Ioannis Parissis

Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases

Let B be a homothecy invariant collection of convex sets in R. Given a measure μ, the associated weighted geometric maximal operator MB,μ is defined by MB,μf(x) ∶= sup x∈B∈B 1

The endpoint Fefferman-Stein inequality for the strong maximal function

Abstract Let M n f denote the strong maximal function of f on R n , that is the maximal average of f with respect to n -dimensional rectangles with sides parallel to the coordinate axes. For any dimension n ⩾ 2 we prove the natural endpoint Fefferman–Stein inequality for M n and any strong Muckenhoupt weight w : w ( { x ∈ R n : M n f ( x ) > λ } ) ≲ w , n ∫ R n | f ( x ) | λ ( 1 + ( log + | f ( x ) | λ ) n − 1 ) M n w ( x ) d x . This extends the corresponding two-dimensional result of T. Mitsis.

Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

Let denote a weight in which belongs to the Muckenhoupt class and let denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure . The sharp Tauberian constant of with respect to , denoted by , is defined by In this paper, we show that the Solyanik estimate

Recurrent Linear Operators

\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}

A sharp estimate for the Hilbert transform along finite order lacunary sets of directions

limα→1-Cw(α)=1holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator and a weight : We show that we have if and only if . As a corollary of our methods we obtain a quantitative embedding of into .

Sharp inequalities for one-sided Muckenhoupt weights

Let \(\mathcal{B}\) denote a collection of open bounded sets in \(\mathbb{R}^{n}\), and define the associated maximal operator \(M_{\mathcal{B}}\) by

Dynamics of tuples of matrices in Jordan form

\displaystyle{M_{\mathcal{B}}f(x)\,:=\,\sup _{x\in R\in \mathcal{B}} \frac{1} {\vert R\vert }\int _{R}\vert f\vert.}