A class of weighted inequalities. (English) Zbl 0772.46011

Emara, Salah A. A.

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

Keywords:

weighted inequalities; interpolation couples; quasi-linear operator; weighted intermediate spaces; Peetre \(K\)-functional