Let $X_0, X_1,..., X_k$ be left-invariant vector fields on a Lie group and let $L = ∑_{i=1}^k X_i^2 + X_0$. Then L is the infinitesimal generator of a semigroup ${p_t}_{t≥0}$ of probability measures on G. Let $P*f(x) = ∑_{0
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Let L be the full laplacian on the Heisenberg group $ℍ^{n}$ of arbitrary dimension n. Then for $f ∈ L^{2}(ℍ^{n})$ such that $(I-L)^{s/2}f ∈ L^{2}(ℍ^{n})$, s > 3/4, for a $ϕ ∈ C_{c}(ℍ^{n})$ we have $ʃ_{ℍ^{n}} |ϕ(x)| sup_{0 < t≤1} |e^{(√-1)tL}f(x)|^{2} dx ≤ C_{ϕ} ∥f∥_{W^{s}}^{2}$. On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group $ℍ^{n}$, then for every s < 1 there exists a sequence $f_{n} ∈ L^{2}(ℍ^{n})$ and $C_{n} > 0$ such that $(I-L)^{s/2} f_{n} ∈ L^{2}(ℍ^{n})$ and for a $ϕ ∈ C_{c}(ℍ^{n})$ we have $ʃ_{ℍ^{n}} |ϕ(x)| sup_{0 < t≤1} |e^{(√-1)tΔ} f_{n}(x)|^{2} dx ≥ C_{n} ∥f_{n}∥_{W^{s}}^{2}, lim_{n→∞}C_{n} = +∞$.
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Let G be the simplest nilpotent Lie group of step 3. We prove that the densities of the semigroup generated by the sublaplacian on G are not real-analytic.
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Let L be the full laplacian on the Heisenberg group ℍⁿ of arbitrary dimension n. Then for f ∈ L²(ℍⁿ) such that $(I-L)^{s/2} f ∈ L²(ℍⁿ)$ for some s > 1/2 and for every $ϕ ∈ C_{c}(ℍⁿ)$ we have $∫_{ℍⁿ} |ϕ(x)| sup_{0
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We prove the dimension free estimates of the $L^{p} → L^{p}$, 1< p ≤ ∞, norms of the Hardy-Littlewood maximal operator related to the optimal control balls on the Heisenberg group ℍⁿ.
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A simple proof of the existence of solutions for the two-dimensional Keller-Segel model with measures with all the atoms less than 8π as the initial data is given. This result was obtained by Senba and Suzuki (2002) and Bedrossian and Masmoudi (2014) using different arguments. Moreover, we show a uniform bound for the existence time of solutions as well as an optimal hypercontractivity estimate.
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Let ${K_{t}}_{t>0}$ be the semigroup of linear operators generated by a Schrödinger operator -L = Δ - V with V ≥ 0. We say that f belongs to $H¹_{L}$ if $||sup_{t>0} |K_{t}f(x)| ||_{L¹(dx)} < ∞$. We state conditions on V and $K_{t}$ which allow us to give an atomic characterization of the space $H¹_{L}$.
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For a Schrödinger operator A = -Δ + V, where V is a nonnegative polynomial, we define a Hardy $H_A^1$ space associated with A. An atomic characterization of $H_A^1$ is shown.
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We prove that on a product of generalized Heisenberg groups, a Hörmander type multiplier theorem for Rockland operators is true with the critical index n/2 + ϵ, ϵ>0, where n is the euclidean (topological) dimension of the group.
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Let A = -Δ + V be a Schrödinger operator on $ℝ^{d}$, d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of $H^{p}_{A}$ if the maximal function $sup_{t>0} |T_{t}f(x)|$ belongs to $L^{p}(ℝ^{d})$, where ${T_{t}}_{t>0}$ is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space $H^{p}_{A}$ admits a special atomic decomposition.
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We consider the two-parameter maximal operator $Mf(x)= sup_{a,b>0}$ ʃ_{|s| < 1} |f(x-(as,bΓ(s)))|ds$ on a homogeneous surface $x_3 = Γ(x_1,x_2)$ in $ℝ^3$. We assume that the curvature of the level set $Γ(x_1,x_2) = 1$ has a degeneracy of finite order k at a given point. We prove that the operator M is bounded on $L^p$ if and only if $p > max{3/2, 2k/(k+1)}$.
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Let L be a positive Rockland operator of homogeneous degree d on a graded homogeneous group G and let $p_t$ be the convolution kernels of the semigroup generated by L. We prove that if τ(x) is a Riemannian distance of x from the unit element, then there are constants c>0 and C such that $|p_1(x)| ≤ Cexp(-cτ(x)^{d/(d-1)})$. Moreover, if G is not stratified, more precise estimates of $p_1$ at infinity are given.
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For rank one solvable Lie groups of the type NA estimates for the Poisson kernels and their derivatives are obtained. The results give estimates on the Poisson kernel and its derivatives in a natural parametrization of the Poisson boundary (minus one point) of a general homogeneous, simply connected manifold of negative curvature.
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