund. As a consequence of our method, we give new endpoint estimates of the embedding Sobolev theorem for an arbitrary domain Omega.

If C is a capacity on a measurable space, we prove that the restriction of the K-functional $K(t,f;{L}^{p}\left(C\right),{L}^{\infty}\left(C\right))$ to quasicontinuous functions f ∈ QC is equivalent to $K(t,f;{L}^{p}\left(C\right)\cap QC,{L}^{\infty}\left(C\right)\cap QC)$. We apply this result to identify the interpolation space ${({L}^{p\u2080,q\u2080}\left(C\right)\cap QC,{L}^{p\u2081,q\u2081}\left(C\right)\cap QC)}_{\theta ,q}$.

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