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Extrapolation theory: new results and applications. (English) Zbl 1081.46018
In [B. Jawerth and 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.

##### MSC:
 46B70 Interpolation between normed linear spaces 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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