Related papers: $A^p_\alpha$ classes in the Dirichlet range: inner…

The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Pel\'{a}ez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces. However, their characterizations for…

We obtain Schwarz-Pick lemma for $(\alpha, \beta)$-harmonic functions u in the disc, where $\alpha$ and $\beta$ are complex parameters satisfying $\Re \alpha + \Re \beta > -1$. We prove sharp estimate of derivative at the origin for such…

For $f$ analytic on the unit disc let $r_t(f)(z)=f(e^{it}z)$ and $f_r(z)=f(rz)$, rotations and dilations respectively. We show that for $f$ in the Bergman space $A^p$ and $0<\alpha\leq 1$ the following are equivalent. \begin{itemize}…

We prove that a nonlocal functional approximating the standard Dirichlet $p$-norm fails to decrease under two-point rearrangement. Furthermore, we get other properties related to this functional such as decay and compactness, and the…

This work explores several aspects of interpolating sequences for $\ell^p_A$, the space of analytic functions on the unit disk with $p$-summable Maclaurin coefficients. Much of this work is communicated through a Carlesonian lens. We…

The class $M^p$ $(1<p<\infty)$ consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ for which $$ \int_0^{2\pi}\left(\log^+Mf(\theta)\right)^p\,\frac{d\theta}{2\pi}<\infty, $$ where $Mf(\theta)=\sup_{0\leqslant…

Let X be a Hilbert C^*-module on C^*-algebra A and p in A. We denote by Dp(A;X) the set of all continuous functions f on A, which are Frechet differentiable on a open neighborhood U of p. Then, we introduce some generalized semi-inner…

For $0<p<\infty$ and $-2\le\alpha\le0$ we show that the $L^p$ integral mean on $rD$ of analytic function in the unit disk $D$ with respect to the weighted area measure $(1-|z|^2)^\alpha dA(z)$ is a logarithmically convex function of $r$ on…

We study conditions for containment of a given space $X$ of analytic functions on the unit disk $\mathbb{D}$ in the de Branges-Rovnyak space $\mathcal{H}(b)$. We deal with the non-extreme case in which $b$ admits a Pythagorean mate $a$, and…

We prove sharp weak and strong type weighted estimates for a class of dyadic operators that includes majorants of both standard singular integrals and square functions. Our main new result is the optimal bound…

We prove a Bernstein-type inequality involving the Bergman and the Hardy norms, for rational functions in the unit disc \mathbb{D} having at most n poles all outside of \frac{1}{r}\mathbb{D}, 0

We define an inner product on a vector space of adelic measures over a number field. We find that the norm induced by this inner product governs weak convergence at each place of $K$. The canonical adelic measure associated to a rational…

For a norm $F$ on $\mathbb{R}^2$, we consider the set of $F$-Dirichlet improvable numbers $\mathbf{DI}_F$. In the most important case of $F$ being an $L_p$-norm with $p=\infty$, which is a supremum norm, it is well-known that $\mathbf{DI}_F…

We study $p$-adic $L$-functions $L_p(s,\chi)$ for Dirichlet characters $\chi$. We show that $L_p(s,\chi)$ has a Dirichlet series expansion for each regularization parameter $c$ that is prime to $p$ and the conductor of $\chi$. The expansion…

Consider the family of locally univalent analytic functions $h$ in the unit disk $|z|<1$ with the normalization $h(0)=0$, $h'(0)=1$ and satisfying the condition $${\real} \left( \frac{z h''(z)}{\alpha h'(z)}\right) <\frac{1}{2} ~\mbox{ for…

We prove two compactness results for function spaces with finite Dirichlet energy of half-space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic…

We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift $f(z) \mapsto \frac{f(z)-f(0)}{z}$ is a contraction on the space. We present a model for this operator and…

The pro-isomorphic zeta function of a finitely generated nilpotent group $\Gamma$ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of $\Gamma$. Such zeta functions…

If $({\cal E}, {\cal D})$ is a symmetric, regular, strongly local Dirichlet form on $L^2 (X,m)$, admitting a carr\'{e} du champ operator $\Gamma$, and $p>1$ is a real number, then one can define a nonlinear form ${\cal E}^p$ by the formula…

We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,\alpha)$-summable to the function in the norm of the space, provided that $\alpha>1/2$. We…