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Register a new userRegularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms
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We study the behaviour of linear partial differential operators with polynomial coefficients via a Wigner type transform. In particular, we obtain some results of regularity in the Schwartz space $mathcal S$ and in the space ${mathcal S}_omega$ as introduced by Bjorck for weight functions $omega$. Several examples are discussed in this new setting.
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In this paper we study regularity of partial differential equations with polynomial coefficients in non isotropic Beurling spaces of ultradifferentiable functions of global type. We study the action of transformations of Gabor and Wigner type in such spaces and we prove that a suitable representation of Wigner type allows to prove regularity for classes of operators that do not have classical hypoellipticity properties.
In this paper we characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension $2$. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree $2$. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential operators, which can be completely characterized in terms of the asymptotic behavior of the Weyl symbol. In conclusion we observe that we have obtained a new class of globally regular partial differential operators which is disjoint from the class of hypo-elliptic operators in the sense of Shubin.
For a constant coefficient partial differential operator $P(D)$ with a single characteristic direction such as the time-dependent free Schrodinger operator as well as non-degenerate parabolic differential operators like the heat operator we characterize when open subsets $X_1subseteq X_2$ of $mathbb{R}^d$ form a $P$-Runge pair. The presented condition does not require any kind of regularity of the boundaries of $X_1$ nor $X_2$. As part of our result we prove that for a large class of non-elliptic operators $P(D)$ there are smooth solutions $u$ to the equation $P(D)u=0$ on $mathbb{R}^d$ with support contained in an arbitarily narrow slab bounded by two parallel characteristic hyperplanes for $P(D)$.
We study {em $ abla$-Sobolev spaces} and {em $ abla$-differential operators} with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate free approach that uses connections (which are typically denoted $ abla$). These concepts arise naturally from Partial Differential Equations, including some that are formulated on plain Euclidean domains, such as the weighted Sobolev spaces used to study PDEs on singular domains. We prove several basic properties of the $ abla$-Sobolev spaces and of the $ abla$-differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and independence of the $ abla$-Sobolev spaces on the choices of the connection $ abla$ with respect to totally bounded perturbations. We introduce a {em Frechet finiteness condition} (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our $ abla$-Sobolev spaces and of our $ abla$-differential operators. We examine in more detail the particular case of domains in the Euclidean space, including the case of weighted Sobolev spaces. We also introduce and study the notion of a {em $ abla$-bidifferential} operator (a bilinear version of differential operators), obtaining results similar to those obtained for $ abla$-differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.
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 $u$.
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