Let $f$ be a non-constant complex-valued analytic function defined on a connected, open set containing the $L^p$-spectrum of the Laplacian $\mathcal L$ on a homogeneous tree. In this paper we give a necessary and sufficient condition for the semigroup $T(t)=e^{tf(\mathcal{L})}$ to be chaotic on $L^{p}$-spaces. We also study the chaotic dynamics of the semigroup $T(t)=e^{t(a\mathcal{L}+b)}$ separately and obtain the sharp range of $b$ for which $T(t)$ is chaotic on $L^{p}$-spaces. It includes some of the important semigroups, such as the heat semigroup and the Schrödinger semigroup.
We describe a procedure to introduce Sobolev spaces and the semigroup generated by the fractional Dirichlet Laplacian on an arbitrary domain of $\R^d$. In particular, the well-definedness of the spaces of both non-homogeneous and homogeneous type together with their duality properties, embeddings, and Gagliardo-Nirenberg inequalities will be discussed. We also show the continuity and the smoothing property of the semigroup.
We show that for graph Laplacians $Δ_G$ on a connected locally finite simplicial undirected graph $G$ with countable infinite vertex set $V$ none of the operators $α\,\mathrm{Id}+βΔ_G, α,β\in\mathbb{K},β\ne 0$, generate a strongly continuous semigroup on $\mathbb{K}^V$ when the latter is equipped with the product topology.
For $1<p\leq q$ we show that the Poisson semigroup $e^{-t\sqrt{-Δ}}$ on the $n$-sphere is hypercontractive from $L^{p}$ to $L^{q}$ in dimensions $n \leq 3$ if and only if $e^{-t\sqrt{n}} \leq \sqrt{\frac{p-1}{q-1}}$. We also show that the equivalence fails in large dimensions.
In this paper we study the behavior of some harmonic analysis operators associated with the discrete Laplacian $Δ_d$ in discrete Hardy spaces $\mathcal H^p(\mathbb Z)$. We prove that the maximal operator and the Littlewood-Paley $g$ function defined by the semigroup generated by $Δ_d$ are bounded from $\mathcal H^p(\mathbb Z)$ into $\ell^p(\mathbb Z)$, $0<p\leq 1$. Also, we establish that every $Δ_d$-spectral multiplier of Laplace transform type is a bounded operator from $\mathcal H^p(\mathbb Z)$ into itself, for every $0<p\leq 1$.
In this paper we characterize the discrete Hölder spaces by means of the heat and Poisson semigroups associated to the discrete Laplacian. These characterizations allow us to get regularity properties of fractional powers of the discrete Laplacian and the Bessel potentials along these spaces and also in the discrete Zygmund spaces in a more direct way than using the pointwise definition of the spaces. To obtain our results, it has been crucial to get boundedness properties of the heat and Poisson kernels and their derivatives in both space and time variables. We believe that these estimates ar…
In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by Minakshisundaram's ideas, we find a precise pointwise description of $(-Δ_{\mathbb{S}^{n-1}})^s u(x)$ in terms of fractional powers of the Dirichlet-to-Neumann map on the sphere. The Poisson kernel for the unit ball will be essential for this part of the analysis. On the other hand, by using the heat semigroup on the sphere, addition…
The \textit{method of semigroups} is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators $L$ and their regularity properties in related functional spaces. The approach was introduced by the author and José L.~Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian $L^s=(-Δ)^s$, $0<s<1$. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointwise for…
In this paper we describe the Euler semigroup $\{e^{-t\mathbb{E}^{*}\mathbb{E}}\}_{t>0}$ on homogeneous Lie groups, which allows us to obtain various types of the Hardy-Sobolev and Gagliardo-Nirenberg type inequalities for the Euler operator $\mathbb{E}$. Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and $|\cdot|$-radial weighted Hardy-Sobolev type inequality are established.
In this paper, we prove that it is always possible to define a realization of the Laplacian $Δ_{κ,θ}$ on $L^2(Ω)$ subject to nonlocal Robin boundary conditions with general jump measures on arbitrary open subsets of $\mathbb R^N$. This is made possible by using a capacity approach to define admissible pair of measures $(κ,θ)$ that allows the associated form $\mathcal E_{κ,θ}$ to be closable. The nonlocal Robin Laplacian $Δ_{κ,θ}$ generates a sub-Markovian $C_0-$semigroup on $L^2(Ω)$ which is not dominated by Neumann Laplacian semigroup unless the jump measure $θ$ vanishes. Finally, the converg…
We study the one-dimensional Laplace operator with point interactions on the real line identified with two copies of the half-line $[0,\infty)$. All possible boundary conditions that define generators of $C_0$-semigroups on $L^2\big([0,\infty)\big)\oplus L^2\big([0,\infty)\big)$ are characterized. Here, the Cayley transform of the boundary conditions plays an important role and using an explicit representation of the Green's functions, it allows us to study invariance properties of semigroups.
