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A class of weighted inequalities. (English) Zbl 0772.46011
Harmonic analysis, Proc. Conf., Sendai/Jap. 1990, ICM-90 Satell. Conf. Proc., 106-116 (1991).
Let $$(A_ 0,A_ 1)$$, $$(B_ 0,B_ 1)$$ be interpolation couples, $$1\leq q_ i$$, $$r_ i\leq \infty$$, and $$w_ i$$, $$s_ i$$ be weight functions in some function classes $$(i=0,1)$$. Suppose $$T$$ is a quasi-linear operator bounded from the weighted intermediate spaces $$(A_ 0,A_ 1)_{w_ i,q_ i}$$ to $$(B_ 0,B_ 1)_{s_ i,r_ i}$$ $$(i=0,1)$$. The author gives conditions on pairs of weights $$u$$ and $$v$$ for which $$\bigl( \int_ 0^ \infty [u(t)K(t,Tf;B_ 0,B_ 1)]^ q dt\bigr)^{1/q}\leq C\bigl(\int_ 0^ \infty [u(t)K(t,f;A_ 0,A_ 1)]^ p dt\bigr)^{1/p}$$ holds, where $$K$$ is the Peetre $$K$$-functional. He notes that the reiteration theorem for real interpolation spaces can be covered by his theorem.
For the entire collection see [Zbl 0759.00011].
Reviewer: K.Yabuta (Nara)
##### MSC:
 46B70 Interpolation between normed linear spaces 46M35 Abstract interpolation of topological vector spaces