Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBoundary values of zero solutions of hypoelliptic differential operators in ultradistribution spaces
104
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study ultradistributional boundary values of zero solutions of a hypoelliptic constant coefficient partial differential operator $P(D) = P(D_x, D_t)$ on $mathbb{R}^{d+1}$. Our work unifies and considerably extends various classical results of Komatsu and Matsuzawa about boundary values of holomorphic functions, harmonic functions and zero solutions of the heat equation in ultradistribution spaces. We also give new proofs of various results of Langenbruch about distributional boundary values of zero solutions of $P(D)$.
rate research
Read More
We consider nonnegative solutions $u:Omegalongrightarrow mathbb{R}$ of second order hypoelliptic equations begin{equation*} mathscr{L} u(x) =sum_{i,j=1}^n partial_{x_i} left(a_{ij}(x)partial_{x_j} u(x) right) + sum_{i=1}^n b_i(x) partial_{x_i} u(x) =0, end{equation*} where $Omega$ is a bounded open subset of $mathbb{R}^{n}$ and $x$ denotes the point of $Omega$. For any fixed $x_0 in Omega$, we prove a Harnack inequality of this type $$sup_K u le C_K u(x_0)qquad forall u mbox{ s.t. } mathscr{L} u=0, ugeq 0,$$ where $K$ is any compact subset of the interior of the $mathscr{L}$-propagation set of $x_0$ and the constant $C_K$ does not depend on $u$.
In this paper we consider second order parabolic partial differential equations subject to the Dirichlet boundary condition on smooth domains. We establish weighted $L_{q}$-maximal regularity in weighted Triebel-Lizorkin spaces for such parabolic problems with inhomogeneous boundary data. The weights that we consider are power weights in time and space, and yield flexibility in the optimal regularity of the initial-boundary data, allow to avoid compatibility conditions at the boundary and provide a smoothing effect. In particular, we can treat rough inhomogeneous boundary data.
We obtain Calderon-Zygmund type estimates in generalized Morrey spaces for nonlinear equations of $p$-Laplacian type. Our result is obtained under minimal regularity assumptions both on the operator and on the domain. This result allows us to study asymptotically regular operators. As a byproduct, we obtain also generalized Holder regularity of the solutions under some minimal restrictions of the weight functions.
We prove weighted $L^p$-Liouville theorems for a class of second order hypoelliptic partial differential operators $mathcal{L}$ on Lie groups $mathbb{G}$ whose underlying manifold is $n$-dimensional space. We show that a natural weight is the right-invariant measure $check{H}$ of $mathbb{G}$. We also prove Liouville-type theorems for $C^2$ subsolutions in $L^p(mathbb{G},check{H})$. We provide examples of operators to which our results apply, jointly with an application to the uniqueness for the Cauchy problem for the evolution operator $mathcal{L}-partial_t$.
Let (X j , d j , $mu$ j), j = 0, 1,. .. , m be metric measure spaces. Given 0 < p $kappa$ $le$ $infty$ for $kappa$ = 1,. .. , m and an analytic family of multilinear operators T z : L p 1 (X 1) x $bullet$ $bullet$ $bullet$ L p m (X m) $rightarrow$ L 1 loc (X 0), for z in the complex unit strip, we prove a theorem in the spirit of Steins complex interpolation for analytic families. Analyticity and our admissibility condition are defined in the weak (integral) sense and relax the pointwise definitions given in [9]. Continuous functions with compact support are natural dense subspaces of Lebesgue spaces over metric measure spaces and we assume the operators T z are initially defined on them. Our main lemma concerns the approximation of continuous functions with compact support by similar functions that depend analytically in an auxiliary parameter z. An application of the main theorem concerning bilinear estimates for Schr{o}dinger operators on L p is included.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
