Related papers: Maximal Operators Associated to Multiplicative Cha…

A method of estimating sums of multiplicative functions braided with Dirichlet characters is demonstrated, leading to a taxonomy of the characters for which such sums are large.

We make progress on an interesting problem on the boundedness of maximal modulations of the Hilbert transform along the parabola. Namely, if we consider the multiplier arising from it and restrict it to lines, we prove uniform $L^p$ bounds…

We study arithmetic inequalities for multiplicative, sub(super)-multiplicative, sub(super)-homogeneous functions. Applications for the classical arithmetic functions are pointed out.

Strengthening of two Clarkson-McCarthy inequalities with several operators is established. These not only confirm a conjecture of the author in [Israel J. Math. 2024], but also improve results of Hirazallah-Kittaneh in [Integral Equations…

We prove $L^p$ bounds for partial polynomial Carleson operators along monomial curves $(t,t^m)$ in the plane $\mathbb{R}^2$ with a phase polynomial consisting of a single monomial. These operators are "partial" in the sense that we consider…

We prove that a variety of oscillatory and polynomial Carleson operators are uniformly bounded on the family of parameters under considerations. As a particular application of our techniques, we prove uniform bounds for oscillatory Carleson…

We investigate the relation between Carleson sequence and balayage, and use this to give an easy proof of the equivalence of the L1-norms of the maximal function and the square function in non-honogeneous martingale settings.

We obtain $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show that, exactly as for the Hilbert transform, $\|{\mathcal C}\|_{L^p(w)}$ is bounded…

We prove weak $(2,2)$ bounds for maximally modulated anisotropically homogeneous smooth multipliers on $\mathbb{R}^n$. These can be understood as generalizing the classical one-dimensional Carleson operator. For the proof we extend the…

Carleson's theorem on the pointwise convergence of Fourier series provides bounds for a maximal operator, with the maximum taken over all choices of linear functions of a phase argument. We extend this to all quadratic choices of phase…

We present a new proof of the compactness of bilinear paraproducts with CMO symbols. By drawing an analogy to compact linear operators, we first explore further properties of compact bilinear operators on Banach spaces and present examples.…

C. Stockdale, P. Villarroya, and B. Wick introduced the $\epsilon$-maximal operator to prove the Haar multiplier is bounded on the weighted spaces $L^p(w)$ for a class of weights larger than $A_p$. We prove the $\epsilon$-maximal operator…

We show that there is no analog of Kirszbraun's extension theorem for Almgren's multiple valued functions.

Let $x=a+ib$ be a complex number, so we have the following inequality $$(1/\sqrt{2})|a+b|\leq |x|\leq |a|+|b|$$ We give an operator version of above inequality. Also we obtain some results for normal operators.

In this paper we prove and discuss some new $\left(H_{p},weak-L_{p}\right) $ type inequalities of maximal operators of Vilenkin-N\"orlund means with monotone coefficients. We also apply these results to prove a.e. convergence of such…

We prove that any $q$-automatic multiplicative function $f:\mathbb{N}\to\mathbb{C}$ either essentially coincides with a Dirichlet character, or vanishes on all sufficiently large primes. This confirms a strong form of a conjecture of J.…

We provide a description for the Bellman function related to the Carleson Imbedding theorem, first mentioned in [4], with the use of the Hardy operator.

We prove $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$, its lacunary version $\mathcal C_{lac}$, and its analogue for the Walsh series $\W$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show…

In this paper, we first prove two new identities for multiplicative differentiable functions. Based on this identity, we establish a midpoint and trapezoid type inequalities for multiplicatively convex functions. Applications to special…

Given Mikhlin-H\"ormander multipliers $m_i$, $i=1,..., N$, with uniform estimates we prove an optimal $\sqrt{\log(N+1)}$ bound in $L^p$ for the maximal function $\sup_i|\cF^{-1}[m_i\hat f]|$ and related bounds for maximal functions…