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Initial value problem for the time dependent Schrödinger equation on the Heisenberg group

Jacek Zienkiewicz — 1997

Studia Mathematica

Let L be the full laplacian on the Heisenberg group n of arbitrary dimension n. Then for f L 2 ( n ) such that ( I - L ) s / 2 f L 2 ( n ) , s > 3/4, for a ϕ C c ( n ) we have ʃ n | ϕ ( x ) | s u p 0 < t 1 | e ( - 1 ) t L f ( x ) | 2 d x 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 n , then for every s < 1 there exists a sequence f n L 2 ( n ) and C n > 0 such that ( I - L ) s / 2 f n L 2 ( n ) and for a ϕ C c ( n ) we have ʃ n | ϕ ( x ) | s u p 0 < t 1 | e ( - 1 ) t Δ f n ( x ) | 2 d x C n f n W s 2 , l i m n C n = + .

H p spaces associated with Schrödinger operators with potentials from reverse Hölder classes

Jacek DziubańskiJacek Zienkiewicz — 2003

Colloquium Mathematicae

Let A = -Δ + V be a Schrödinger operator on 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 A p if the maximal function s u p t > 0 | T t f ( x ) | belongs to L p ( 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 A p admits a special atomic decomposition.

Hardy space H associated to Schrödinger operator with potential satisfying reverse Hölder inequality.

Jacek DziubanskiJacek Zienkiewicz — 1999

Revista Matemática Iberoamericana

Let {T} be the semigroup of linear operators generated by a Schrödinger operator -A = Δ - V, where V is a nonnegative potential that belongs to a certain reverse Hölder class. We define a Hardy space H by means of a maximal function associated with the semigroup {T}. Atomic and Riesz transforms characterizations of H are shown.

Existence of Solutions for the Keller-Segel Model of Chemotaxis with Measures as Initial Data

Piotr BilerJacek Zienkiewicz — 2015

Bulletin of the Polish Academy of Sciences. Mathematics

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.

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

Jacek DziubańskiWaldemar HebischJacek Zienkiewicz — 1994

Studia Mathematica

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 ) | C e x p ( - c τ ( x ) d / ( d - 1 ) ) . Moreover, if G is not stratified, more precise estimates of p 1 at infinity are given.

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