Related papers: On the Hardy-Littlewood majorant problem for rando…

Let $S^{(\Lambda)}$ denote the classical Littlewood-Paley square function formed with respect to a lacunary sequence $\Lambda$ of positive integers. Motivated by a remark of Pichorides, we obtain sharp asymptotic estimates of the behaviour…

Let $T$ be a finite tree graph, $T^N$ be the Cartesian power graph of $T$, and $d^N$ be the graph distance metric on $T^N$. Also let \[ \mathbb S_r^N(x) := \{v \in T^N: d^N(x,v) = r\} \] be the sphere of radius $r$ centered at $x$ and $M$…

Let M denote the maximal function along the polynomial curve p(t)=(t,t^2,...,t^d) in R^d: M(f)=sup_{r>0} (1/2r) \int_{|t|<r} |f(x-p(t))| dt. We show that the L^2-norm of this operator grows at most logarithmically with the parameter d:…

The regularity of the Hardy-Littlewood maximal function, in both discrete and continuous contexts, and for both centered and noncentered variants, has been subjected to intense study for the last two decades. But efforts so far have…

In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients. A polynomial is called dominant if it has a simple root whose modulus is strictly greater than the moduli of its remaining roots. In…

We study growth of absolute and homological $k$-dimensional systoles of arithmetic $n$-manifolds along congruence coverings. Our main interest is in the growth of systoles of manifolds whose real rank $r > 1$. We observe, in particular,…

We prove an existence result of weak solutions $u\in W_{0}^{1,p}\left( \Omega \right) \cap W_{\mathrm{loc}}^{1,q}\left( \Omega \right) $, to a Dirichlet problem for a second order elliptic equation in divergence form, under general and…

The paper focuses on the behaviour of unimodular Fourier multipliers with exponential growth in the context of weighted $L^p$-spaces. Our main result shows that much of the general theory of multipliers is approachable through the theory of…

We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We derive this by…

We study holomorphic functions attaining weighted norms and its connections with the classical theory of norm attaining holomorphic functions. We prove that there are polynomials on $\ell_p$ which attain their weighted but not their…

Let $\{\mathbb{P}_t\}_{t>0}$ be the classical Poisson semigroup on $\mathbb{R}^d$ and $G^{\mathbb{P}}$ the associated Littlewood-Paley $g$-function operator: $$G^{\mathbb{P}}(f)=\Big(\int_0^\infty t|\frac{\partial}{\partial t}…

We study two questions. When does a function belong to the union of Lebesgue spaces and when does a function have an $A_1$ majorant? We show these questions are fundamentally related. For functions restricted to a fixed cube we prove that…

We study the regularity and well-posedness of the local, first-order forward-backward mean field games system, assuming a polynomially growing cost function and a Hamiltonian of quadratic growth. We consider systems and terminal data that…

We extend recent work of Gurel-Gurevich--Jerison--Nachmias (2020) and Bou-Rabee--Gwynne (2024) by showing that as the mesh of our lattice tends to $0$, we have a polynomial rate of convergence for the Dirichlet problem on orthodiagonal maps…

In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term which depends on some Lorentz norms of $u$ or of its gradient and we find the best values of the constants…

We introduce and study the median maximal function \mathcal{M} f, defined in the same manner as the classical Hardy-Littlewood maximal function, only replacing integral averages of f by medians throughout the definition. This change has a…

We present an account of different problems that arise in relation with cyclicity problems in Dirichlet-type spaces, in particular with polynomials $p$ that minimize the norm $\|pf-1\|$.

We analyze the Fourier growth, i.e. the $L_1$ Fourier weight at level $k$ (denoted $L_{1,k}$), of various well-studied classes of "structured" $\mathbb{F}_2$-polynomials. This study is motivated by applications in pseudorandomness, in…

Majorization and trumping are two partial orders which have proved useful in quantum information theory. We show some relations between these two partial orders and generalized Dirichlet polynomials, Mellin transforms, and completely…

We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a…