arXiv subjects

Marcin Preisner

Publications and source records attributed to Marcin Preisner.

16 recordsLinked to original sources

arXiv2024

Harmonic functions for Bessel operators

We verify the continuity of the Riesz transform from the operator related Hardy space to $L^1$ - Lebesgue space of integrable functions. For the standard Euclidean Laplace operator, this is a classical result that plays a significant role in harmonic analysis and theory of singular integral operators. Here, we consider a one-dimensional model of manifolds with ends and external Dirichlet boundary operators. This setting extends the work of Hassell and the third author. Specifically, we examine the real line with the measure $|x|^{n-1}dx$ leading to various versions of Bessel operators. For integer $n$, this mimics the measure on Euclidean $n$-dimensional space and the obtained results are expected to provide good predictions for a class of Riemannian manifolds with Euclidean ends.

math.FA↗

arXiv2022

Hardy spaces meet harmonic weights revisited

We investigate Hardy spaces $H^1_L(X)$ corresponding to self-adjoint operators $L$. Our main aim is to obtain a description of $H^1_L(X)$ in terms of atomic decompositions similar to such characterisation of the classical Hardy spaces $H^1(\mathbb{R}^d)$. Under suitable assumptions, such a description was obtained by Yan and the authors in [Trans. Amer. Math. Soc. 375 (2022), no. 9, 6417-6451], where the atoms associated with an $L$-harmonic function are considered. Here we continue this study and modify the previous definition of atoms. The modified approach allows us to investigate settings, when the generating operator is related to a system of linearly independent harmonic functions. In this context, the cancellation condition for atoms is adjusted to fit this system. In an explicit example, we consider a symmetric manifold with ends $\mathbb{R}^d \# \mathbb{R}^d$. For this manifold the space of bounded harmonic functions is two-dimensional. Any element from the Hardy space $H^1_L(X)$ has to be orthogonal to all of the harmonic functions in the system.

math.FA↗

arXiv2021

Riesz transform characterizations for multidimensional Hardy spaces

We study Hardy space $H^1_L(X)$ related to a self-adjoint operator $L$ defined on Euclidean domain $X \subseteq \mathbb{R}^d$. Under certain assumptions on the heat semigroup $\exp(-tL)$ we prove characterization of $H^1_L(X)$ by the Riesz transforms related to $L$. As an application, we prove the Riesz transform characterization for multidimensional Bessel and Laguerre operators.

math.FA↗

arXiv2019

Hardy spaces meet harmonic weights

We investigate the Hardy space $H^1_L$ associated with a self-adjoint operator $L$ defined in a general setting in [S. Hofmann, et. al., Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc. 214 (2011), no. 1007, vi+78.]. We assume that there exists an $L$-harmonic non-negative function $h$ such that the semigroup $\exp(-tL)$, after applying the Doob transform related to $h$, satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space $H^1_L$ in terms of a simple atomic decomposition associated with the $L$-harmonic function $h$. Our approach also yields a natural characterisation of the $BMO$-type space corresponding to the operator $L$ and dual to $H^1_L$ in the same circumstances. The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in $\mathbb{R}^d$, Schr\"odinger operators with certain potentials, and Bessel operators.

math.FA↗

arXiv2018

Local atomic decompositions for multidimensional Hardy spaces

We consider a nonnegative self-adjoint operator $L$ on $L^2(X)$, where $X\subseteq \mathbb{R}^d$. Under certain assumptions, we prove atomic characterizations of the Hardy space $$H^1(L) = \l \{f\in L^1(X) \ : \ \ {\|}\sup_{t>0} \ |\exp(-tL)f \ | \ {\|}_{L^1(X)}<\infty\ \}.$$ We state simple conditions, such that $H^1(L)$ is characterized by atoms being either the classical atoms on $X\subseteq \mathbb{R}^d$ or local atoms of the form $|Q|^{-1}\chi_Q$, where $Q\subseteq X$ is a cube (or cuboid). One of our main motivation is to study multidimensional operators related to orthogonal expansions. We prove that if two operators $L_1, L_2$ satisfy the assumptions of our theorem, then the sum $L_1 + L_2$ also does. As a consequence, we give atomic characterizations for multidimensional Bessel, Laguerre, and Schr\"odinger operators. As a by-product, under the same assumptions, we characterize $H^1(L)$ also by the maximal operator related to the subordinate semigroup $\exp(-tL^\nu)$, where $\nu\in(0,1)$.

