Extrapolation theory: new results and applications
In [\textit{B.\,Jawerth} and \textit{M.\,Milman}, Mem. Am. Math. Soc. 440 (1991; Zbl 0733.46040)], the \(\Sigma^{(p)}\) and \(\Delta^{(p)}\) extrapolation methods were introduced and used to construct end point extrapolation spaces and to prove new extrapolation estimates. However, only the \(\Sigma^{(1)}\) and \(\Delta^{(\infty)}\) methods were studied in detail there. The aim of this paper is to give a more extensive study of the \(\Sigma^{(p)}\) and \(\Delta^{(p)}\) methods of extrapolation for \(p>0\) and to present new applications of these methods. To illustrate results of the paper, we mention the following Yano type extrapolation theorems which are consequences of the \(\Sigma^{(p)}\) and \(\Delta^{(p)}\) methods of extrapolation. (i) Let \(0<s\leq 1\). Put \(\| f\| _{L^{q,s}} = \{sq^{-1} \int^{\infty}_0 [t^{1/q} f^*(t)]^sdt/t\}^{1/s}\), and \(L^{q,\infty}:= (L^s,L^{\infty})_{\theta, \infty}\), \(1/q = (1-\theta)/s\). If \(T\) is a sublinear operator satisfying \[ \| Tf\| _{L^{q,\infty}} \leq c (q-s)^{-a} \| f\| _{L^{q,s}}, \quad 0 < s < q < p, \quad a > 0, \] then \[ T\: L^s (\log L)_a + L^{p,s} \rightarrow L^s + L^{p,s}. \] (ii) Let \(L^{q,\infty}:= (L^r,L^{\infty})_{\theta,\infty}\), \(1/q = (1-\theta)/r\), \(r= \min (p,s)\). If \(T\) is a sublinear operator satisfying \[ \| Tf\| _{L^{q,\infty}} \leq c q^a\| f\| _{L^{q,s}}, \quad 0 < p \leq q < \infty, \;s > 0, \;a > 0, \] then \[ T\: L^p\cap L^{\infty} \rightarrow L^r \cap L^{\infty} (\log L)_{-a}, \quad p \leq s, \] and \[ T\: L^{p,\infty} \cap L^{\infty} \rightarrow L^{p,\infty} \cap L^{\infty} (\log L)_{-a}, \quad p > s. \] Two types of applications are given. First, the authors show that spaces currently appearing in analysis (e.g., Lorentz--Zygmund spaces, Donaldson--Sullivan spaces, logarithmic Sobolev spaces, etc.) are in fact extrapolation spaces for the \(\Sigma^{(p)}\) and \(\Delta^{(p)}\) methods. Second, the authors consider in detail certain classical operators (e.g., semigroups associated with the theory of logarithmic Sobolev inequalities) to give concrete applications of extrapolation theory to classical analysis.
scientific article; zbMATH DE number 2165388
scientific article