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We consider a (possibly discrete) unimodular locally compact group $G$ with Haar measure $\mu_G$, and a compact $A\subseteq G$ of positive measure with $\mu_G(A^2)\leq K\mu_G(A)$. Let $H$ be a closed normal subgroup of G and $\pi: G…

We prove two new exceptional set estimates for radial projections in the plane. If $K \subset \mathbb{R}^{2}$ is a Borel set with $\dim_{\mathrm{H}} K > 1$, then $$\dim_{\mathrm{H}} \{x \in \mathbb{R}^{2} \, \setminus \, K :…

We study self-similar measures in $\mathbb{R}$ satisfying the weak separation condition along with weak technical assumptions which are satisfied in all known examples. For such a measure $\mu$, we show that there is a finite set of concave…

Let $S$ be a connected surface possibly with boundary, $\mu$ a finite Borel measure which is positive on open sets and $f:S\to S$ a homeomorphism preserving $\mu$. We prove that if $K$ is a compact connected subset of $S$ and $L$ is a…

Let $T$ be a bounded quaternionic normal operator on a right quaternionic Hilbert space $\mathcal{H}$. We show that $T$ can be factorized in a strongly irreducible sense, that is, for any $\delta >0$ there exist a compact operator $K$ with…

A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform…

Finite rank perturbations $T=N+K$ of a bounded normal operator $N$ on a separable Hilbert space are studied thanks to a natural functional model of $T$; in its turn the functional model solely relies on a perturbation matrix/ characteristic…

If $\mu$ is a smooth measure supported on a real-analytic submanifold of $\mathbb{R}^{2n}$ which is not contained in any affine hyperplane, then the Weyl transform of $\mu$ is a compact operator.

Let $\theta$ be an inner function satisfying the connected level set condition of B. Cohn, and let $K^{1}_{\theta}$ be the shift-coinvariant subspace of the Hardy space $H^1$ generated by $\theta$. We describe the dual space to…

In this paper we prove that if $\phi:\C\to\C$ is a $K$-quasiconformal map, with $K>1$, and $E\subset \C$ is a compact set contained in a ball $B$, then $$\frac{\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}(E)}{\diam(B)^{\frac2{K+1}}} \geq…

Given a bounded symmetric domain $D$ in $\mathbb C^n$, we consider the Clark measures $\mu_\alpha$, $\alpha\in \mathbb T$, associated with a rational inner function $\varphi$ from $D$ into the unit disc in $\mathbb C$. We show that…

If $G$ is a locally compact groupoid with a Haar system $\lambda$, then a positive definite function $p$ on $G$ has a form $p(x)=< L(x)\xi(d(x)),\xi(r(x))>$, where $L$ is a representation of $G$ on a Hilbert bundle ${\h}=(G^0,\{H_u\},\mu)$,…

Given a positive Borel measure $\mu$ on $[0,1)$ and a parameter $\beta>0$, we consider the Ces\`aro-type operator $\mathcal C_{\mu,\beta}$ acting on the analytic function $f(z)=\sum_{n=0}^\infty a_n z^n$ on the unit disc of the complex…

Let $G/K$ be an irreducible symmetric space where $G$ is a non-compact, connected Lie group and $K$ is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$…

We shall consider the truncated singular integral operators T_{\mu, K}^{\epsilon}f(x)=\int_{\mathbb{R}^{n}\setminus B(x,\epsilon)}K(x-y)f(y)d\mu y and related maximal operators $T_{\mu,K}^{\ast}f(x)=\underset{\epsilon >0}{\sup}|…

Given a probability measure space $(X,\Sigma,\mu)$, it is well known that the Riesz space $L^0(\mu)$ of equivalence classes of measurable functions $f: X \to \mathbf{R}$ is universally complete and the constant function $\mathbf{1}$ is a…

Let $\mu$ be a radial compactly supported distribution on a harmonic $NA$ group. We prove that the right convolution operator $c_{\mu}:f \mapsto f* \mu$ maps the space of smooth $\mathfrak{v}$-radial functions onto itself if and only if the…

Weyl-von Neumann Theorem asserts that two bounded self-adjoint operators $A,B$ on a Hilbert space $H$ are unitarily equivalent modulo compacts, i.e., $uAu^*+K=B$ for some unitary $u\in \mathcal{U}(H)$ and compact self-adjoint operator $K$,…

We consider the generalised Krein-Feller operator $\Delta_{\nu, \mu} $ with respect to compactly supported Borel probability measures $\mu$ and $\nu$ with the natural restrictions that $\mu$ is atomless, the supp$(\nu)\subseteq$supp$(\mu)$…

Let $\mu$ be a Borel probability measure with compact support. We consider exponential type orthonormal bases, Riesz bases and frames in $L^2(\mu)$. We show that if $L^2(\mu)$ admits an exponential frame, then $\mu$ must be of pure type. We…