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1
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Maximal estimates for nonsymmetric semigroups

100%
Zienkiewicz J.
Studia Mathematica
|
1994
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tom 109
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nr 1
41-49
EN
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
2
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Initial value problem for the time dependent Schrödinger equation on the Heisenberg group

100%
Zienkiewicz J.
Studia Mathematica
|
1997
|
tom 122
|
nr 1
15-37
EN
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 $ʃ_{ℍ^{n}} |ϕ(x)| sup_{0 < t≤1} |e^{(√-1)tΔ} f_{n}(x)|^{2} dx ≥ C_{n} ∥f_{n}∥_{W^{s}}^{2}, lim_{n→∞}C_{n} = +∞$.
3
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Note on analytic regularity of heat kernels on nilpotent Lie groups

100%
Zienkiewicz J.
Colloquium Mathematicum
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2003
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tom 97
|
nr 2
163-168
EN
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.
4
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Schrödinger equation on the Heisenberg group

100%
Zienkiewicz J.
Studia Mathematica
|
2004
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tom 161
|
nr 2
99-111
EN
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
5
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Estimates for the Hardy-Littlewood maximal function on the Heisenberg group

100%
Zienkiewicz J.
Colloquium Mathematicum
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2005
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tom 103
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nr 2
199-205
EN
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.
6
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Existence of Solutions for the Keller-Segel Model of Chemotaxis with Measures as Initial Data

64%
Biler P., Zienkiewicz J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2015
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tom 63
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nr 1
41-51
EN
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.
7
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Hardy spaces H¹ for Schrödinger operators with certain potentials

64%
Dziubański J., Zienkiewicz J.
Studia Mathematica
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2004
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tom 164
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nr 1
39-53
EN
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 $H¹_{L}$.
8
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Smoothness of densities of semigroups of measures on homogeneous groups

64%
Dziubański J., Zienkiewicz J.
Colloquium Mathematicum
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1993
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tom 66
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nr 2
227-242
9
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Hardy spaces associated with some Schrödinger operators

64%
Dziubański J., Zienkiewicz J.
Studia Mathematica
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1997
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tom 126
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nr 2
149-160
EN
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.
10
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Multiplier theorem on generalized Heisenberg groups II

64%
Hebisch W., Zienkiewicz J.
Colloquium Mathematicum
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1996
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tom 69
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nr 1
29-36
EN
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.
11
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$H^{p}$ spaces associated with Schrödinger operators with potentials from reverse Hölder classes

64%
Dziubański J., Zienkiewicz J.
Colloquium Mathematicum
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2003
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tom 98
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nr 1
5-38
EN
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.
12
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Two-parameter maximal functions associated with degenerate homogeneous surfaces in ℝ³

52%
Marletta G., Ricci F., Zienkiewicz J.
Studia Mathematica
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1998
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tom 130
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nr 1
67-75
EN
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 $p > max{3/2, 2k/(k+1)}$.
13
Content available remote

Note on semigroups generated by positive Rockland operators on graded homogeneous groups

52%
Dziubański J., Hebisch W., Zienkiewicz J.
Studia Mathematica
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1994
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tom 110
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nr 2
115-126
EN
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.
14
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Estimates for the Poisson kernels and their derivatives on rank one NA groups

52%
Damek E., Hulanicki A., Zienkiewicz J.
Studia Mathematica
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1997
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tom 126
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nr 2
115-148
EN
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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