We obtain necessary and sufficient conditions on weights for the generalized Fourier-type transforms to be bounded between weighted $L^p$-$L^q$ spaces. As an important example, we investigate transforms with kernel of power type, as for instance the sine, Hankel, or $\mathscr{H}_\alpha$ transforms. The obtained necessary and sufficient conditions are given in terms of weights, but not in terms of their decreasing rearrangements, as in several previous investigations.

Let $\Omega$ be a collection of disjoint dyadic squares $\omega$, let $\pi_\omega$ denote the non-smooth bilinear projection onto $\omega$ \[ \pi_\omega (f,g)(x):=\iint 𝟙_{\omega}(\xi,\eta) \widehat{f}(\xi) \widehat{g}(\eta) e^{2\pi i (\xi + \eta) x} \,\mathrm{d} \xi\, \mathrm{d}\eta , \] and let $r>2$. We show that the bilinear Rubio de Francia operator \[ \biggl(\sum_{\omega\in\Omega} |\pi_{\omega} (f,g)|^r \biggr)^{1/r} \] is $L^p \times L^q \rightarrow L^s$ bounded with constant independent of $\Omega$ whenever $1/p + 1/q = 1/s$, $r'\lt p,q \lt r$, and $r'/2 \lt s \lt r/2$.

Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb{Z})$ and $\chi=\chi_1\chi_2$ a Dirichlet character with $\chi_i$ primitive modulo $M_i$. Suppose that $M_1$, $M_2$ are primes such that $\max\{\!(M|t|)^{\!1/3+2\delta/3\!},M^{2/5}|t|^{-9/20\!}, M^{1/2+2\delta}|t|^{-3/4+2\delta}\}(M|t|)^{\varepsilon\!}\!\lt\!M_1\!\lt\! \min\{ (M|t|)^{2/5\!},$ $(M|t|)^{1/2-8\delta}\}(M|t|)^{-\varepsilon}$ for any $\varepsilon\!>\!0$, where $M\!=\!M_1M_2$, $|t|\!\geq\! 1$, and $0\lt\delta\lt 1/52$. Then we have $$ L\left(\frac{1}{2}+it,\pi\otimes \chi\right)\ll_{\pi,\varepsilon} (M|t|)^{3/4-\delta+\varepsilon}. $$

In this article we study the family of $\mathit{BMO}^p$ spaces, $p \geq 1$, in the general context of metric measure spaces. We give a characterization theorem that allows to describe all possible relations between these spaces considered as sets of functions. Examples illustrating the obtained cases and some additional results related to the John-Nirenberg inequality are also included.

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,\mu)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of degree up to eleven and several properties of those. Besides, we exhibit several results on Hopf Galois structures inspired by the program output. We prove that if $(H,\mu)$ is an almost classically Hopf Galois structure, then it is the unique Hopf Galois structure with underlying Hopf algebra $H$ up to isomorphism. For $p$ an odd prime, we prove that a separable extension of degree $p^2$ may have only one type of Hopf Galois structure and determine those of cyclic type; we determine as well the Hopf Galois structures on separable extensions of degree $2p$. We highlight the richness of the results obtained for extensions of degree $8$ by computing an explicit example and presenting some tables which summarize these results.

In their seminal paper Erdős and Szemerédi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this paper we show that this strong form of the Erdős-Szemerédi conjecture does not hold. We give upper and lower bounds on the cardinalities of sumsets, product sets, and ratio sets along the edges of graphs.

Let $A_{1},\dotsc,A_{r}$ be linear partial differential operators in $N$ variables, with constant coefficients in a field $\mathbb{K}$ of characteristic $0$. With $\mathbf{A}:=(A_{1},\dotsc,A_{r})$, a polynomial $u$ is $\mathbf{A}$-harmonic if $\mathbf{A}u=0$, that is, $A_{1}u=\dotsb =A_{r}u=0$.

Denote by $m_{i}$ the order of the first nonzero homogeneous part of $A_{i}$ (initial part). The main result of this paper is that if $r\leq N$, the dimension over $\mathbb{K}$ of the space of $\mathbf{A}$-harmonic polynomials of degree at most $d$ is given by an explicit formula depending only upon $r$, $N$, $d$, and $m_{1},\dotsc,m_{r}$ (but not $\mathbb{K}$) provided that the initial parts of $A_{1},\dotsc,A_{r}$ satisfy a simple generic condition. If $r>N$ and $ A_{1},\dotsc,A_{r}$ are homogeneous, the existence of a generic formula is closely related to a conjecture of Fröberg on Hilbert functions.

The main result holds even if $A_{1},\dotsc,A_{r}$ have infinite order, which is unambiguous since they act only on polynomials. This is used to prove, as a corollary, the same formula when $A_{1},\dotsc,A_{r}$ are replaced with finite difference operators. Another application, when $\mathbb{K}=\mathbb{C}$ and $A_{1},\dotsc,A_{r}$ have finite order, yields dimension formulas for spaces of $\mathbf{A}$-harmonic polynomial-exponentials.

We consider which groups $G$ are nilpotent if they have a nilpotent normal subgroup $N$ with $G/N$ a restricted soluble group and if $G$ is the only contranormal subgroup of $G$. This supplements Kurdachenko, Otal, and Subbotin work of 2009, where they consider the corresponding question but with $G/N$ nilpotent and $N$ a restricted soluble normal subgroup.

Let $K$ be a field. For any valuation $\mu$ on $K[x]$ admitting key polynomials we determine the structure of the whole set of key polynomials in terms of a fixed key polynomial of minimal degree. We deduce a canonical bijection between the set of $\mu$-equivalence classes of key polynomials and the maximal spectrum of the subring of elements of degree zero in the graded algebra of $\mu$.

We show that, in any right-angled Artin group whose defining graph has chromatic number $k$, every non-trivial element has stable commutator length at least $1/(6k)$. Secondly, if the defining graph does not contain triangles, then every non-trivial element has stable commutator length at least $1/20$. These results are obtained via an elementary geometric argument based on earlier work of Culler.

We prove sharp decoupling inequalities for a class of two dimensional non-degenerate surfaces in $\mathbb{R}^5$, introduced by Prendiville [13]. As a consequence, we obtain sharp bounds on the number of integer solutions of the Diophantine systems associated with these surfaces.

We investigate ergodic properties of Markov semigroups in von Neumann algebras with the help of the notion of constrictor, which expresses the idea of closeness of the orbits of the semigroup to some set, as well as the notion of ‘generalised averages’, which generalises to arbitrary abelian semigroups the classical notions of Cesàro, Borel, or Abel means. In particular, mean ergodicity, asymptotic stability, and structure properties of the fixed-point space are analysed in some detail.

In this work we study when neighborhoods of curves in holomorphic surfaces with positive self-intersection number can be embedded in the projective plane.

The usual one third trick allows to reduce problems involving general cubes to a countable family. Moreover, this covering lemma uses only dyadic cubes, which allows to use nice martingale properties in harmonic analysis problems. We consider alternatives to this technique in spaces equipped with nonhomogeneous measures. This entails additional difficulties which force us to consider martingale filtrations that are not regular. The dyadic covering that we find can be used to clarify the relationship between martingale BMO spaces and the most natural BMO space in this setting, which is the space RBMO introduced by Tolsa.