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.


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.)
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