math.FA↗

arXiv2018

Sharp multiplier theorem for multidimensional Bessel operators

Consider the multidimensional Bessel operator $$B f(x) = -\sum_{j=1}^N \left(\partial_j^2 f(x) +\frac{\alpha_j}{x_j} \partial_j f(x)\right), \quad x\in(0,\infty)^N. $$ Let $d = \sum_{j=1}^N \max(1,\alpha_j+1)$ be the homogeneous dimension of the space $(0,\infty)^N$ equipped with the measure $x_1^{\alpha_1}... x_N^{\alpha_N} dx_1...dx_N$. In the general case $\alpha_1,...,\alpha_N >-1$ we prove multiplier theorems for spectral multipliers $m(B)$ on $L^{1,\infty}$ and the Hardy space $H^1$. We assume that $m$ satisfies the classical H\"ormander condition $$\sup_{t>0} \left||\eta(\cdot) m(t\cdot)\right||_{W^{2,\beta}(\mathbb{R})}<\infty$$ with $\beta > d/2$. Furthermore, we investigate imaginary powers $B^{ib}$, $b\in \mathbb{R}$, and prove some lower estimates on $L^{1,\infty}$ and $L^p$, $1<p<2$. As a consequence, we deduce that our multiplier theorem is sharp.

math.FA↗

arXiv2017

Kinetic energy represented in terms of moments of vorticity and applications

We study 2d vortex sheets with unbounded support. First we show a version of the Biot- Savart law related to a class of objects including such vortex sheets. Next, we give a formula associating the kinetic energy of a very general class of ows with certain moments of their vorticities. It allows us to identify a class of vortex sheets of unbounded support being only ?-finite measures (in particluar including measures \omega such that \omega(R2) = \infty), but with locally finite kinetic energy. One of such examples are celebrated Kaden approximations. We study them in details. In particular our estimates allow us to show that the kinetic energy of Kaden approximations in the neighbourhood of an origin is dissipated, actually we show that the energy is pushed out of any ball centered in the origin of the Kaden spiral. The latter result can be interpreted as an artificial viscosity in the center of a spiral.

math-ph↗

arXiv2016

Hardy spaces for semigroups with Gaussian bounds

Let T_t=e^{-tL} be a semigroup of self-adjoint linear operators acting on L^2(X,mu), where (X,d mu) is a space of homogeneous type. We assume that T_t has an integral kernel T_t(x,y) which satisfies the upper and lower Gaussian bounds: \frac{C_1}{mu(B(x,\sqrt{t}))} \exp(-c_1d(x,y)^2/t)\leq T_t(x,y) \leq \frac{C_2}{\mu(B(x,\sqrt{t}))} \exp(-c_2 d(x,y)^2/t). By definition, f belongs to H^1_L if \| f\|_{H^1_L}=\|\sup_{t>0}|T_t f(x)|\|_{L^1(X,\mu)} <\infty. We prove that there is a function \omega(x), 0<c \leq \omega(x) \leq C, such that H^1_L admits an atomic decomposition with atoms satisfying: supp a \subset B, \|a\|_{L^\infty} \leq mu(B)^{-1}, and the weighted cancellation condition \int a(x)\omega(x) dmu(x)=0.

math.FA↗

arXiv2016

Hardy spaces for Bessel-Schr\"odinger operators

Consider the Bessel operator with a potential on L^2((0,infty), x^a dx), namely Lf(x) = -f"(x) - a/x f'(x) + V(x)f(x). We assume that a>0 and V\in L^1_{loc}((0,infty), x^a dx) is a non-negative function. By definition, a function f\in L^1((0,infty), x^a dx) belongs to the Hardy space H^1(L) if sup_{t>0} |e^{-tL} f| \in L^1((0,infty), x^a dx). Under certain assumptions on V we characterize the space H^1(L) in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to L for a \in (0,1) with no additional assumptions on the potential V.

math.CA↗

arXiv2014

Atomic decompositions for Hardy spaces related to Schr\"odinger operators

Let L_U = -Delta+U be a Schr\"odinger operator on R^d, where U\in L^1_{loc}(R^d) is a non-negative potential and d\geq 3. The Hardy space H^1(L_U) is defined in terms of the maximal function for the semigroup K_{t,U} = exp(-t L_U), namely H^1(L_U) = {f\in L^1(R^d): \|f\|_{H^1(L_U)}:= \|sup_{t>0} |K_{t,U} f| \|_{L^1(R^d)} < \infty. Assume that U=V+W, where V\geq 0 satisfies the global Kato condition sup_{x\in R^d} \int_{R^d} V(y)|x-y|^{2-d} < \infty. We prove that, under certain assumptions on W\geq 0, the space H^1(L_U) admits an atomic decomposition of local type. An atom a for H^1(L_U) is either of the form a(x)=|Q|^{-1}\chi_Q(x), where Q are special cubes determined by W, or a satisfies the cancellation condition \int a(x)w(x) dx = 0, where w is an (-Delta+V)-harmonic function given by w(x) = lim_{t\to \infty} K_{t,V} 1(x). Furthermore, we show that, in some cases, the cancellation condition \int_{R^d} a(x)w(x) dx = 0 can be replaced by the classical one \int_{R^d} a(x) dx = 0. However, we construct another example, such that the atomic spaces with these two cancellation conditions are not equivalent as Banach spaces.

