## Rescaled extrapolation for vector-valued functions.(English)Zbl 1412.42045

The authors prove several extensions of Rubio de Francia’s extrapolation theorem [J. L. Rubio de Francia, Lect. Notes Math. 1221, 195–202 (1986; Zbl 0615.60041)] for vector-valued functions, namely if $$p_0>0$$ and $$X\subset L^0(\Omega)$$ is a Banach function space such that $$X^{p_0}\in \mathrm{UMD}$$ and one assumes that $$\mathcal F\subset L^0(\mathbb R^d, X)\times L^0(\mathbb R^d, X)$$ satisfies that for all $$p>p_0$$, $$(f,g)\in \mathcal F$$ and $$w\in A_{p/p_0}$$ we have the estimate $\|f(\cdot,\omega)\|_{L^p(w)}\le \phi_{p,p_0}([w]_{A_{p/p_0}})\|g(\cdot, \omega)\|_{L^p(w)},\quad \mu\text{-a.e. }\omega\in \Omega.$ Then for all $$p>p_0$$, $$(f,g)\in \mathcal F$$ and $$w\in A_{p/p_0}$$, we have $\|f\|_{L^p(w,X)}\le \phi_{p,p_0}([w]_{A_{p/p_0}})\|g\|_{L^p(w,X)}.$ Such a general result can be applied to get a number of consequences. In particular, if $$X$$ is a Banach function space and $$T$$ is a mapping acting on simple functions and with values in measurable functions defined on $$\mathbb R^d$$ such that for any $$X$$-valued simple function $$f$$ the function $$\tilde Tf(x,\omega)= (Tf(\cdot,\omega))(x)$$, for $$x\in \mathbb R^d$$ and $$\omega\in \Omega$$, is well-defined and strongly measurable, then under the assumptions that $$|T(f)-T(g)|\le |T(f-g)|$$ for simple functions $$f$$ and $$g$$ and that it extends to a bounded operator on $$L^p(\mathbb R^d, w)$$ for all $$p>p_0$$ and all Muckenhoupt weights $$w\in A_{p/p_0}$$, and that $$X$$ is a $$p_0$$-convex Banach function space such that $$X^{p_0}\in \mathrm{UMD}$$ then one obtains that $$\tilde T$$ extends to a bounded operator on $$L^p(\mathbb R^d, w, X)$$ for all $$p>p_0$$ and $$w\in A_{p/p_0}$$.
Recall that if $$\mathcal I$$ is a family of intervals in $$\mathbb R$$ and we denote $$S_I f:= \mathcal F^{-1}(1_I \mathcal F f)$$ for a given interval $$I\subset \mathbb R$$ and $$\mathcal S_{\mathcal I,q}(f)= (\sum_{I\in \mathcal I} |S_I f|^q)^{1/q}$$ then $$\|\mathcal S_{\mathcal I,2}(f)\|_{L^p}\approx \|f\|_{L^p}$$ for $$1<p<\infty$$, where $$\mathcal I$$ is the collection of dyadic intervals (due to Littlewood and Paley), and also for a general collection of disjoint intervals, due to J. L. Rubio de Francia [Rev. Mat. Iberoam. 1, No. 2, 1–14 (1985; Zbl 0611.42005)].
A lot of effort has been made to analyse the vector-valued setting in the case $$q=2$$ (see for instance [D. Potapov et al., Rev. Mat. Iberoam. 28, No. 3, 839–856 (2012; Zbl 1253.46011)]). In the paper the case $$q\ge 2$$ is handled and in the weighted setting, showing that if $$X$$ is $$q$$-convex and $$X^{q'}\in \mathrm{UMD}$$ then there exists $$\phi$$ (depending on $$X, p, q$$) such that $$\|\mathcal S_{\mathcal I,q}(f)\|_{L^p(w,X)}\le \phi([w]_{A_{p/q'}}) \|f\|_{L^p(w,X)}$$ for all $$q'<p<\infty$$, all $$w\in A_{p/q'}$$ and all $$f\in L^p(w,X)$$. Also the boundedness of vector-valued Carleson operators and some Fourier multipliers on vector-valued functions are obtained in the paper.

### MSC:

 42B25 Maximal functions, Littlewood-Paley theory 42A20 Convergence and absolute convergence of Fourier and trigonometric series 42B15 Multipliers for harmonic analysis in several variables 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

### Citations:

Zbl 0615.60041; Zbl 0611.42005; Zbl 1253.46011
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