Related papers: Bilinear maximal functions associated with degener…

$L^p$ boundedness of the circular maximal function $\mathcal M_{\mathbb{H}^1}$ on the Heisenberg group $\mathbb{H}^1$ has received considerable attentions. While the problem still remains open, $L^p$ boundedness of $\mathcal…

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb{R}) \times W^{1,q}(\mathbb{R}) \to W^{1,r}(\mathbb{R})$ with $1 <p,q < \infty$ and $r\geq 1$, boundedly and…

In this paper, for general plane curves $\gamma$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p(\mathbb{R}^2)$-boundedness of the Hilbert transforms $H^\infty_{U,\gamma}$ along the variable…

We develop subrepresentation inequalities for infinitely degenerate metrics, and obtain corresponding Poincare and Sobolev inequalities. We then derive conditions on the degenerate metric under which weak solutions to associated infinitely…

This work is devoted to studying the boundedness on Lebesgue spaces of bilinear multipliers on $\R$ whose symbol is narrowly supported around a curve (in the frequency plane). We are looking for the optimal decay rate (depending on the…

We derive sharp lower bounds for L^p-functions on the n-dimensional unit hypercube in terms of their p-th marginal moments. Such bounds are the unique solutions of a system of constrained nonlinear integral equations depending on the…

Conditional on Fourier restriction estimates for elliptic hypersurfaces, we prove optimal restriction estimates for polynomial hypersurfaces of revolution for which the defining polynomial has non-negative coefficients. In particular, we…

In this article, we aim to study decoupling inequality for a specific degenerate hypersurface in $\mathbb{R}^4$. Inspired by the work of Bourgain--Demeter and Li--Zheng, we consider the hypersurface…

In this paper, we introduce a criterion for maximal operators associated with Fourier multipliers to be bounded on $L^p(\mathbb{R}^d)$. Noteworthy examples satisfying the criterion are multipliers of the Mikhlin type or limited decay which…

In this paper, we prove \( L^p \) boundedness results for lacunary elliptic maximal operators on the Heisenberg group. Furthermore, we extend these \( L^p \) estimates from skew-symmetric matrices, which naturally arise in Heisenberg group…

In this article, we continue the study of $L^p$-boundedness of the maximal operator $\mathcal M_S$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ in 3-dimensional Euclidean space. We focus here on small…

We consider the $L^p$ mapping properties of maximal averages associated to families of curves, and thickened curves, in the plane. These include the (planar) Kakeya maximal function, the circular maximal functions of Wolff and Bourgain, and…

In this paper, we study the restriction problem for one class of hypersurfaces with vanishing curvature in $\mathbb{R}^n$ with $n$ being odd. We obtain an $L^2-L^p$ restriction estimate, which is optimal except at the endpoint. Furthermore,…

We study the $L^p$ mapping properties of the strong spherical maximal function, which is a multiparameter generalisation of Stein's spherical maximal function. We show that this operator is bounded on $L^p$ for $p > 2$ in all dimensions $n…

We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following:…

There are two parts for this paper. In the first part, we extend some results in a recent paper by Du, Guth, Li and Zhang to a more general class of phase functions. The main methods are Bourgain-Demeter's $l^2$ decoupling theorem and…

In this paper we introduce and study a bilinear spherical maximal function of product type in the spirit of bilinear Calder\'{o}n-Zygmund theory. This operator is different from the bilinear spherical maximal function considered by Geba et…

We prove old and new $L^p$ bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, uniformly in the parameter that models degeneration of the bilinear Hilbert transform. We obtain the full range of exponents that…

We establish $L^{p_1}\times\cdots\times L^{p_k}\to L^r$ and $\ell^{p_1}\times\cdots\times \ell^{p_k}\to \ell^r$ type bounds for multilinear maximal operators associated to averages over isometric copies of a given non-degenerate $k$-simplex…

We study the linearized maximal operator associated with dilates of the hyperbolic cross multiplier in dimension two. Assuming a Lipschitz condition and a lower bound on the linearizing function, we obtain $L^{p}(\mathbb{R}^{2}) \to…