该研究提出定量多线性有限范围外推定理，拓展已有外推结果，发展适配多子线性哈代-李特尔伍德极大算子的多线性Rubio de Francia算法，得到多类空间相关估计及推论。
摘要：
本文证明了一个定量多线性有限范围外推定理，该定理允许我们从包含部分指数为无穷大情况的加权估计出发进行外推，拓展了Li、Martell和Ombrosi近期的外推结果。我们还得到了包括ℓ^∞空间在内的向量值估计，尤其能够重新证明通过Benea和Muscalu的螺旋法得到的双线性希尔伯特变换的所有向量值界。此外，我们的结果是定量的，特别地可将巴拿赫空间框架下由稀疏控制得到的定量估计推广到拟巴拿赫空间框架。我们的证明不依赖任何非对角外推结果，还发展了适配多子线性哈代-李特尔伍德极大算子的Rubio de Francia算法的多线性版本。作为推论，我们得到了弱型空间和BMO空间中部分上下端点估计的多线性外推结果。
英文摘要：
In this paper we prove a quantitative multilinear limited range extrapolation theorem which allows us to extrapolate from weighted estimates that include the cases where some of the exponents are infinite. This extends the recent extrapolation result of Li, Martell, and Ombrosi. We also obtain vector-valued estimates including $\ell^\infty$ spaces and, in particular, we are able to reprove all the vector-valued bounds for the bilinear Hilbert transform obtained through the helicoidal method of Benea and Muscalu. Moreover, our result is quantitative and, in particular, allows us to extend quantitative estimates obtained from sparse domination in the Banach space setting to the quasi-Banach space setting.
Our proof does not rely on any off-diagonal extrapolation results and we develop a multilinear version of the Rubio de Francia algorithm adapted to the multisublinear Hardy-Littlewood maximal operator.
As a corollary, we obtain multilinear extrapolation results for some upper and lower endpoints estimates in weak-type and BMO spaces.
In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes $A_{\vec{p}}$, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calderón-Zygmund operators. Furthermore, we go beyond the classes $A_{\vec{p}}$ and extrapolate within the classes $A_{\vec{p},\vec{r}}$ which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several a…
In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing so, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding ``finite'' case obtained by Duoandikoetxea, which wa…
We prove a limited range, off-diagonal extrapolation theorem that generalizes a number of results in the theory of Rubio de Francia extrapolation, and use this to prove a limited range, multilinear extrapolation theorem. We give two applications of this result to the bilinear Hilbert transform. First, we give sufficient conditions on a pair of weights $w_1,\,w_2$ for the bilinear Hilbert transform to satisfy weighted norm inequalities of the form \[ BH : L^{p_1}(w_1^{p_1}) \times L^{p_2}(w_2^{p_2}) \longrightarrow L^p(w^p), \] where $w=w_1w_2$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}<\frac…
We prove norm estimates for multilinear fractional integrals acting on weighted and variable Hardy spaces. In the weighted case we develop ideas we used for multilinear singular integrals [7]. For the variable exponent case, a key element of our proof is a new multilinear, off-diagonal version of the Rubio de Francia extrapolation theorem.
This note is devoted to establishing two-weight estimates for commutators of singular integrals. We combine multilinearity with product spaces. A new type of two-weight extrapolation result is used to yield the quasi-Banach range of estimates.
We prove genuinely multilinear weighted estimates for singular integrals in product spaces. The estimates complete the qualitative weighted theory in this setting. Such estimates were previously known only in the one-parameter situation. Extrapolation gives powerful applications -- for example, a free access to mixed-norm estimates in the full range of exponents.
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 establish the boundedness of the multilinear Calderón-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton (Collect. Math. 2001) and recent work by the third author, Grafakos, Nakamura, and Sawano. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multil…
We prove a sparse bound for the $m$-sublinear form associated to vector-valued maximal functions of Fefferman-Stein type. As a consequence, we show that the sparse bounds of multisublinear operators are preserved via $\ell^r$-valued extension. This observation is in turn used to deduce vector-valued, multilinear weighted norm inequalities for multisublinear operators obeying sparse bounds, which are out of reach for the extrapolation theory recently developed by Cruz-Uribe and Martell. As an example, vector-valued multilinear weighted inequalities for bilinear Hilbert transforms are deduced fr…
In this article, we address pointwise sparse domination for multilinear Calderón-Zygmund operators on upper doubling, geometrically doubling metric measure spaces. As a consequence, we have obtained sharp quantitative weighted estimates for multilinear Calderón-Zygmund operators.
