Related papers: Optimal $L^p$ regularity for $\bar\partial$ on the…

In this paper, optimal $L^p-L^q$ estimates are obtained for operators which average functions over polynomial submanifolds, generalizing the $k$-plane transform. An important advance over previous work is that full $L^p-L^q$ estimates are…

In this article we prove $L^p$ estimates for resolvents of Laplace-Beltrami operators on compact Riemannian manifolds, generalizing results of Kenig, Ruiz and Sogge in the Euclidean case and Shen for the torus. We follow Sogge and construct…

We give two weighted norm estimates for higher order commutator of classical operators such as singular integral and fractional type operators, between weighted $L^p$ and certain spaces that include Lipschitz, BMO and Morrey spaces. We also…

We obtain large time decay estimates on weighted $L^p$ spaces for solutions to the wave equation with real-valued potential $V(x)=O(|x|^{-2-a})$, $a>0$, for $|x|>1$.

This paper gives necessary conditions and slightly stronger sufficient conditions for a holomorphic function to be the Segal-Bargmann transform of a function in L^p(R^d) with respect to a Gaussian measure. The proof relies on a family of…

The Boltzmann equation without an angular cutoff in a three-dimensional periodic domain is considered. The global-in-time existence of solutions in a function space $ W_k^{\zeta,p}L^\infty_TL^2_v $ with $p>1$ and $\zeta>3(1-\frac{1}{p})$ is…

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $\partial_tu - \mbox{div}(A\nabla u)=0$ on the domain $\mathbb R^{n+1}_+\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the…

The aim of this paper is to propose weak assumptions to prove maximal L^q regularity for Cauchy problem: du/dt - Lu(t)=f(t). Mainly we only require "off-diagonal" estimates on the real semigroup (e^{tL})_{t>0} to obtain maximal L^q…

We present a short, direct proof of the uniform convexity of L^p spaces for 1<p<\infty.

In the paper, we develop an $L^1_k\cap L^p_k$ approach to construct global solutions to the Cauchy problem on the non-cutoff Boltzmann equation near equilibrium in $\mathbb{R}^3$. In particular, only smallness of…

We obtain uniform estimates for the canonical solution to $\bar\partial u=f$ on the Cartesian product of bounded planar domains with $C^2$ boundaries, when $f$ is continuous up to the boundary. This generalizes Landucci's result for the…

In this paper, positive solutions to the Laplace equation with 1-dimensional circular singularities are investigated. First, we establish $L^p$ integrability estimates for such solutions $u$ near the singularities, in comparison with…

We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called…

Let $A = -{\rm div} \,a(\cdot) \nabla$ be a second order divergence form elliptic operator on $\R^n$ with bounded measurable real-valued coefficients and let $W$ be a cylindrical Brownian motion in a Hilbert space $H$. Our main result…

The global existence issue in critical regularity spaces for the full Navier-Stokes equationssatisfied by compressible viscous and heat-conductive gases has been first addressed in \cite{D2}, then recently extended to the general $L^p$…

Let $G$ be the group $\mathbb{R}_+\ltimes \mathbb{R}^n$ endowed with Riemannian symmetric space metric $d$ and the right Haar measure $\mathrm{d} \rho$ which is of $ax+b$ type, and $L$ be the positive definite distinguished left invariant…

It is shown that if a point $x_0$ admits a bounded point derivation on $R^p(X)$, the closure of rational function with poles off $X$ in the $L^p(dA)$ norm, for $p >2$, then there is an approximate derivative at $x_0$. A similar result is…

We consider fractional parabolic equations with variable coefficients and establish maximal $L_{q}$-regularity in Bessel potential spaces of arbitrary nonnegative order. As an application, we show higher order regularity and instantaneous…

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously,…

We study estimates for Hardy space norms of analytic projections. We first find a sufficient condition for the Bergman projection of a function in the unit disc to belong to the Hardy space $H^p$ for $1 < p < \infty$. We apply the result to…