We develop a theory of eventually positive $C_0$-semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on $L^p$-spaces, the Dirichlet-to-Neuma…
We combine dyadic analysis through Haar type wavelets defined on Christ's families of generalized cubes, and Lax-Milgram theorem, in order to prove existence of Green's functions for fractional Laplacians on some function spaces of vanishing small resolution in spaces of homogeneous type.
The purpose of this paper is to establish bilinear estimates in Besov spaces generated by the Dirichlet Laplacian on a domain of Euclidian spaces. These estimates are proved by using the gradient estimates for heat semigroup together with the Bony paraproduct formula and the boundedness of spectral multipliers.
We investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain $Ω\subseteq\mathbb{R}^d$ with Lipschitz boundary $Γ$. More precisely, using form methods, we show that the associated operator on the ground space $L^2(Ω)\times L^2(Γ)$ has compact resolvent and generates a holomorphic and strongly continuous real semigroup of self-adjoint operators. Furthermore, we give a full characterization of the domain in terms of Sobolev spaces, also proving Hölder regularity of solutions, allowing classical interpretation of the boundary condition. Finally, we investigate spectrum and…
In this paper, we prove that the generator of any bounded analytic semigroup in $(θ,1)$-type real interpolation of its domain and underlying Banach space has maximal $L^1$-regularity, using a duality argument combined with the result of maximal continuous regularity. As an application, we consider maximal $L^1$-regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous $B^s_{q,1}$-type Besov spaces on domains of $\mathbb R^n$, $n\geq 2$.
Let $\mathbb T_{q+1}$ denote the homogeneous tree of degree $q+1$ with the standard graph distance $d$ and the canonical flow measure $μ$. The metric measure space $(\mathbb T_{q+1},d,μ)$ is of exponential growth. Let $\mathcal{L}$ denote the flow Laplacian, which is a probabilistic Laplacian self-adjoint on $L^2(μ)$. In this note, we prove some weighted $L^1$-estimates for the heat kernel associated with $\mathcal{L}$ and its gradient. As a consequence, we show that the first order Riesz transform associated with the flow Laplacian on $\mathbb T_{q+1}$ is bounded on $L^p(μ)$, for $p \in (1,2]…
We establish convolution inequalities for Besov spaces $B_{p,q}^s$ and Triebel--Lizorkin spaces $F_{p,q}^s$. As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces $A_{p,q}^s$, $A \in \{B,F\}$. Our results apply to a wide class of convolution semigroups including the Gauß--Weierstraß semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian $(-Δ)^m$, and we can derive various caloric smoothing estimates.
On a semi-homogeneous tree, we study the $\ell^p$-spectrum of the Laplace operator $μ_1$ (the isotropic nearest-neighbor transition operator); the known results in the much simpler setting of homogeneous trees are obtained as particular cases. The spectrum is given by the eigenvalues of spherical functions, i.e., eigenfunctions of $mu_1$ that are radial with respect to a reference vertex $v_0$ and normalized there. We show that spherical functions are boundary integrals of generalized Poisson kernels that, unlike the homogeneous setting, are not complex powers of the usual Poisson kernel. We c…
We give sharp two-sided estimates of the semigroup generated by the fractional Laplacian plus the Hardy potential on $\mathbb{R}^d$, including the case of the critical constant. We use Davies' method back-to-back with a new method of integral analysis of Duhamel's formula.
We investigate in a systematic way hypercontractivity property in Orlicz spaces for Markov semi-groups related to homogeneous and non homogeneous diffusions in $\mathbb{R}^{n}$. We provide an explicit construction of a family of Orlicz functions for which we prove that the associated hypercontractivity property is equivalent to a suitable functional inequality.
A positive correlation inequality is established for circular-invariant plurisubharmonic functions, with respect to complex Gaussian measures. The main ingredients of the proofs are the Ornstein-Uhlenbeck semigroup, and another natural semigroup associated to the Gaussian $\partial$-Laplacian.
We study $L^p$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. The main tool is the theory of heat semigroup based Besov classes in Dirichlet spaces that was introduced by the authors in previous works.
Let $G$ be a $2$-step stratified group of topological dimension $d$ and homogeneous dimension $Q$. Let $L$ be a homogeneous sub-Laplacian on $G$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in (1,\infty)$ whenever the multiplier $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$. It is known that, for several $2$-step groups and sub-Laplacians, the threshold $Q/2$ in the smoothness condition is not sharp and in many cases it is possible to push it down to $d/2$. Here we show …
We study the fractional Laplacian $(-Δ)^{σ/2}$ on the $n$-dimensional torus $\mathbb{T}^n$, $n\geq1$. First, we present a general extension problem that describes \textit{any} fractional power $L^γ$, $γ>0$, where $L$ is a general nonnegative selfadjoint operator defined in an $L^2$-space. This generalizes to all $γ>0$ and to a large class of operators the previous known results by Caffarelli and Silvestre. In particular it applies to the fractional Laplacian on the torus. The extension problem is used to prove interior and boundary Harnack's inequalities for $(-Δ)^{σ/2}$, when $0<σ<2$. We dedu…
