We establish the boundedness of the Riesz transform from the Hardy space associated with the operator to the Lebesgue space L^1 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 exterior Dirichlet boundary conditions. This setting extends the work of Hassell and the third author. Specifically, we examine the real line with the measure |x|^d-1dx leading to various versions of Bessel operators. For integer d, this mimics the measure on Euclidean d-dimensional space and the obtained results are expected to provide good predictions for a class of Riemannian manifolds with Euclidean ends.

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-1dx 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.

We study Hardy space $$H^1_L(X)$$ related to a self-adjoint operator L defined on an Euclidean subspace X of $${{\mathbb {R}}^d}$$ . We continue study from [27], where, under certain assumptions on the heat semigroup $$\exp (-tL)$$ , the atomic characterization of local type for $$H^1_L(X)$$ was proved. In this paper we provide additional assumptions that lead to another characterization of $$H^1_L(X)$$ by the Riesz transforms related to L. As an application, we prove the Riesz transform characterization of $$H^1_L(X)$$ for multidimensional Bessel and Laguerre operators, and the Dirichlet Laplacian on $${\mathbb {R}}^d_+$$ .

We investigate the Hardy space H L 1 H^1_L associated with a self-adjoint operator L L defined in a general setting by Hofmann, Lu, Mitrea, Mitrea, and Yan [Mem. Amer. Math. Soc. 214 (2011), pp. vi+78]. We assume that there exists an L L -harmonic non-negative function h h such that the semigroup exp ⁡ ( − t L ) \exp (-tL) , after applying the Doob transform related to h h , satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space H L 1 H^1_L in terms of a simple atomic decomposition associated with the L L -harmonic function h h . Our approach also yields a natural characterisation of the B M O BMO -type space corresponding to the operator L L and dual to H L 1 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 R n {\mathbb {R}^n} , Schrödinger operators with certain potentials, and Bessel operators.

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 Schrodinger 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 $\nu\in(0,1)$.

Consider the multidimensional Bessel operator B f(x) = -∑ _j=1^N ( ∂ _j^2 f(x) +α _j/x_j∂ _j f(x) ) , x∈ (0,∞ )^N. Let d = ∑ _j=1^N max (1,α _j+1) be the dimension of the space (0,∞ )^N equipped with the measure x_1^α _1… x_N^α _N dx_1… dx_N . In the general case α _1,… ,α _N >-1 we prove multiplier theorems for spectral multipliers m ( B ) on L^1,∞ and the Hardy space H^1 . We assume that m satisfies the classical Hörmander condition sup _t>0η (· ) m(t· ) _W^2,β(ℝ)<∞ with β > d/2 . Furthermore, we investigate imaginary powers B^ib , b∈ℝ , and prove some lower estimates on L^1,∞ and L^p , 1

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.

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

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.

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.

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 H 1 as the sets of L 1-functions whose maximal functions for the corresponding Poisson semigroups belong to L 1. Atomic characterizations are obtained.

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.

We investigate the Hardy space HL1 associated with the Schrödinger operator L=−Δ+V on Rn, where V=∑j=1dVj. We assume that each Vj depends on variables from a linear subspace Vj of Rn, dimVj≥3, and Vj belongs to Lq(Vj) for certain q. We prove that there exist two distinct isomorphisms of HL1 with the classical Hardy space. We deduce as a corollary a specific atomic characterization of HL1. We also prove that the space HL1 can be described by means of the Riesz transforms RL,i=∂iL−1/2.

For alpha > 0 we consider the system psi((alpha-1)/2)(k)(x) of the Laguerre functions which are eigenfunctions of the differential operator Lf = -d(2)/dx(2) f - alpha/x d/dx f + x(2)f. We define an atomic Hardy space H-at(1) (X), which is subspace of L-1((0, infinity), x(alpha) dx). Then we prove that the space H-at(1) (X) is also characterized by the Riesz transform Rf = root pi partial derivative/partial derivative x L-1/2 f in the sense that f is an element of H-at(1)/ (X) if and only if f, Rf is an element of L-1 ((0, infinity), x(alpha) dx). (C) 2011 Elsevier Inc. All rights reserved.

Let ℒf(x)=-Δ f (x)+V(x)f(x) , V ≥ 0, V∈ L^1_loc(ℝ^d) , be a non-negative self-adjoint Schrödinger operator on ℝ^d . We say that an L 1 -function f is an element of 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=∂/∂ x_jℒ^-1 2 , j = 1,..., d , associated with ℒ . As an example of such a potential V one can take any V ≥ 0, V∈ L^1_loc , in one dimension.

Let L=−Δ+V be a Schrödinger operator on ℝ d , d≥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 \(H^{1}_{L}\) associated with L if sup t>0|K t f| belongs to L 1(ℝ d ). We prove that \(f\in H^{1}_{L}\) if and only if R j f∈L 1(ℝ d ) for j=1,…,d, where R j =(∂/∂ x j )L −1/2 are the Riesz transforms associated with L.

Let L be a non-negative, self-adjoint operator on L-2(Omega), where (Omega, d, mu) is a space of homogeneous type. Assume that the semigroup {Tt}t> 0 generated by - L satisfies Gaussian bounds, or more generally Davies-Gaffney estimates. We say that f belongs to the Hardy space H-L(1) if the square functionS(h)f(x) = integral integral(Gamma(x)) vertical bar t(2) Le(-t2L)f(y)vertical bar(2)d mu(y)/mu(b(d)(x,t))dt/t)(1/2)belongs to L-1(Omega, d mu), where Gamma(x) = {( y, t) is an element of Omega x (0, infinity) : d(x, y) < t}. We prove spectral multiplier theorems for L on H-L(1).

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).