Hypoelliptic heat kernel inequalities on the Heisenberg group

We study the existence of “L^p-type” gradient estimates for the heat kernel of the natural hypoelliptic “Laplacian” on the real three-dimensional Heisenberg Lie group. Using Malliavin calculus methods, we verify that these estimates hold in the case p>1. The gradient estimate for p=2 implies a corresponding Poincaré inequality for the heat kernel. The gradient estimate for p=1 is still open; if proved, this estimate would imply a logarithmic Sobolev inequality for the heat kernel.     

Heat kernel analysis on semi-infinite Lie groups

This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the Lp norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds in this setting.     

Characterization of Carleson measures by the Hausdorff-Young property

It is shown that the Laplace transform of an L ^p (1 < p ≤ 2) function defined on the positive semiaxis satisfies the Hausdorff-Young type inequality with a positive weight in the right complex half-plane if and only if the weight is a Carleson measure. In addition, Carleson’s weighted L ^p inequality for the harmonic extension is given with a numeric constant.     

Kato class measures of symmetric Markov processes under heat kernel estimates

We establish the coincidence of two classes of Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of Kato class measures is defined by way of the heat kernel, another is defined in terms of the Green kernel depending on some exponents related to the heat kernel estimates. We also prove that pth integrable functions on balls with radius 1 having a uniformity of its norm with respect to centers are of Kato class if p is greater than a constant related to the estimate under the same conditions. These are complete extensions of some results for the Brownian motion on Euclidean space by Aizenman and Simon. Our result can be applicable to many examples, for instance, symmetric (relativistic) stable processes, jump processes on d-sets, Brownian motions on Riemannian manifolds, diffusions on fractals and so on.     

Littlewood–Paley and Spectral Multipliers on Weighted L p Spaces

Let L be a non-negative self-adjoint operator acting on L ²(X), where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e ^?tL whose kernel p _t (x,y) has a Gaussian upper bound but there is no assumption on the regularity in variables x and y. In this article we study weighted L ^p -norm inequalities for spectral multipliers of L. We show that a weighted Hörmander-type spectral multiplier theorem follows from weighted L ^p -norm inequalities for the Lusin and Littlewood–Paley functions, Gaussian heat kernel bounds, and appropriate L ² estimates of the kernels of the spectral multipliers.     

Weighted Estimates for Singular Integral Operators Satisfying Hörmander’s Conditions of Young Type

The following open question was implicit in the literature: Are there singular integrals whose kernels satisfy the L^r-Hörmander condition for any r > 1 but not the L^∞-Hörmander condition? We prove that the one-sided discrete square function, studied in ergodic theory, is an example of a vector-valued singular integral whose kernel satisfies the L^r-Hörmander condition for any r > 1 but not the L^∞-Hörmander condition. For a Young function A we introduce the notion of L^A-Hörmander. We prove that if an operator satisfies this condition, then one can dominate the L^p(w) norm of the operator by the L^p(w) norm of a maximal function associated to the complementary function of A, for any weight w in the A_∞ class and 0 < p < ∞. We use this result to prove that, for the one-sided discrete square function, one can dominate the L^p(w) norm of the operator by the L^p(w) norm of an iterate of the one-sided Hardy-Littlewood Maximal Operator, for any w in the A⁺ _∞ class.     

Weighted L p estimates for the area integral associated to self-adjoint operators

This article is concerned with some weighted norm inequalities for the so-called horizontal (i.e., involving time derivatives) area integrals associated to a non-negative self-adjoint operator satisfying a pointwise Gaussian estimate for its heat kernel, as well as the corresponding vertical (i.e., involving space derivatives) area integrals associated to a non-negative self-adjoint operator satisfying in addition a pointwise upper bounds for the gradient of the heat kernel. As applications, we obtain sharp estimates for the operator norm of the area integrals on ${L^p(\mathbb{R}^N)}$ as p becomes large, and the growth of the A _p constant on estimates of the area integrals on the weighted L ^p spaces.     

Finite-dimensional subspaces of <Emphasis Type="Italic">L</Emphasis><Subscript>p</Subscript> with Lipschitz metric projection

We prove that the metric projection onto a finite-dimensional subspace Y ? L _p, p ∈ (1, 2) ∪ (2, ∞), satisfies the Lipschitz condition if and only if every function in Y is supported on finitely many atoms. We estimate the Lipschitz constant of such a projection for the case in which the subspace is one-dimensional.     

Quasi-antichain Chermak–Delgado lattices of finite groups

The Chermak–Delgado lattice of a finite group is a dual, modular sublattice of the subgroup lattice of the group. This paper considers groups with a quasi-antichain interval in the Chermak–Delgado lattice, ultimately proving that if there is a quasi-antichain interval between subgroups L and H with L ≤ H then there exists a prime p such that H/L is an elementary abelian p-group and the number of atoms in the quasi-antichain is one more than a power of p. In the case where the Chermak–Delgado lattice of the entire group is a quasi-antichain, the relationship between the number of abelian atoms and the prime p is examined; additionally, several examples of groups with a quasi-antichain Chermak–Delgado lattice are constructed.     

