Related papers: $L^p$-estimates for parabolic systems with unbound…

In this paper we derive $W^{1,\infty}$ and piecewise $C^{1,\alpha}$ estimates for solutions, and their $t-$derivatives, of divergence form parabolic systems with coefficients piecewise H\"older continuous in space variables $x$ and smooth…

We find a minimal notion of non-degeneracy for bilinear singular integral operators $T$ and identify testing conditions on the multiplying function $b$ that characterize the $L^p\times L^q\to L^r,$ $1<p,q<\infty$ and $r>\frac{1}{2},$…

In this paper, we establish Liouville-type theorems for parabolic differential inequalities with $(p,q)-$Laplacian operator on Riemannian manifolds. By a test function argument, we establish nonexistence results under suitable weighted…

In this paper, we continue the investigation on the connection between observability and inverse problems for a class of parabolic equations with unbounded first order coefficients. We prove new logarithmic stability estimates for a class…

We define a p-norm in the context of quantum random variables, measurable operator-valued functions with respect to a positive operator-valued measure. This norm leads to a operator-valued L^p space that is shown to be complete. Various…

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…

In this paper we obtain a new boundedness criterion for the maximal operator $M$ on variable exponent spaces $L^{p(\cdot)}$. It is formulated in terms of the variable exponent analogue of the well known weighted $A_{\infty}$ condition.

We consider structured optimisation problems defined in terms of the sum of a smooth and convex function, and a proper, l.s.c., convex (typically non-smooth) one in reflexive variable exponent Lebesgue spaces $L_{p(\cdot)}(\Omega)$. Due to…

We prove well posedness and stability in $\mathbf{L}^1$ for a class of mixed hyperbolic-parabolic non linear and non local equations in a bounded domain with no flow along the boundary. While the treatment of boundary conditions for the…

Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, in…

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…

Given $A,B\in M_n(\mathbb R)$, we consider the Cauchy problem for partially dissipative hyperbolic systems having the form \begin{equation*} \partial_{t}u+A\partial_{x}u+Bu=0, \end{equation*} with the aim of providing a detailed description…

We study higher-order elliptic operators on one-dimensional ramified structures (networks). We introduce a general variational framework for fourth-order operators that allows us to study features of both hyperbolic and parabolic equations…

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…

Let $G$ be a finitely generated infinite group and let $p > 1$. In this paper we make a connection between the first $L^p$-cohomology space of $G$ and $p$-harmonic functions on $G$. We also describe the elements in the first…

We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficient may depend on space and time. Pointwise convergence of the…

For weak solutions $u \in W^{m,1}(\Omega;\R^N)$ of higher order systems of the type \int_\Omega < A(x,D^m u),D^m \phi > dx = \int_\Omega < |F|^{p(x)-2}F,D^m \phi> dx, for all $\phi \in C^{\infty}_c(\Omega;\R^N), m > 1$ with variable growth…

In this paper we deal with a strongly ill-posed second-order degenerate parabolic problem in the unbounded open set $\Omega\times {\mathcal O}\subset \mathbb R^{M+N}$, related to a linear equation with unbounded coefficients, with no…

We show that noncommutative $L_p$-spaces satisfy the axioms of the (nonunital) operator system with a dominating constant $2^{1 \over p}$. Therefore, noncommutative $L_p$-spaces can be embedded into $B(H)$ $2^{1 \over p}$-completely…

In this short note, we prove that the restriction conjecture for the (hyperbolic) paraboloid in $\mathbb{R}^d$ implies the $l^p$-decoupling theorem for the (hyperbolic) paraboloid in $\mathbb{R}^{2d-1}$. In particular, this gives a simple…