In $R^d$, define a maximal function in the directions $v\in \directions\subset\{x \mid \abs x=1\}$ by $$ M^\directions f(x)=\sup_{v\in\directions} \sup_{\zve} \int_{-\ze}^\ze \abs{f(x-vy)} dy. $$ For a function $f$ on $\ZR^d$, let $S_\zw f$ denote the Fourier restriction of $f$ to a region $\zw$. We are especially interested taking \zw to be a sector of $R^d$ with base points at the origin. A sector is a product of the interval $(0,\infty)$ with respect to a choice of (non orthogonal) basis. What is most important is that the basis is a subset of $\directions$. Consider a collection $\zW$ of pairwise disjoint sectors $\zw$ as above. Assume that $M^\directions $ maps $L^p$ into $L^p$, for some $1<p<\zI $. Then we have the following Littlewood--Paley inequality $$ \NORm \Bigl[\sum_{\zw\in\zW}\abs{S_\zw f}^2\Bigr]^{1/2}.q.\lesssim{}\norm f.q., \qquad 2\le q<2 \frac p{p-1}. $$ The one dimensional analogue of this inequality is due to Rubio de Francia.
The conclusion when the set of vectors is a fixed basis is known, is due to Journé. Our method of proof relies on a phase plane analysis. We introduce a notion of Carleson measures adapted to $\directions$, and demonstrate a John Nirenberg inequality for these measures. The John Nirenberg inequality, and an obvious $L^2$ estimate will prove the Theorem.
The one-sided Littlewood--Paley inequality for arbitrary intervals was proved by Rubio de Francia. Later, N. Osipov proved its analogue for the system of Walsh functions. In this paper, this inequality is proved for more general Vilenkin systems.
Rubio de Francia's Littlewood Paley inequality is an extension of the classical Littlewood Paley inequality to one that holds for a decomposition of frequency space into arbitrary disjoint intervals. We survey this inequality, its higher dimensional analogs, and the implications for multipliers.
Let $Ω$ be a collection of disjoint dyadic squares $ω$, let $π_ω$ denote the non-smooth bilinear projection onto $ω$ \[ π_ω(f,g)(x):=\int\int \mathbf{1}_ω(ξ,η) \widehat{f}(ξ) \widehat{g}(η) e^{2πi (ξ+ η) x} d ξdη\] and let $r>2$. We show that the bilinear Rubio de Francia operator \[ \Big(\sum_{ω\inΩ} |π_ω (f,g)|^r \Big)^{1/r} \] is $L^p \times L^q \rightarrow L^s$ bounded with constant independent of $Ω$ whenever $1/p + 1/q = 1/s$, $r'<p,q<r$, $r'/2 < s < r/2$.
We consider the Rubio de Francia's Littlewood--Paley square function associated with an arbitrary family of intervals in $\mathbb{R}$ with finite overlapping. Quantitative weighted estimates are obtained for this operator. The linear dependence on the characteristic of the weight $[w]_{A_{p/2}}$ turns out to be sharp for $3\le p<\infty$, whereas the sharpness in the range $2<p<3$ remains as an open question. Weighted weak-type estimates in the endpoint $p=2$ are also provided. The results arise as a consequence of a sparse domination shown for these operators, obtained by suitably adapting the…
Let $\mathscr{R}$ be a collection of disjoint dyadic rectangles $R$ with sides parallel to the axes, let $π_R$ denote the non-smooth bilinear projection onto $R$ \[ π_R (f,g)(x):=\iint \mathbf{1}_{R}(ξ,η) \widehat{f}(ξ) \widehat{g}(η) e^{2πi (ξ+ η) x} dξdη\] and let $r>2$. We show that the bilinear Rubio de Francia operator associated to $\mathscr{R}$ given by \[ f,g \mapsto \Big(\sum_{R\in\mathscr{R}} |π_{R} (f,g)|^r \Big)^{1/r} \] is $L^p \times L^q \rightarrow L^s$ bounded whenever $1/p + 1/q = 1/s$, $r'<p,q<r$. This extends from squares to rectangles a previous result by the same authors, …
Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2 \le p < \infty$. In this article, his methods are developed and employed to prove an analogue of such an inequality "beyond the index $p=\infty$", i.e., for spaces of Hölder functions and BMO.
We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane\[\left(f, g \right)\mapsto \left( \sum\_{ω\in Ω}\left| \int\_{\mathbb{R}^2} \hat{f}(ξ) \hat{g}(η) Φ\_ω(ξ, η) e^{2 πi x\left(ξ+η\right)} d ξd η\right|^r \right)^{1/r},\] provided $r\textgreater{}2$. More exactly, we show that the above operator maps $L^p \times L^q \to L^s$ whenever $p, q, s'$ are in the "local $L^{r'}$" range, i.e. $\displaystyle \frac{1}{p}+\frac{1}{q}+\frac{1}{s'}=1$, $\displaystyle0 \leq \frac{1}{…
Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2 \le p < \infty$. In this article, such an inequality is proved for the Walsh system.