math.FA↗

arXiv2011

Multivariate H\"ormander-type multiplier theorem for the Hankel transform

Let H(f)(x)=\int_{(0,infty)^d} f(v) E_{x}(v) d\nu(v), be the multivariable Hankel transform, where E_{x}(v)=\prod_{k=1}^d (x_k v_k)^{-a_k+1/2} J_{a_k-1/2}(x_k v_k), d\nu(v)=v^a dv, a=(a_1,...,a_d). We give sufficient conditions on a bounded continuous function m(v) which guarantee that the operator H(m Hf) is bounded on L^p(d\nu) and of weak-type (1,1), or bounded on the Hardy space H^1((0,infty)^d, d\nu) in the sense of Coifman-Weiss.

math.FA↗

arXiv2011

Hardy spaces related to Schr\"odinger operators with potentials which are sums of L^p-functions

We investigate the Hardy space H^1_L associated to the Schr\"odinger operator L=-\Delta+V on R^n, where V=\sum_{j=1}^d V_j. We assume that each V_j depends on variables from a linear subspace VV_j of \Rn, dim VV_j \geq 3, and V_j belongs to L^q(VV_j) for certain q. We prove that there exist two distinct isomorphisms of H^1_L with the classical Hardy space. As a corollary we deduce a specific atomic characterization of H_L^1. We also prove that the space H_L^1 is described by means of the Riesz transforms R_{L,i} = \partial_i L^{-1/2}.

math.FA↗

arXiv2010

On Riesz transforms characterization of H^1 spaces associated with some Schr\"odinger operators

Let Lf(x)=-\Delta f(x) + V(x)f(x), V\geq 0, V\in L^1_{loc}(R^d), be a non-negative self-adjoint Schr\"odinger operator on R^d. We say that an L^1-function f belongs to the Hardy space H^1_L if the maximal function M_L f(x)=\sup_{t>0} |e^{-tL} f(x)| belongs to L^1(R^d). We prove that under certain assumptions on V the space H^1_L is also characterized by the Riesz transforms R_j=\frac{\partial}{\partial x_j} L^{-1/2}, j=1,...,d, associated with L. As an example of such a potential V one can take any V\geq 0, V\in L^1_{loc}, in one dimension.

math.FA↗

arXiv2010

Remarks on spectral multiplier theorems on Hardy spaces associated with semigroups of operators

Let L be a non-negative, self-adjoint operator on L^2(Ω), where (Ω, d μ) is a space of homogeneous type. Assume that the semigroup {T_t}_{t>0} generated by -L satisfies Gaussian bounds, or more generally Davies-Gaffney estimates. We say that f belongs to the Hardy space H^1_L if the square function S_h f(x)=(\iint_{Γ(x)} |t^2 L e^{-t^2 L} f(y)|^2 \frac{dμ(y)}{μ(B_d(x,t))} \frac{dt}{t})^{1/2} belongs to L^1(Ω, dμ), where Γ(x)={(y,t) \in Ω\times (0,\infty): d(x,y)<t}. We prove spectral multiplier theorems for L on H^1_L.

math.FA↗

arXiv2010

Riesz transform characterization of H^1 spaces associated with certain Laguerre expansions

For alpha>0 we consider the system l_k^{(alpha-1)/2}(x) of the Laguerre functions which are eigenfunctions of the differential operator Lf =-\frac{d^2}{dx^2}f-\frac{alpha}{x}\frac{d}{dx}f+x^2 f. We define an atomic Hardy space H^1_{at}(X), which is a subspace of L^1((0,infty), x^alpha dx). Then we prove that the space H^1_{at}(X) is also characterized by the Riesz transform Rf=\sqrt{\pi}\frac{d}{dx}L^{-1/2}f in the sense that f\in H^1_{at}(X) if and only if f,Rf \in L^1((0,infty),x^alpha dx).

math.FA↗

arXiv2009

Riesz transform characterization of Hardy spaces associated with Schr\"odinger operators with compactly supported potentials

Let L=-\Delta+V be a Schr\"odinger operator on R^d, d\geq 3. We assume that V is a nonnegative, compactly supported potential that belongs to L^p(R^d), for some p>d/2. Let K_t be the semigroup generated by -L. We say that an L^1(R^d)-function f belongs to the Hardy space H_L^1 associated with L if sup_{t>0} |K_t f| belongs to L^1(R^d). We prove that f\in H_L^1 if and only if R_j f \in L^1(R^d) for j=1,...,d, where R_j= \frac{d}{dx_j} L^{-1/2} are the Riesz transforms associated with L.

math.FA↗