Related papers: Integral means and boundary limits of Dirichlet se…

We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In…

We prove an inequality of Hardy type for functions in Triebel-Lizorkin spaces. The distance involved is being measured to a given Ahlfors d-regular set in R^n, with n-1<d<n. As an application of the Hardy inequality, we consider boundedness…

The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs…

This paper is devoted to Hardy type inequalities with remainders for compactly supported smooth functions on open sets in the Euclidean space. We establish new inequalities with weight functions depending on the distance function to the…

We study the Hardy spaces $\mathcal{H}^p$, $0<p<\infty$ of quasiregular mappings on the unit ball $\mathbb{B}^n$ in ${\mathbb{R}}^n$ under the appropriate growth and multiplicity conditions. Our focus is on the averaged derivatives of maps…

We study Hardy spaces $H^p_\nu$ of the conjugate Beltrami equation $\bar{\partial} f=\nu\bar{\partial f}$ over Dini-smooth finitely connected domains, for real contractive $\nu\in W^{1,r}$ with $r>2$, in the range $r/(r-1)<p<\infty$. We…

For $p>\frac{2\lambda}{2\lambda+1}$ with $\lambda>0$, the Hardy spaces $H_{\lambda}^{p}(\mathbb{R}^{2}_+)$ associated with the Dunkl transform $\mathscr{F}_\lambda$ and the Dunkl operator $D_x$ on the line, where…

A theorem of Henry Helson shows that for every ordinary Dirichlet series $\sum a_n n^{-s}$ with a square summable sequence $(a_n)$ of coefficients, almost all vertical limits $\sum a_n \chi(n) n^{-s}$, where $\chi: \mathbb{N} \to…

Let $L$ be a one to one operator of type $\omega$ having a bounded $H_\infty$ functional calculus and satisfying the $k$-Davies-Gaffney estimates with $k\in{\mathbb N}$. In this paper, the authors introduce the Hardy space…

In the first part of this study we consider the boundedness and compactness properties of Cauchy-Fantappie type operators on Poletsky-Stessin Hardy spaces $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$ of complex ellipsoids. We show that boundedness…

We establish a complete theory of the flag Hardy space on the Heisenberg group $\mathbb H^{n}$ with characterisations via atomic decompositions, area functions, square functions, maximal functions and singular integrals. We introduce…

The $L^p$ convergence of eigenfunction expansions for the Laplacian on planar domains is largely unknown for $p\neq 2$. After discussing the classical Fourier series on the 2-torus, we move onto the disc, whose eigenfunctions are explicitly…

We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and…

It is well known that the Hilbert matrix operator $\mathcal {H}$ is bounded from $H^{\infty}$ to the mean Lipschitz spaces $\Lambda^{p}_{\frac{1}{p}}$ for all $1<p<\infty$. In this paper, we prove that the range of Hilbert matrix operator…

We give necessary and sufficient conditions for a composition operator with Dirichlet series symbol to belong to the Schatten classes $S_p$ of the Hardy space $\mathcal{H}^2$ of Dirichlet series. For $p\geq 2$, these conditions lead to a…

For $0<p<\infty $ and $\alpha >-1$ the space of Dirichlet type $\mathcal D^p_\alpha $ consists of those functions $f$ which are analytic in the unit disc $\mathbb D$ and satisfy $\int_{\mathbb D}(1-| z| )^\alpha| f^\prime (z)|…

Let $(M,F)$ be a $C^\infty$ Finsler manifold, $p\geq 1$ a real number, $k$ a positive integer and $H_k^p (M)$ a certain Sobolev space determined by a Finsler structure $F$. Here, it is shown that the set of all real $C^{\infty}$ functions…

Given a sequence of points in the unit disk, a well known result due to Carleson states that if given any point of the sequence it is possible to interpolate the value one in that point and zero in all the other points of the sequence, with…

We state and discuss several interrelated results, conjectures, and questions regarding contractive inequalities for classical $H^p$ spaces of the unit disc. We study both coefficient estimates in terms of weighted $\ell^2$ sums and the…

Let $({\mathcal X},d,\mu)$ be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Let $\rho\in (1,\infty)$, $0<p\le1\le q\le\infty$, $p\neq q$, $\gamma\in[1,\infty)$ and…