J. L. Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2\le p<\infty$ and later N. N. Osipov proved the similar inequality for Walsh functions. In this paper we investigate some properties of Banach spaces $X$ such that the latter inequality holds for $X$-valued functions.
We obtain one variant of the extrapolation theorem of Rubio de Fracia for variable exponent Lebesgue spaces. As a consequence we obtain conditions guarantee boundedness of strongly singular integral operators, singular integral operators with rough kernels, fractional maximal operators related to spherical means, Bochner-Riesz operators in variable Lebesgue spaces.
Let $T$ be an arbitrary operator bounded from $L^{p_0}(w)$ into $L^{p_0, \infty}(w)$ for every weight $w$ in the Muckenhoupt class $A_{p_0}$. It is proved in this article that the distribution function of $Tf$ with respect to any weight $u$ can be essentially majorized by the distribution function of $Mf$ with respect to $u$ (plus an integral term easy to control). As a consequence, well-known extrapolation results, including results in a multilinear setting, can be obtained with very simple proofs. New applications in extrapolation for two-weight problems and estimates on rearrangement invari…
We extend Rubio de Francia's extrapolation theorem for functions valued in UMD Banach function spaces, leading to short proofs of some new and known results. In particular we prove Littlewood-Paley-Rubio de Francia-type estimates and boundedness of variational Carleson operators for Banach function spaces with UMD concavifications.
The purpose of this note is to correct an error in a paper of M. Cowling, G. Fendler and J.J.F. Fournier, and to give a counterexample to a conjecture of J.-L. Rubio de Francia.
We prove various extensions of the Coifman-Rubio de Francia-Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call $\ell^{r}(\ell^{s})$-boundedness, which implies $\mathcal{R}$-boundedness in many cases. The proofs are based on new Littlewood-Paley-Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors.
We extend the theory of Rubio de Francia extrapolation, including off-diagonal, limited range, and $A_{\infty}$ extrapolation, to the weighted variable Lebesgue spaces. As a consequence we are able to show that a number of different operators from harmonic analysis are bounded on these spaces. The proofs of our extrapolation results are developed in a way that outlines a general approach to proving extrapolation theorems on other Banach function spaces.
Results analogous to those proved by Rubio de Francia are obtained for a class of maximal functions formed by dilations of bilinear multiplier operators of limited decay. We focus our attention to $L^2\times L^2\to L^1$ estimates. We discuss two applications: the boundedness of the bilinear maximal Bochner-Riesz operator and of the bilinear spherical maximal operator. For the latter we improve the known results by reducing the dimension restriction from $n\ge 8$ to $n\ge 4$.
Let $Ω\in L^1{({\mathbb S^{n-1}})}$, be a function of homogeneous of degree zero, and $M_Ω$ be the Hardy-Littlewood maximal operator associated with $Ω$ defined by $M_Ω(f)(x) = \sup_{r>0}\frac1{r^n}\int_{|x-y|<r}|Ω(x-y)f(y)|dy.$ It was shown by Christ and Rubio de Francia that $\|M_Ω(f)\|_{L^{1,\infty}({\mathbb R^n})} \le C(\|Ω\|_{L\log L({\mathbb S^{n-1}})}+1)\|f\|_{L^1({\mathbb R^n})}$ provided $Ω\in L\log L {({\mathbb S^{n-1}})}$. In this paper, we show that, if $Ω\in L\log L({\mathbb S^{n-1}})$, then for all $f\in L^1({\mathbb R^n})$, $M_Ω$ enjoys the limiting weak-type behaviors that $$\l…
We generalize the extrapolation theory of Rubio de Francia to the context of Banach function spaces and modular spaces. Our results are formulated in terms of some natural weighted estimates for the Hardy-Littlewood maximal function and are stated in measure spaces and for general Muckenhoupt bases. Finally, we give several applications in analysis and partial differential equations.
This paper is devoted to studying the extrapolation theory of Rubio de Francia on general function spaces. We present endpoint extrapolation results including $A_1$, $A_p$, and $A_\infty$ extrapolation in the context of Banach function spaces, and also on modular spaces. We also include several applications that can be easily obtained using extrapolation: local decay estimates for various operators, Coifman--Fefferman inequalities that can be used to show some known sharp $A_1$ inequalities, Muckenhoupt--Wheeden and Sawyer's conjectures are also presented for many operators, which go beyond Ca…
The following subexponential estimate for commutators is proved |[|\{x\in Q: |[b,T]f(x)|>tM^2f(x)\}|\leq c\,e^{-\sqrt{α\, t\|b\|_{BMO}}}\, |Q|, \qquad t>0.\] where $c$ and $α$ are absolute constants, $T$ is a Calderón--Zygmund operator, $M$ is the Hardy Littlewood maximal function and $f$ is any function supported on the cube $Q$. It is also obtained \[|\{x\in Q: |f(x)-m_f(Q)|>tM_{1/4;Q}^#(f)(x) \}|\le c\, e^{-α\,t}|Q|,\qquad t>0,\] where $m_f(Q)$ is the median value of $f$ on the cube $Q$ and $M_{1/4;Q}^#$ is Strömberg's local sharp maximal function. As a consequence it is derived Karagulyan'…
