We obtain characterizations of positive Borel measures$μ$on$B$^{$n$}so that some weighted holomorphic Besov spaces$B$_{$s$}^{$p$}($ω$,$B$^{$n$}) are embedded in$L$^{$p$}($d$$μ$).

We consider weighted ray-transforms PW (weighted Radon transforms along oriented straight lines) in Rd,d≥2, with strictly positive weights W. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on Rd. In addition, the constructed weight W is rotation-invariant continuous and is infinitely smooth almost everywhere on Rd×Sd−1. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of W is slightly relaxed. We also give examples of continous strictly positive W such that dimkerPW≥n in the space of infinitely smooth compactly supported functions on Rd for arbitrary n∈N∪{∞}, where W are infinitely smooth for d=2 and infinitely smooth almost everywhere for d≥3.

We prove sharp weighted weak type (1,1) estimates for rough singular integral operators on homogeneous groups. Similar results are shown for singular integrals on $\mathbb{R}^{2}$ with the generalized homogeneity.

We prove the existence of a resonance-free region in scattering by a strictly convex obstacle $\mathcal{O}$ with the Robin boundary condition $\partial_{\nu}u+\gamma u|_{\partial\mathcal{O}}=0$ . More precisely, we show that the scattering resonances lie below a cubic curve ℑ$ζ$=−$S$|$ζ$|^{1/3}+$C$. The constant$S$is the same as in the case of the Neumann boundary condition$γ$=0. This generalizes earlier results on cubic pole-free regions obtained for the Dirichlet boundary condition.

In a previous paper, we studied certain random walks on the hyperbolic graphs $X$ associated with the self-similar sets $K$, and showed that the discrete energy ${\mathcal E}_X$ on $X$ has an induced energy form ${\mathcal E}_K$ on $K$. The domain of ${\mathcal E}_K$ is a Besov space $\Lambda^{\alpha, \beta/2}_{2,2}$ where $\alpha$ is the Hausdorff dimension of $K$ and $\beta$ is a parameter determined by the ``return ratio" of the random walk. In this paper, we consider the functional relationship of ${\mathcal E}_X$ and ${\mathcal E}_K$. In particular, we investigate the critical exponents of the $\beta$ in the domain $\Lambda^{\alpha, \beta/2}_{2,2}$ in order for ${\mathcal E}_K$ to be a Dirichlet form. We provide some criteria to determine the critical exponents through the effective resistance of the random walk on $X$, and make use of certain electrical network techniques to calculate the exponents for some concrete examples.