Related papers: Maximal Operators Associated to Multiplicative Cha…

We prove some Sawyer-type characterizations for multilinear fractional maximal function for the upper triangle case. We also provide some two-weight norm estimates for this operator. As one of the main tools, we use an extension of the…

In this paper, we prove a large sieve inequality for quartic Dirichlet characters. The result is analogous to large sieve inequalities for the quadratic and cubic Dirichlet characters.

In this paper we prove and discuss some new $\left( H_{p},L_{p}\right)$ type inequalities of maximal operators of $T$ means with respect to the Vilenkin systems with monotone coefficients. We also apply these inequalities to prove strong…

We provide a Fefferman-Stein type weighted inequality for maximally modulated Calder\'on-Zygmund operators that satisfy \textit{a priori} weak type unweighted estimates. This inequality corresponds to a maximally modulated version of a…

In this article, we obtain an explicit version of Heath-Brown's large sieve inequality for quadratic characters and discuss its applications to $L$-functions and quadratic fields.

We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack…

In this paper, we establish Newton-Maclaurin type inequalities for functions arising from linear combinations of primitively symmetric polynomials. This generalization extends the classical Newton-Maclaurin inequality to a broader class of…

We prove an inequality for polynomials applied in a symmetric way to non-commuting operators.

We solve the Stechkin problem about approximation of generally speaking unbounded hypersingular integral operators by bounded ones. As a part of the proof, we also solve several related and interesting on their own problems. In particular,…

We give a direct proof of the operator valued Hardy-Littlewood maximal inequality for $2<p<\infty$.

This note contains a plethora of counterexamples to attempts to generalize the results of bi-parameter embedding from $p=2$ case to either $p>2$ or $p<2$. This is in striking difference to $p=2$ case that was fully understood in the series…

We prove that any $q$-automatic completely multiplicative function $f:\mathbb{N}\to\mathbb{C}$ essentially coincides with a Dirichlet character. This answers a question of J. P. Allouche and L. Goldmakher and confirms a conjecture of J.…

We show that maximal operators formed by dilations of Mikhlin-H"ormander multipliers are typically not bounded on $L^p(R^d)$. We also give rather weak conditions in terms of the decay of such multipliers under which $L^p$ boundedness of the…

We characterize the matrices $U=(u_{ij})$ for which the operator square-sum identity $$\sum_{i=1}^m\Big|\sum_{j=1}^n u_{ij}A_j\Big|^2=\sum_{j=1}^n|A_j|^2$$ holds for all Schatten-class operators $A_1,\ldots,A_n$; this happens exactly when…

Let $T$ be a polynomially bounded operator, and let $\mathcal M$ be its invariant subspace. Suppose that $P_{\mathcal M^\perp}T|_{\mathcal M^\perp}$ is similar to a contraction, while $\theta(T|_{\mathcal M})=0$, where $\theta$ is a finite…

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case: \mu=n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: \mu=n and p=q, by limiting the inequality…

We show that the maximal Nevanlinna counting function and the Carleson function of analytic self-maps of the unit disk are equivalent, up to constants.

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

We obtain a new upper bound for binary sums with multiplicative characters over variables belong to some sets, having small additive doubling.

We give an asymptotic formula for the number of biquadratic extensions of the rationals of bounded discriminant that fail the Hasse norm principle.