On generalized Heisenberg-type groups G ( 2 n , m , U , W ) $\mathbb{G}\left(2n,m,\mathbb{U},\mathbb{W} ight)$ , we establish uniform volume estimates for the ball associated with a large class of Carnot-Carath & eacute;odory distances. As an application, we prove weak (1, 1) O(C m n)-estimates for the corresponding centered Hardy-Littlewood maximal functions, extending the results in [C. Bi, H.-Q. Li, and Y. Zhang, "Centered Hardy-Littlewood maximal functions on H-type groups revisited," Math. Ann., vol. 391, no. 3, pp. 3765-3797, 2025]. As a by-product, we obtain uniformly volume doubling property on Heisenberg groups for a natural family of left-invariant Riemannian metrics.
Let G be a compact connected Lie group and P-e,P-a (G) = C([0, 1] -> G vertical bar gamma(0) = e, gamma(1) = a) be the pinned path space with a pinned Brownian motion measure nu(lambda,a) defined by the heat kernel p(lambda(-1) t, x, y), where lambda is a positive parameter. We consider a Witten Laplacian - L-lambda,L-D acting on functions with the Dirichlet boundary condition on a certain domain D subset of P-e,P-a(G) which includes finitely many geodesics {l(1), ..., l(N)} between e and a. nu(lambda,a) has the formal path integral expression nu(lambda,a)(d gamma ) = Z(lambda)(-1) exp (-lambda E(gamma)d gamma where E(gamma ) = 1/2 integral(1)(0) |gamma(t)|(2) dt and E is a Morse function when a is not a point of the cut-locus of e. Hence, by the analogy of finite dimensional cases, one may expect that the lowlying spectrum of -lambda(-1) L-lambda,L-D can be approximated by the spectral sets of Ornstein-Uhlenbeck type operators which approximate - lambda(-1) L-lambda,L-D in each small neighborhood of critical points {l(i)} when lambda -> infinity. However, in contrast to the finite dimensional case, the spectral sets of the approximate Ornstein-Uhlenbeck type operators contain essential spectrum. It may be difficult to analyze the behavior of the spectrum of -lambda(-1) L-lambda,L-D near the set of the essential spectrum. In this paper, we study the asymptotic behavior of the lowlying discrete spectrum of -lambda L--1(lambda,D) in the complement of the neighborhood of the set of essential spectrum of the approximate Ornstein-Uhlenbeck type operators at {l(i) } as lambda -> infinity.
Let (A, tr) be a von Neumann algebra with a faithful, normal trace tr : A ->& Copf;. For each a is an element of A, define S(lambda, epsilon) = tr[log((a-lambda)(& lowast;)(a-lambda) + E)], lambda is an element of & Copf;, epsilon > 0, so that the limit as epsilon -> 0(+) of S is the log potential of the Brown measure of a. Suppose that for a fixed lambda is an element of & Copf;, the function epsilon bar right arrow partial derivative S/partial derivative epsilon (lambda, epsilon) = tr[log((a-lambda)(& lowast;)(a-lambda) + E)(-1)] admits a real analytic extension to a neighborhood of 0 in & Ropf;. Then we will show that lambda is outside the spectrum of a. We will apply this result to several examples involving circular and elliptic elements, as well as free multiplicative Brownian motions. In most cases, we will show that the spectrum of the relevant element a coincides with the support of its Brown measure.
This paper focuses on the global-in-time existence and singularity formation of smooth solutions for the pressure-gradient system with radial symmetry. The system owns one positive and one negative wave characteristics, which result that both expansion and convergence effects exist simultaneously for any initial data. We define the rarefaction and compression characters for the system and establish the a priori estimates of solutions by using the methods of characteristic decompositions and invariant regions. It is verified that the smooth solution exists globally for rarefaction initial data with vacuum at the origin, while singularities can develop in finite time for the initial data including strong compression somewhere. Moreover, the numerical simulation results verified the analysis in this paper.
We consider the existence of nontrivial solution u is an element of H-1 (& Ropf;(N) ) for the following nonlinear Choquard equation with an almost periodic term -Delta u + u = ( I-mu & lowast; (alpha( y)F(u))) alpha(x) f (u) in & Ropf;(N), where N >= 3, mu is an element of (0,N), I-mu is the Riesz potential, F is the primitive function of f , and alpha is almost periodic. As one will see, almost periodic functions of several variables lack some important features of periodic functions and almost periodic functions on & Ropf;. This makes our problem more intriguing and challenging.
