Related papers: Sparse bounds for the bilinear spherical maximal f…

For a given set of dilations $E\subset [1,2]$, Lebesgue space mapping properties of the spherical maximal operator with dilations restricted to $E$ are studied when acting on radial functions. In higher dimensions, the type set only depends…

For a class of sparse operators including majorants of singular integral, square function, and fractional integral operators in a uniform manner, we prove off-diagonal two-weight estimates of mixed type in the two-weight and…

In this paper we study maximal directional singular integral operators in $ \mathbb{R}^n $ given by a H\"ormander--Mihlin multiplier on an $ (n-1)$-dimensional subspace and acting trivially in the perpendicular direction. The subspace is…

The boundedness from $H^p \times L^2$ to $L^r$, $1/p+1/2=1/r$, and from $H^p \times L^{\infty}$ to $L^p$ of bilinear pseudo-differential operators is proved under the assumption that their symbols are in the bilinear H\"ormander class…

We introduce a notion of maximal potentials and we prove that they form bounded operators from $L^p$ to the homogeneous Sobolev space $\dot{W}^{1,p}$ for all $n/(n-1)<p<n$. We apply this result to the problem of boundedness of the spherical…

We give new lower bounds for $L^p$ estimates of the Schr\"odinger maximal function by generalizing an example of Bourgain.

The paper is devoted to two-weight estimates for the fractional maximal operators $\mathcal{M}^\alpha$ on general probability spaces equipped with a tree-like structure. For given $1<p\leq q<\infty$, we study the sharp universal upper bound…

We prove $L^p$ estimates for the shifted bilinear Hilbert transform, with a polylogarithmic bound in the size of the shift. As applications, we obtain $r$-variation estimates for bilinear ergodic averages in the sharp range $r > 2$, a sharp…

We establish general weighted $L^2$ inequalities for pseudodifferential operators associated to the H\"ormander symbol classes $S^m_{\rho,\delta}$. Such inequalities allow to control these operators by fractional "non-tangential" maximal…

We prove L^p estimates for a two-dimensional bilinear operator of paraproduct type. This result answers a question posed by Demeter and Thiele in [3].

We establish some new $L^p$-improving bounds for the $k$-simplex averaging operators $S^k$ that hold in dimensions $d \geq k$. As a consequence of these $L^p$-improving bounds we obtain nontrivial bounds $S^k\colon L^{p_1}\times\cdots\times…

Maximal angular operator sends a function defined in a sector of the complex plane to a Maximal angular operator sends a function defined in a sector of the complex plane with vertex at 0 to the function of modulus obtained by maximizing…

We provide quantitative weighted estimates for the $L^p(w)$ norm of a maximal operator associated to cube skeletons in $\mathbb{R}^n$. The method of proof differs from the usual in the area of weighted inequalities since there are no…

We establish some weighted $L^2$ estimates for the Fourier extension operator in $\mathbb{R}^2$ and discuss several applications to $L^p$ problems. These include estimates for the maximal Schr\"odinger operator and the maximal extension…

Assuming the bilinear reverse Holder's condition, we character weighted inequalities for the bilinear maximal operator on filtered measure spaces. We also obtain Hytonen-Perez type weighted estimates for the bilinear maximal operator. Our…

In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy-Littlewood maximal operator associated to certain families of doubling, radial decreasing…

In this article we consider a modification of the Stein's spherical maximal operator of complex order $\alpha$ on ${\mathbb R^n}$: $$ {\mathfrak M}^\alpha_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over \Gamma(\alpha) } \int_{|y|\leq…

We proof pointwise bounds for rough Fourier integral operators by the $L^p$ Hardy-Littlewood maximal function. We assume the Fourier integral operators have amplitudes in $L^\infty S^m_\rho$ and phases $\varphi$ such that $\varphi(x,\xi) -…

Consider spherical means on the Heisenberg group with a codimension two incidence relation, and associated spherical local maximal functions $M_Ef$ where the dilations are restricted to a set $E$. We prove $L^p\to L^q$ estimates for these…

We extend the main result of Harrow, Kolla, and Schulman -- the existence of dimension-free $L^2$-bounds for the spherical maximal function in the hypercube -- to all $L^p, p > 1$. Our approach is motivated by the spectral technique…