On Riesz Transforms Characterization of <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> Spaces Associated with Some Schrödinger Operators

Let \(\mathcal Lf(x)=-\Delta f (x)+V(x)f(x)\), V?≥?0, \(V\in L^1_{loc}(\mathbb R^d)\), be a non-negative self-adjoint Schrödinger operator on \(\mathbb R^d\). We say that an L ¹-function f is an element of the Hardy space \(H^1_{\mathcal L}\) if the maximal function

$ \mathcal M_{\mathcal L} f(x)=\sup\limits_{t>0}|e^{-t\mathcal L} f(x)| $

belongs to \(L^1(\mathbb R^d)\). We prove that under certain assumptions on V the space \(H^1_{\mathcal L}\) is also characterized by the Riesz transforms \(R_j=\frac{\partial}{\partial x_j}\mathcal L^{-1\slash 2}\), j?=?1,...,d, associated with \(\mathcal L\). As an example of such a potential V one can take any V?≥?0, \(V\in L^1_{loc}\), in one dimension.

     

Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials

Let L=?Δ+V be a Schrödinger operator on ? ^d , d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L ^p (? ^d ), for some p>d /2. Let K _t be the semigroup generated by ?L. We say that an L ¹(? ^d )-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup?_t>0|K _t f| belongs to L ¹(? ^d ). We prove that \(f\in H^{1}_{L}\) if and only if R _j f∈L ¹(? ^d ) for j=1,…,d, where R _j =(?/? x _j )L ^?1/2 are the Riesz transforms associated with L.     

Multivariate Hörmander-Type Multiplier Theorem for the Hankel transform

Let $\mathcal{H}(f)(x)=\int_{(0,\infty)^{d}} f(\lambda) E_{x}(\lambda) d\nu(\lambda )$ , be the multivariate Hankel transform, where $E_{x}(\lambda)=\prod_{k=1}^{d} (x_{k} \lambda_{k})^{-\alpha _{k}+1/2}J_{\alpha_{k}-1/2}(x_{k} \lambda_{k})$ , with dν(λ)=λ ^2α dλ, α=(α ₁,…,α _d ). We give sufficient conditions on a bounded function m(λ) which guarantee that the operator $\mathcal{H}(m\mathcal{H} f)$ is bounded on L ^p (dν) and of weak-type (1,1), or bounded on the Hardy space H ¹((0,∞) ^d ,dν) in the sense of Coifman-Weiss.     

Hardy spaces for Fourier-Bessel expansions

We study Hardy spaces for Fourier-Bessel expansions associated with Bessel operators on \(((0,1),{x^{2\nu + 1}}dx)\) and ((0, 1), dx). We define Hardy spaces H¹ as the sets of L¹-functions whose maximal functions for the corresponding Poisson semigroups belong to L¹. Atomic characterizations are obtained.     

Sharp Multiplier Theorem for Multidimensional Bessel Operators

Journal of Fourier Analysis and Applications - Consider the multidimensional Bessel operator $$\begin{aligned} B f(x) = -\sum _{j=1}^N \left( \partial _j^2 f(x) +\frac{\alpha _j}{x_j} \partial _j...     

Multiplier theorem for Hankel transform on Hardy spaces

The aim of this paper is to prove a multiplier theorem for the Hankel transform on the atomic Hardy space H ¹(X), where X = ((0, ∞), x ^α dx) is the space of homogeneous type in the sense of Coifman–Weiss. The main tool is a maximal function characterization of H ¹(X).     

Multiplier theorem for Hankel transform on Hardy spaces

The aim of this paper is to prove a multiplier theorem for the Hankel transform on the atomic Hardy space H ¹(X), where X = ((0, ∞), x ^α dx) is the space of homogeneous type in the sense of Coifman–Weiss. The main tool is a maximal function characterization of H ¹(X).     

Fourier transform infrared (FT-IR) spectroscopy in bacteriology: towards a reference method for bacteria discrimination

Rapid and reliable discrimination among clinically relevant pathogenic organisms is a crucial task in microbiology. Microorganism resistance to antimicrobial agents increases prevalence of infections. The possibility of Fourier transform infrared (FT-IR) spectroscopy to assess the overall molecular composition of microbial cells in a non-destructive manner is reflected in the specific spectral fingerprints highly typical for different microorganisms. With the objective of using FT-IR spectroscopy for discrimination between diverse microbial species and strains on a routine basis, a wide range of chemometrics techniques need to be applied. Still a major issue in using FT-IR for successful bacteria characterization is the method for spectra pre-processing. We analyzed different spectra pre-processing methods and their impact on the reduction of spectral variability and on the increase of robustness of chemometrics models. Different types of the Enterococcus faecium bacterial strain were classified according to chromosomal DNA restriction patterns produced by pulsed-field gel electrophoresis (PFGE). Samples were collected from human patients. Collected FT-IR spectra were used to verify if the same classification was obtained. In order to further optimize bacteria classification we investigated whether a selected combination of the most discriminative spectral regions could improve results. Two different variable selection methods (genetic algorithms (GAs) and bootstrapping) were investigated and their relative merit for bacteria classification is reported by comparing with results obtained using the entire spectra. Discriminant partial least-squares (Di-PLS) models based on corrected spectra showed improved predictive ability up to 40% when compared to equivalent models using the entire spectral range. The uncertainty in estimating scores was reduced by about 50% when compared to models with all wavelengths. Spectral ranges with relevant chemical information for Enterococcus faecium bacteria discrimination were outlined.