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In this paper, we study the structural properties of Nevanlinna measures, that is, Borel measures that arise in the integral representation of Herglotz–Nevanlinna functions. In particular, we give a characterization of these measures in terms of their Fourier transform, characterize measures supported on hyperplanes including extremal measures, describe the structure of the singular part of the measures when some variable are set to a fixed value, and provide estimates for the measure of expanding and shrinking cubes. Corresponding results are stated also in the setting of the polydisc where applicable, and some of our proofs are actually performed via the polydisc. [ABSTRACT FROM AUTHOR]

The Lp$L_p$ Busemann–Petty problem for 00$\varepsilon >0$ small enough, if K is an Lp$L_p$ intersection body and L is a star body such that for any u∈Sn−1$u\in S^{n-1}$, the following inequality: ∥u∥IpK−p≤∥u∥IpL−p+ε$$\begin{equation*} \Vert u\Vert _{I_pK}^{-p}\le \Vert u\Vert _{I_pL}^{-p}+\varepsilon \end{equation*}$$implies V(K)n−pn≤V(L)n−pn+Cε,$$\begin{equation*} V(K)^{\frac{n-p}{n}}\le V(L)^{\frac{n-p}{n}}+C\varepsilon , \end{equation*}$$where ∥·∥IpK$\Vert \cdot \Vert _{I_pK}$ is the Minkowski functional of IpK$I_pK$, and C is a positive number depending only on p and n. Moreover, we also prove the linear stability and separation results for the negative answer of the Lp$L_p$ Busemann–Petty problem. [ABSTRACT FROM AUTHOR]

For an expansive real matrix M=ρ−1C0ρ−1$M= \def\eqcellsep{&}\begin{bmatrix} \rho ^{-1} & \mathcal {C}\\ 0& \rho ^{-1} \end{bmatrix}$ and a noncollinear integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t}$D=\lbrace (0,0)^t,(\alpha _1,\alpha _2)^t,(\beta _1,\beta _2)^t\rbrace$ with α2−2β2∉3Z$\alpha _2-2\beta _2\notin 3\mathbb {Z}$, let μM,D$\mu _{M,D}$ be the self‐affine measure defined by μM,D(·)=13∑d∈DμM,D(M(·)−d)$\mu _{M,D}(\cdot)=\frac{1}{3}\sum _{d\in D}\mu _{M,D}(M(\cdot)-d)$. In this paper, some necessary and sufficient conditions for L2(μM,D)$L^2(\mu _{M,D})$ contains an infinite orthogonal exponential set or μM,D$\mu _{M,D}$ to be a spectral measure are given. [ABSTRACT FROM AUTHOR]

Let n,b≥2 be two positive integers. For D={0,1,⋯,b−1}, let the self‐similar measure μbn,D be defined by μbn,D=1b∑d∈Dμbn,D(bnx−d). It is known [18] that μbn,D is a spectral measure with a spectrum Λ(bn,C)=∑j=0finiteajbnj:aj∈C,where C=bn−1{0,1,⋯,b−1}. In this paper, we give some conditions on τ∈Z under which the scaling set τΛ(bn,C) is also a spectrum of μbn,D. [ABSTRACT FROM AUTHOR]

Abstract: Paley–Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundles on a compact Riemannian symmetric space U / K. It characterizes f with small support in terms of holomorphic extendability and exponential growth of their χ‐spherical Fourier transforms, where χ is a character of K. An important tool in our proof is a generalization of Opdam's estimate for the hypergeometric functions associated to multiplicity functions that are not necessarily positive. At the same time the radius of the domain where this estimate is valid is increased. This is done in an appendix. [ABSTRACT FROM AUTHOR]

Abstract: With the help of Van der Corput lemmas, decay estimates are proven for Fourier transforms of mixed homogeneous hypersurface measures with densities that can be quite irregular. The primary results are local in nature, but can be extended to global theorems in an appropriate sense. The estimates are sharp for a certain range of indices in the theorems. [ABSTRACT FROM AUTHOR]

In this sequel to the article , a criterion for the regularity of fundamental solutions of differential operators with positive symbol is proved in analogy to Hörmander's criteria, see . It implies that the temperate fundamental solution of the operator [ABSTRACT FROM AUTHOR]

We obtain necessary and sufficient conditions for the uniform convergence of sine integrals [ABSTRACT FROM AUTHOR]

The Stokes equations in a layer domain are investigated. For the case of a two-dimensional layer it is shown by resolvent estimates that there exists a semigroup of angle [ABSTRACT FROM AUTHOR]

We study general (not necessarily Hamiltonian) first-order symmetric systems [ABSTRACT FROM AUTHOR]