Families Of Pseudodifferential Operators Research Articles

Some Results of Cubic Family of Pseudo Differential Operators

Some Results of Cubic Family of Pseudo Differential Operators

Pseudodifferential Operators and Markov Processes on Adèles

In this article a class of Markov processes on the ring of finite adeles of the rational numbers are introduced. A class of non-Archimedean metrics on \(\mathbb{A}_{f}\) are chosen in order to describe this ring as a general polyadic ring and to introduce a family of pseudodifferential operators and parabolic-type equations on \({L^2}(\mathbb{A}_{f})\). The fundamental solutions of these parabolic equations determine transition functions of time and space homogeneous Markov processes on \(\mathbb{A}_{f}\) which are invariant under multiplication by units. Considering the infinite place ℝ, we extend these results to the complete ring of adeles.

Boundary Harnack principle for $Δ+ Δ^{𝛼/2}$

For d � 1 and � 2 (0,2), consider the family of pseudo differential operators f �+b� �/2 ;b 2 (0,1)g on R d that evolves continuously fromto � + � �/2 . In this paper, we establish a boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to �+b� �/2 (or equivalently, the sum of a Brownian motion and an independent symmetric�-stable process with constant multipleb 1/� ) inC 1,1 open sets. Here a uniform BHP means that the comparing constant in the BHP is independent of b 2 (0,1). Along the way, a Carleson type estimate is established for nonnegative functions which are harmonic with respect to � +b� �/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.

Heat kernel estimates for Δ+Δ α /2 in C 1, 1 open sets

We consider a family of pseudo differential operators {Δ+aα Δα/2; a∈(0, 1]} on ℝd for every d ⩾ 1 that evolves continuously from Δ to Δ+Δα/2, where α∈(0, 2). It gives rise to a family of Lévy processes {Xa, a∈(0, 1]} in ℝd, where Xa is the sum of a Brownian motion and an independent symmetric α-stable process with weight a. We establish sharp two-sided estimates for the heat kernel of Δ + aα Δα/2 with zero exterior condition in a family of open subsets, including bounded C1, 1 (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric α-stable process with weight a in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in a in the sense that the constants in the estimates are independent of a∈(0, 1] so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking a → 0. Integrating the heat kernel estimates in time t, we recover the two-sided sharp uniform Green function estimates of Xa in bounded C1, 1 open sets in ℝd, which were recently established in (Z.-Q. Chen, P. Kim, R. Song and Z. Vondracek, ‘Sharp Green function estimates for Δ+Δα/2 in C1, 1 open sets and their applications’, Illinois J. Math., to appear) using a completely different approach.

Sharp Green function estimates for $\Delta+ \Delta^{\alpha /2}$ in $C^{1,1}$ open sets and their applications

We consider a family of pseudo differential operators $\{\Delta+ a^\alpha\Delta^{\alpha/2}$; $a\in[0, 1]\}$ on ${\mathbb R}^d$ that evolves continuously from $\Delta$ to $\Delta+ \Delta^{\alpha/2}$, where $d\geq1$ and $\alpha\in(0, 2)$. It gives rise to a family of Levy processes $\{X^a, a\in[0, 1]\}$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$. Using a recently obtained uniform boundary Harnack principle with explicit decay rate, we establish sharp bounds for the Green function of the process $X^a$ killed upon exiting a bounded $C^{1,1}$ open set $D\subset{\mathbb R}^d$. Our estimates are uniform in $a\in(0, 1]$ and taking $a\to0$ recovers the Green function estimates for Brownian motion in $D$. As a consequence of the Green function estimates for $X^a$ in $D$, we identify both the Martin boundary and the minimal Martin boundary of $D$ with respect to $X^a$ with its Euclidean boundary. Finally, sharp Green function estimates are derived for certain Levy processes which can be obtained as perturbations of $X^a$.

Dirichlet heat kernel estimates for $\Delta^{\alpha/2}+ \Delta^{\beta/2}$

For $d\geq1$ and $0<\beta<\alpha<2$, consider a family of pseudo differential operators $\{\Delta^{\alpha} + a^\beta \Delta^{\beta/2}$; $a \in[0, 1]\}$ on $\mathbb{R}^d$ that evolves continuously from $\Delta^{\alpha/2}$ to $ \Delta^{\alpha/2}+ \Delta^{\beta/2}$. It gives arise to a family of Levy processes $\{X^a, a\in[0, 1]\}$ on $\mathbb{R}^d$, where each $X^a$ is the independent sum of a symmetric $\alpha$-stable process and a symmetric $\beta$-stable process with weight $a$. For any $C^{1,1}$ open set $D\subset\mathbb{R}^d$, we establish explicit sharp two-sided estimates, which are uniform in $a\in(0, 1]$, for the transition density function of the subprocess $X^{a, D}$ of $X^a$ killed upon leaving the open set~$D$. The infinitesimal generator of $X^{a, D}$ is the nonlocal operator $\Delta^{\alpha} + a^\beta\Delta^{\beta/2}$ with zero exterior condition on $D^c$. As consequences of these sharp heat kernel estimates, we obtain uniform sharp Green function estimates for $X^{a, D}$ and uniform boundary Harnack principle for $X^a$ in $D$ with explicit decay rate.

Pseudodifferential p-adic vector fields and pseudodifferentiation of a composite p-adic function

We discuss transformation of p-adic pseudodifferential operators (in the one-dimensional and multidimensional cases) with respect to p-adic maps which correspond to automorphisms of the tree of balls in the corresponding p-adic spaces. In the dimension one we find a rule of transformation for pseudodifferential operators. In particular we find the formula of pseudodifferentiation of a composite function with respect to the Vladimirov p-adic fractional operator. We describe the frame of wavelets for the group of parabolic automorphisms of the tree T (O p ) of balls in O p . In many dimensions we introduce the group of mod p-affine transformations, the family of pseudodifferential operators corresponding to pseudodifferentiation along vector fields on the tree T (O p ) and obtain a rule of transformation of the introduced pseudodifferential operators with respect to mod p-affine transformations.

Multidimensional ultrametric pseudodifferential equations

We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and the space D′0(X) of generalized functions on a multidimensional ultrametric space. We also consider some family of pseudodifferential operators on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of ultrametric characteristics are introduced. We prove an existence theorem and describe all solutions for the Cauchy problem (an analog of the Kovalevskaya theorem).

Boundary value problems on manifolds with fibered boundary

AbstractWe define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah–Patodi–Singer problems in subspaces (it contains both as special cases). The boundary conditions in this theory are taken as elements of the C *‐algebra generated by pseudodifferential operators and families of pseudodifferential operators in the fibers. We prove the Fredholm property for elliptic boundary value problems and compute a topological obstruction (similar to Atiyah–Bott obstruction) to the existence of elliptic boundary conditions for a given elliptic operator. Geometric operators with trivial and nontrivial obstruction are given. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A family of pseudo-differential operators and a stability theorem for the Friedrichs scheme

A family of pseudo-differential operators and a stability theorem for the Friedrichs scheme