×

Real interpolation with logarithmic functors. (English) Zbl 1041.46011

The interpolation spaces in question consist of the vectors \(f\in X_0 +X_1\) satisfying \[ \Biggl(\int_0 ^\infty (t^{-\theta} \phi (t) K(x,\,y;\,X_0,\,X_1))^q t^{-1} \,dt \Biggr)^{1/q}<\infty \] (with the usual modification if \(q=\infty\)); here \(K(\dots)\) is the \(K\)-functional, and \(\phi\) is a function that behaves like some power of \(| \log t| \) near \(0\) and like some other power of \(\log t\) near \(\infty\). (In another version, a double logarithm is also incorporated in the definition of \(\phi\).) The emphasis is on Hardy-type inequalities, analogs of the Holmstedt formula, and reiteration theorems.

MSC:

46B70 Interpolation between normed linear spaces
47B38 Linear operators on function spaces (general)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47G10 Integral operators
26D10 Inequalities involving derivatives and differential and integral operators
PDFBibTeX XMLCite
Full Text: DOI