### 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...

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 ${({L}^{p\u2080,q\u2080}\left(C\right)\cap QC,{L}^{p\u2081,q\u2081}\left(C\right)\cap QC)}_{\theta ,q}$.

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.

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