Using finer majorizing sequences we approximate solutions of nonlinear equations using inexact Newton methods on a Banach and a Hilbert setting. It turns out that under weaker conditions and the same computational cost we can provide finer error bounds on the distances involved and a more precise information on the location of the solution than in earlier results already in the literature.

We establish a large sieve bound for expressions of the form $$\sum\limits_{r=1}^R \left\vert \sum\limits_{M < n\le M+N} a_ne\left(\alpha_rf(n)\right)\right\vert^2,$$ where $f(x)=\alpha x^2+\beta x+\theta\in \mathbb{R}[x]$ is a quadratic polynomial with $\alpha>0$ and $\beta\ge 0$. We also consider the case when $f(x)=x^d$ with $d\in \mathbb{N}$, $d\ge 3$.

We get some inequalities concerning the modular distance $I^\varphi_G[Tf -f]$ for bounded functions $f:G\rightarrow \mathbb{R}.$ Here $G$ is a locally compact Hausdorff topological space provided with a regular and $\sigma$-finite measure $\mu_G,$ $I^\varphi_G$ is the modular functional generating the Orlicz spaces $L^\varphi(G)$ and $T$ is a nonlinear integral operator of the form $$(Tf)(s) = \int_H K(s,t, f(t)) d\mu_H(t),$$ where $H$ is a closed subset of $G$ endowed with another regular and $\sigma$-finite measure $\mu_H.$ As a consequence we obtain a convergence theorem for a net of such operators. Some applications to discrete operators are given.

Within the study of arithmetical Dirichlet series, those that have a functional equation and Euler product are of particular interest. In 1989 Selberg described a class $\mathcal{S}$ of Dirichlet series through a set of four axioms which possibly contain all of these interesting Dirichlet series and made a number of interesting conjectures. In particular, he conjectured the Riemann Hypothesis for this class. We prove that one consequence of the Riemann Hypothesis for functions in $\mathcal{S}$ is the generalized Lindelöf Hypothesis. Moreover, we give an example of a function $D$ which satisfies the first three of Selberg's axioms but fails the Lindelöf Hypothesis in the $Q$ aspect.

In this article we study variable exponent Sobolev spaces on metric measure spaces. We employ two definitions: a Hajłasz type definition, which uses a pointwise maximal inequality, and a Newtonian type definition, which uses an upper gradient. We prove that these spaces are Banach, that Lipschitz functions are dense as well as other basic properties. We also study when these spaces coincide.

The present paper is devoted to the study of traces on fractals of weighted Besov spaces $B^s_{pq}({\mathbb{R}}^n,w_{\varkappa}^{\Gamma})$ as well as weighted Triebel-Lizorkin spaces $F^s_{pq}({\mathbb{R}}^n,w_{\varkappa}^{\Gamma})$ with $w_{\varkappa}^{\Gamma}(x)={\rm dist}(x,\Gamma)^{\varkappa}$, where $\Gamma$ is some $d$-set, $0<d<n$, $\varkappa>-(n-d)$.

In this paper we study the Omega theorems for a class of general $L$-functions satisfying certain conditions and as an important application, we obtain the Omega theorems for the Rankin-Selberg zeta-functions $Z(s_0)$ attached to holomorphic cusp forms of fixed weight for the full modular group when $\frac {1}{2}\le \sigma_0<1$.