Related papers: $H^1$ and dyadic $H^1$

We demonstrate that the failure of $L^1$ regularity in Calder\'on-Zygmund theory is a universal phenomenon: every non-constant holomorphic function in $\C^n$ generates a counterexample to the Poisson equation. In order to achieve this goal,…

Let $\mu$ be a Radon measure on $R^d$, which may be non doubling. The only condition that $\mu$ must satisfy is $\mu(B(x,r))\leq C r^n$, for all $x,r$ and for some fixed $0<n\leq d$. Recently we introduced spaces of type $BMO(\mu)$ and…

Given a frequency $\lambda=(\lambda_n)$, we consider the Hardy spaces $ \mathcal{H}_p^\lambda$ of $\lambda$-Dirichlet series $ D = \sum_n a_n e^{-\lambda_n s}$ and study the asymptotic behavior of the upper and lower democracy functions of…

Let (G,+) be a compact, abelian, and metrizable topological group. In this group we take $g\in G$ such that the corresponding automorphism t_g is ergodic. The main result of this paper is a new ergodic theorem for functions in L^1(G,M),…

Let $\Delta$ be the Dunkl Laplacian on $\mathbb R^N$ associated with a normalized root system $R$ and a multiplicity function $k(\alpha)\geq 0$. We say that a function $f$ belongs to the Hardy space $H^1_{\Delta}$ if the nontangential…

We consider self-adjoint semigroups $T_t = \exp(-tA)$ acting on $L^2(\Omega)$ and satisfying (generalised) Gaussian estimates, where $\Omega$ is a metric measure space of homogeneous type of dimension $d$. The aim of the article is to show…

We explore the optimality of the constants making valid the recently established Little Grothendieck inequality for JB$^*$-triples and JB$^*$-algebras. In our main result we prove that for each bounded linear operator $T$ from a…

We introduce a parafermionic version of the Jaynes Cummings Hamiltonian, by coupling $k$ Fock parafermions (nilpotent of order $F$) to a 1D harmonic oscillator, representing the interaction with a single mode of the electromagnetic field.…

Every diagonalmatrix D yields an endomorphism on the n-dimensional complex vectorspace. If one provides this space with Hoelder norms, we can compute the operator norm of D. We define homogeneous weighted spaces as a generalization of…

Let $H$ be a Hilbert space that can be embedded as a dense subspace of a Banach space $X$ such that the norm of the embedding is equal to $1$. We consider the following statements for a nonzero vector $\varphi$ in $H$: (A) $\|\varphi\|_X =…

Let $b$ be a $BMO$-function. It is well-known that the linear commutator $[b, T]$ of a Calder\'on-Zygmund operator $T$ does not, in general, map continuously $H^1(\mathbb R^n)$ into $L^1(\mathbb R^n)$. However, P\'erez showed that if…

We establish two global subellipticity properties of positive symmetric second-order partial differential operators on $L_2(\Ri^d)$. First, if $m \in \Ni$ then we consider operators $H_0$ with coefficients in $W^{m+1,\infty}(\Ri^d)$ and…

Given a bounded open set $\Omega\subset \mathbb{R}^n$, we study sequences of quadratic functionals on the Sobolev space $H^1_0(\Omega)$, perturbed by sequences of bounded linear functionals. We prove that their $\Gamma$-limits, in the weak…

Let $\Omega$ be a regular Koenigs domain in the complex plane $\mathbb{C}$. We prove that the Hardy number of $\Omega$ is greater or equal to $1/2$. That is, every holomorphic function in the unit disc $f \colon \mathbb{D} \to \Omega$…

It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p$-space, $1\leq p<\infty$, or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge…

It is a classical result that dyadic partial sums of the Fourier series of functions $f \in L^p(\mathbb{T})$ converge almost everywhere for $p \in (1, \infty)$. In 1968, E. A. Bredihina established an analogous result for functions…

A unified approach to the concept of a Hausdorff operator is proposed in such a way that a number of classical and new operators feet into the given definition. Conditions are given for the boundedness of the operators under consideration…

Let ${\mathcal X}$ be a metric space with doubling measure, $L$ a nonnegative self-adjoint operator in $L^2({\mathcal X})$ satisfying the Davies-Gaffney estimate, $\omega$ a concave function on $(0,\infty)$ of strictly lower type…

In this paper, we investigate the boundedness of a square function from ergodic theory on noncommutative $L_{p}$-spaces. The main result is a weak $(1,1)$ type estimate of this square function. We also show the $(L_{\infty},\mathrm{BMO})$…

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n-1$ in $\mathbb R^n$, and later this result has been…