Related papers: Weighted decoupling estimates and the Bochner-Ries…

We improve an $L^2\times L^2\to L^2$ estimate for a certain bilinear operator in the finite field of size $p$, where $p$ is a prime sufficiently large. Our method carefully picks the variables to apply the Cauchy-Schwarz inequality. As a…

We study $L^p$-boundedness of the Bochner-Riesz means for critical magnetic Schr\"odinger operators $\mathcal{L}_{\bf A}$ in ${\mathbb{R}^2}$, which involve the physcial Aharonov-Bohm potential. We show that for $1\leq p\leq +\infty$ and…

We study weighted $(L^p, L^q)$-boundedness properties of Riesz potentials and fractional maximal functions for the Dunkl transform. In particular, we obtain the weighted Hardy-Littlewood-Sobolev type inequality and weighted week $(L^1,…

In this paper we obtain sharp weighted estimates for solutions of the $\partial$-equation in a lineally convex domains of finite type. Precisely we obtain estimates in spaces of the form L p ({\Omega},$\delta$ $\gamma$), $\delta$ being the…

Let $R$ be the vector of Riesz transforms on $\mathbb{R}^n$, and let $\mu,\lambda \in A_p$ be two weights on $\mathbb{R}^n$, $1 < p < \infty$. The two-weight norm inequality for the commutator $[b, R] : L^p(\mathbb{R}^n;\mu) \to…

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].

Let $L=-\Delta +V$ with non-negative potential $V$ satisfying some appropriate reverse H\"older inequality. In this paper, we study the boundedness of the commutators of some singular integrals associated to $L$ such as Riesz transforms and…

We extend Loeper's $L^2$-estimate relating the electric fields to the densities for the Vlasov-Poisson system to $L^p$, with $1 < p < +\infty$, based on the Helmholtz-Weyl decomposition. This allows us to generalize both the classical…

We obtain matching direct and inverse theorems for the degree of weighted $L_p$-approximation by polynomials with the Jacobi weights $(1-x)^\alpha (1+x)^\beta$. Combined, the estimates yield a constructive characterization of various…

For weighted $L^1$ space on the unit sphere of $\RR^{d+1}$, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal…

This article focuses on Lp-estimates for the square root of elliptic systems of second order in divergence form on a bounded domain. We treat complex bounded measurable coefficients and allow for mixed Dirichlet/Neumann boundary conditions…

We study the pointwise convergence of solutions to the free Schr\"{o}dinger equation with initial data in the Bessel potential spaces $L_s^p(\mathbb{R}^n)$. We establish new sufficient regularity indices for pointwise convergence across the…

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second order equation in divergence form with discontinuous coefficient. Our concern is to estimate the solutions with explicit constants,…

We use decoupling theory to estimate the number of solutions for quadratic and cubic Parsell--Vinogradov systems in two dimensions.

In this paper, we establish a standard $L^p$-theory of solutions to one dimensional nonlinear Schr\"odinger equations with the power like nonlinearity. More precisely, we extend the following three well-known results in the $L^2$ space into…

We establish L^p bounds on L^2 normalized spectral clusters for self-adjoint elliptic Dirichlet forms with Lipschitz coefficients. In two dimensions we obtain best possible bounds for all p between $2 and infinity, up to logarithmic losses…

We study the $L^p$-convergence of Fourier expansions in terms of non-symmetric Heckman-Opdam polynomials of type $A_1$. Using kernel estimates and duality arguments, we prove that the partial sums converge in $ L^p([-\pi,\pi],dm_k)$ for…

We generalize the respective ``double recurrence'' results of Bourgain and of the second author, which established for pairs of $L^{\infty}$ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages…

We prove new $L^p$-$L^q$-estimates for solutions to elliptic differential operators with constant coefficients in $\mathbb{R}^3$. We use the estimates for the decay of the Fourier transform of particular surfaces in $\mathbb{R}^3$ with…

We prove $L_p$ estimates of solutions to a conormal derivative problem for divergence form complex-valued higher-order elliptic systems on a half space and on a Reifenberg flat domain. The leading coefficients are assumed to be merely…