Heat kernel analysis on infinite-dimensional Heisenberg groups

We introduce a class of non-commutative Heisenberg-like infinite-dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, {νt}_t>0, are also studied. We show that these heat kernel measures admit: (1) Gaussian like upper bounds, (2) Cameron-Martin type quasi-invariance results, (3) good Lp-bounds on the corresponding Radon-Nikodym derivatives, (4) integration by parts formulas, and (5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.     

A local solvability result for operators with characteristics having odd order multiplicity

It is shown that an operator L with the canonical form L = D_t^{2p + 1} + a(t, D_x) is locally solvable if and only if a(t, D_x) satisfies a Nirenberg-Treves-type condition.     

Estimates for the maximal multilinear singular integral operators

In this paper, some mapping properties are considered for the maximal multilinear singular integral operator whose kernel satisfies certain minimum regularity condition. It is proved that certain uniform local estimate for doubly truncated operators implies the Lp(Rn) (1 < p < ∞) boundedness and a weak type L log L estimate for the corresponding maximal operator.     

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L²(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L² estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.     

Noyaux de la chaleur et estimations mixtes Lp — Lq optimales: le cas non autonome

Consider the non-autonomous initial value problem u′(t) + A(t)u(t) = f(t), u(0) = 0, where −A(t) is for each t [0,T], the generator of a bounded analytic semigroup on L²(Ω). We prove maximal L^p — L^q a priori estimates for the solution of the above equation provided the semigroups T_t are associated to kernels which satisfies an upper Gaussian bound and A(t), t [0, T] fulfills a Acquistapace-Terreni commutator condition.     

Large time behavior and L−L estimate of solutions of 2-dimensional nonlinear damped wave equations

We show the asymptotic behavior of the solution to the Cauchy problem of the two-dimensional damped wave equation. It is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the L^p−L^q type estimate. This is a two-dimensional generalization of the three-dimensional result due to Nishihara (Math. Z. 244 (2003) 631). To show this, we use the Fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations. By using the L^p−L^q estimate, we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity |u|^αu. Our result covers the whole super critical case α>1, where the α=1 is well known as the Fujita exponent when n=2.     

On the Lie algebra of skew-symmetric elements of an enveloping algebra

Let L be a restricted Lie algebra over a field of characteristic p>2 and denote by u(L) its restricted enveloping algebra. We establish when the Lie algebra of skew-symmetric elements of u(L) under the principal involution is solvable, nilpotent, or satisfies an Engel condition.     

L p -continuity for Calderón-Zygmund operator

Given a Calderón-Zygmund (C-Z for short) operatorT, which satisfies Hörmander condition, we prove that: ifT maps all the characteristic atoms toWL ¹, thenT is continuous fromL ^p toL ^p (1 <p < ∞). So the study of strong continuity on arbitrary function inL ^p has been changed into the study of weak continuity on characteristic functions.     

On Jodeit’s multiplier extension theorems

Let Δ(x) = max {1 - ¦x¦, 0} for all x ∈ ?, and let ξ_[0,1) be the characteristic function of the interval 0 ≤x < 1. Two seminal theorems of M. Jodeit assert that A and ξ_[0,1) act as summability kernels convertingp-multipliers for Fourier series to multipliers forL ^P (?). The summability process corresponding to Δ extendsL ^P (T)-multipliers from ? to ? by linearity over the intervals [n, n + 1],n ∈ ?, when 1 ≤p < ∞, while the summability process corresponding to ξ_[0,1) extends L^P(T)-multipliers by constancy on the intervals [n, n + 1),n ∈ ?, when 1 <p < ∞. We describe how both these results have the following complete generalization: for 1 ≤p < ∞, an arbitrary compactly supported multiplier forL ^P (?) will act as a summability kernel forL ^P (T)-multipliers, transferring maximal estimates from L^P(T) to L^P(?). In particular, specialization of this maximal theorem to Jodeit’s summability kernel ξ_{[0, 1)} provides a quick structural way to recover the fact that the maximal partial sum operator on L^P(?), 1 <p < ∞, inherits strong type (p,p)-boundedness from the Carleson-Hunt Theorem for Fourier series. Another result of Jodeit treats summability kernels lacking compact support, and we show that this aspect of multiplier theory sets up a lively interplay with entire functions of exponential type and sampling methods for band limited distributions.     

Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey

Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator L^Jf(x) = ∫(f(x) ? f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (?L^J, D) acts in L²(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel p^J (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (?L^J, D) extends in L²(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel p^J(t, x, y), and the function p^J(t, x, y) is uniformly comparable in t, x, y to the function p^J(t, x, y), the heat kernel related to the operator (?L^J,D).     

Characterization of the shape stability for nonlinear elliptic problems

We characterize all geometric perturbations of an open set, for which the solution of a nonlinear elliptic PDE of p-Laplacian type with Dirichlet boundary condition is stable in the L^∞-norm. The necessary and sufficient conditions are jointly expressed by a geometric property associated to the γp-convergence.If the dimension N of the space satisfies N−1<p?N and if the number of the connected components of the complements of the moving domains are uniformly bounded, a simple characterization of the uniform convergence can be derived in a purely geometric frame, in terms of the Hausdorff complementary convergence. Several examples are presented.