Related papers: Uniform Sobolev inequalities for second order non-…

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If ${\mathcal L}_0=\mbox{div}…

Let $n \geq 2$, let $\Omega \subset \mathbf{R}^n$ be a bounded domain with smooth boundary, and let $1 \leq p \leq 2$. We prove a reverse-Holder inequality for functions $u$ realizing the best constant in the Sobolev inequality, that is…

We study the Sobolev regularity of $p$-harmonic functions. We show that $|Du|^{\frac{p-2+s}{2}}Du$ belongs to the Sobolev space $W^{1,2}_{loc}$, $s>-1-\frac{p-1}{n-1}$, for any $p$-harmonic function $u$. The proof is based on an elementary…

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal{A}_{D,\varepsilon}$ with the…

We prove a Harnack inequality for distributional solutions to a type of degenerate elliptic PDEs in $N$ dimensions. The differential operators in question are related to the Kolmogorov operator, made up of the Laplacian in the last $N-1$…

In this work, we have proved a version of the Hardy-Littlewood-Sobolev inequality for variable exponents. After we use the variational method to establish the existence of solution for a class of Choquard equations involving the…

This paper comprises two parts. In the first, we study $L^p$ to $L^q$ bounds for spectral multipliers and Bochner-Riesz means with negative index in the general setting of abstract self-adjoint operators. In the second we obtain the uniform…

This paper is devoted to establish a class of sharp Sobolev inequalities on the unit complex sphere as follows: 1) Case $0<d<Q=2n+2$: for any $f\in C^\infty$ and $2\leq q \leq \frac{2Q}{Q-d}$, \begin{equation*} \|f\|_q^2\leq…

This paper is concerned with establishing uniform weighted $L^p$-$L^q$ estimates for a class of operators generalizing both Radon-like operators and sublevel set operators. Such estimates are shown to hold under general circumstances…

We consider a class of nonvariational degenerate elliptic operators of the kind \[ Lu=\sum_{i,j=1}^{m}a_{ij}\left( x\right) X_{i}X_{j}u \] where $\left\{ a_{ij}\left( x\right) \right\} _{i,j=1}^{m}$ is a symmetric uniformly positive matrix…

In this article we deal with different forms of the unique continuation property for second order elliptic equations with nonlinear potentials of sublinear growth. Under suitable regularity assumptions, we prove the weak and the strong…

We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift \[-u_{t}+Lu=-u_{t}+\sum_{ij}a_{ij}D_{ij}u+\sum b_{i}D_{i}u=0\,(\geq0,\,\leq0)\] in some domain $\Omega\subset…

In this work, we study the existence of weak solution to the following quasi linear elliptic problem involving the fractional $p$-Laplacian operator, a Hardy potential and multiple critical Sobolev nonlinearities with singularities,…

In this paper we consider nonlinear elliptic PDEs of the type $$-\Delta_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }\Omega,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of…

We prove uniform Sobolev estimates for the resolvent of Schr\"odinger operators with large scaling-critical potentials without any repulsive condition. As applications, global-in-time Strichartz estimates including some non-admissible…

We consider elliptic differential operators on either the entire Euclidean space $\mathbb{R}^d$ or on subsets consisting of a cube $\Lambda_L$ of integer length $L$. For eigenfunctions of the operator, and more general solutions of elliptic…

Inspired by the work of Cossetti and D'Arca [CD25], we show that the general weighted $L^{p}$-Hardy type inequalities [CD25, Theorems 1.1 and 1.2] and the corresponding identities hold for all $1<p<\infty$, thus extending their results…

Let $0<\alpha<d$ and $1\leq p<d/\alpha$. We present a proof that for all $f\in W^{1,p}(\mathbb{R}^d)$ both the centered and the uncentered Hardy-Littlewood fractional maximal operator $\mathcal M_\alpha f$ are weakly differentiable and $…

We study both divergence and non-divergence form parabolic and elliptic equations in the half space $\{x_d>0\}$ whose coefficients are the product of $x_d^\alpha$ and uniformly nondegenerate bounded measurable matrix-valued functions, where…

In this paper, we establish a fundamental inequality for fourth order partial differential operator $\cal P=\alpha\partial_s+\beta\partial_{ss}+\Delta^2$ ($\alpha, \beta\in\mathbb{R}$) with an abstract exponential-type weight function. Such…