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Interpolation of quasi-normed spaces involving weights. (English) Zbl 0561.46036
Harmonic analysis, Semin. Montréal/Qué. 1980, CMS Conf. Proc. 1, 245-267 (1981).
[For the entire collection see Zbl 0538.00010.]
Following the work of T. F. Kalugina [Moscow Univ. Math. Bull. 30, No.5/6, 108-116 (1975; Zbl 0333.46021)] and J. Gustavsson [Math. Scand. 42, 289-305 (1978; Zbl 0389.46024)] the author studies the weighted interpolation spaces \[ (A_ 0,A_ 1)_{w,p}=\{a:\quad a\in A_ 0+A_ 1,\{\int^{\infty}_{0}[\frac{K(t,a;A_ 0,A_ 1)}{w(t)}]^ p\frac{dt}{t}\}^{1/p}<\infty \} \] 0\(<p\leq \infty\), where K is the Peetre functional and the weight w is non-negative, non- decreasing and satisfies \(\bar w(s)\Doteq \sup_{t>0}\frac{w(st)}{w(t)}<\infty \quad and\int^{\infty}_{0}\min (1,1/t)\bar w(t)\frac{dt}{t}<\infty.\) The main result shows that if \(E_ i=(A_ 0,A_ 1)_{w_ i,q_ i},\) \(i=0,1\) and \(\tau (t)=w_ 0(t)/w_ t(t)\) satisfies certain conditions, \(then\)
K(t,a;E\({}_ 0,E_ 1)\sim (\int^{\tau^{- 1}(t)}_{0}[\frac{K^{(s,a;A_ 0,A_ 1)}}{w(s)}]^{q_ 0}\frac{ds}{s})^{1/q_ 0}+\)
\(t(\int^{\infty}_{\tau^{-1}(t)}[\frac{K^{(s,a;A_ 0,A_ 1)}}{w(s)}]^{q_ 1}\frac{ds}{s})^{1/q_ 1}.\)
This reduces to a result of T. Holmstedt [Math. Scand. 26, 177-199 (1970; Zbl 0193.088)] for \(w(t)=t^{\theta}\), \(0<\theta <1\). It also yields a reiteration theorem for these weighted spaces.
In addition \((L^ r,L^{\infty})_{w,p}\), \(0<p,r<\infty\), is shown to be the space of functions f for which \(\int^{\infty}_{0}[\frac{f^*(t^ r)t}{w(t)}]^ p\frac{dt}{t}<\infty,\) where \(f^*\) is the decreasing rearrangement of f. For specific w this reduces to the Lorentz-Zygmund spaces \(L^{pq}(Log L)^{\alpha}\).

MSC:
46M35 Abstract interpolation of topological vector spaces
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems