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1
Content available remote

A note on a multi-variable polynomial link invariant

100%
Sikora A.
Colloquium Mathematicum
|
1996
|
tom 69
|
nr 1
53-58
2
Content available remote

Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth (A remark on the paper of M. Cowling et al.)

100%
Sikora A.
Studia Mathematica
|
2002
|
tom 152
|
nr 2
125-130
EN
We study spectral multipliers for a distinguished Laplacian on certain groups of exponential growth. We obtain a stronger optimal version of the results proved in [CGHM] and [A].
3
Content available remote

Riesz meets Sobolev

64%
Coulhon T., Sikora A.
Colloquium Mathematicum
|
2010
|
tom 118
|
nr 2
685-704
EN
We show that the $L^{p}$ boundedness, p > 2, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth.
4
Content available remote

L₁-uniqueness of degenerate elliptic operators

64%
Robinson D., Sikora A.
Studia Mathematica
|
2011
|
tom 203
|
nr 1
79-103
EN
Let Ω be an open subset of $ℝ^{d}$ with 0 ∈ Ω. Furthermore, let $H_{Ω} = -∑^{d}_{i,j=1} ∂_{i}c_{ij}∂_{j}$ be a second-order partial differential operator with domain $C_{c}^{∞}(Ω)$ where the coefficients $c_{ij} ∈ W^{1,∞}_{loc}(Ω̅)$ are real, $c_{ij} = c_{ji}$ and the coefficient matrix $C = (c_{ij})$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If $∫_{0}^{∞} ds s^{d/2}e^{-λμ(s)²} < ∞$ for some λ > 0 where $μ(s) = ∫_{0}^{s} dt c(t)^{-1/2}$ then we establish that $H_{Ω}$ is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup. Moreover these uniqueness conditions are equivalent to the capacity of the boundary of Ω, measured with respect to $H_{Ω}$, being zero. We also demonstrate that the capacity depends on two gross features, the Hausdorff dimension of subsets A of the boundary of the set and the order of degeneracy of $H_{Ω}$ at A.
5
Content available remote

Heat kernels and Riesz transforms on nilpotent Lie groups

51%
ter Elst A. F. M., Robinson D. W., Sikora A.
Colloquium Mathematicum
|
1997
|
tom 74
|
nr 2
191-218
EN
We consider pure mth order subcoercive operators with complex coefficients acting on a connected nilpotent Lie group. We derive Gaussian bounds with the correct small time singularity and the optimal large time asymptotic behaviour on the heat kernel and all its derivatives, both right and left. Further we prove that the Riesz transforms of all orders are bounded on the Lp -spaces with p ∈ (1, ∞). Finally, for second-order operators with real coefficients we derive matching Gaussian lower bounds and deduce Harnack inequalities valid for all times.
6
Content available remote

A smooth subadditive homogeneous norm on a homogeneous group

38%
Hebisch W., Sikora A.
Studia Mathematica
|
1990
|
tom 96
|
nr 3
231-236
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