Recent zbMATH articles in MSC 42B15https://zbmath.org/atom/cc/42B152021-06-15T18:09:00+00:00WerkzeugWeighted estimates for maximal functions associated with finite type curves in \(\mathbb{R}^2\).https://zbmath.org/1460.420282021-06-15T18:09:00+00:00"Manna, Ramesh"https://zbmath.org/authors/?q=ai:manna.ramesh"Shrivastava, Saurabh"https://zbmath.org/authors/?q=ai:shrivastava.saurabh"Shuin, Kalachand"https://zbmath.org/authors/?q=ai:shuin.kalachandThe authors obtain weighted boundedness on Lebesgue spaces \(L^p (w)\) for the maximal functions \(\mathcal{M}_{lac}\) and \(\mathcal{M}\) associated with a finite type curve \(\mathcal{C}\) with type \(m\) on the plane (Theorem 2.1). The exponent \(p\) is within the expected range, and \(w\) belongs to a specific class of Muckenhoupt weights and satisfies a precise reverse Hölder condition. From this they characterize the power weights for which the corresponding weighted \(L^p\) boundedness of \(\mathcal{M}_{lac}\) and \(\mathcal{M}\) hold (Theorem 2.2).
They also obtain a suitable sparse dominations for both \(\mathcal{M}_{lac}\) and \(\mathcal{M}\) (Theorem 2.4). This result is key in the proof of Theorem 2.1.
Additionally, the article contains several consequences of Theorem 2.1 (Section 5) and some further generalisation and applications of the main results (Section 7).
Reviewer: Guillermo Flores (Córdoba)Weighted estimates for bilinear Fourier multiplier operators with multiple weights.https://zbmath.org/1460.420112021-06-15T18:09:00+00:00"Hu, Guoen"https://zbmath.org/authors/?q=ai:hu.guoen"Wang, Zhidan"https://zbmath.org/authors/?q=ai:wang.zhidan"Xue, Qingying"https://zbmath.org/authors/?q=ai:xue.qingying"Yabuta, Kôzô"https://zbmath.org/authors/?q=ai:yabuta.kozoSummary: In the weighted theory of multilinear operators, the weights class which usually has been considered is the product of \(A_p\) weights. However, it is known that \(\prod_{k=1}^2A_{p_k}(\mathbb{R}^n)\varsubsetneq A_{\vec{p}}(\mathbb{R}^{2n})\), and \(\vec{w}=(w_1,\,w_2)\in A_{\vec{p}}(\mathbb{R}^{2n})\) does not imply that \(w_k\in L^1_{\text{loc}}(\mathbb{R}^n)\) for \(k=1,\,2\). Therefore, it is very interesting to study the weighted theory of multilinear operators with the weights in \(A_{\vec{p}}(\mathbb{R}^{2n})\). In this paper, we consider the weights class \(A_{\vec{p}/\vec{r}}(\mathbb{R}^{2n})\), which is more general than \(A_{\vec{p}}(\mathbb{R}^{2n})\). If \(\vec{w}=(w_1,\,w_2)\in A_{\vec{p}/\vec{r}}(\mathbb{R}^{2n})\), we show that the bilinear Fourier multiplier operator \(T_{\sigma}\) is bounded from \(L^{p_1}(w_1)\times L^{p_2}(w_2)\) to \(L^p(\nu_{\vec{w}})\) when the symbol \(\sigma\) satisfies the Sobolev regularity that \(\sup_{\kappa\in\mathbb{Z}}\Vert \sigma_k\Vert_{W^{s_1,s_2}(\mathbb{R}^{2n})}<\infty\) with \(s_1,s_2\in (\frac{n}{2},\,n]\).