arXiv subjects

Ioannis Parissis

Publications and source records attributed to Ioannis Parissis.

At least 37 records · Page 2Linked to original sources

arXiv2016

Asymptotically sharp reverse Hölder inequalities for flat Muckenhoupt weights

We present reverse Hölder inequalities for Muckenhoupt weights in $\mathbb{R}^n$ with an asymptotically sharp behavior for flat weights, namely $A_\infty$ weights with Fujii-Wilson constant $(w)_{A_\infty}\to 1^+$. That is, the local integrability exponent in the reverse Hölder inequality blows up as the weight becomes nearly constant. This is expressed in a precise and explicit computation of the constants involved in the reverse Hölder inequality. The proofs avoid BMO methods and rely instead on precise covering arguments. Furthermore, in the one-dimensional case we prove sharp reverse Hölder inequalities for one-sided and two sided weights in the sense that both the integrability exponent as well as the multiplicative constant appearing in the estimate are best possible. We also prove sharp endpoint weak-type reverse Hölder inequalities and consider further extensions to general non-doubling measures and multiparameter weights.

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arXiv2016

Hölder continuity of Tauberian constants associated with discrete and ergodic strong maximal operators

This paper concerns the smoothness of Tauberian constants of maximal operators in the discrete and ergodic settings. In particular, we define the discrete strong maximal operator $\tilde{M}_S$ on $\mathbb{Z}^n$ by \[ \tilde{M}_S f(m) := \sup_{0 \in R \subset \mathbb{R}^n}\frac{1}{\#(R \cap \mathbb{Z}^n)}\sum_{ j\in R \cap \mathbb{Z}^n} |f(m+j)|,\qquad m\in \mathbb{Z}^n, \] where the supremum is taken over all open rectangles in $\mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. We show that the associated Tauberian constant $\tilde{C}_S(α)$, defined by \[ \tilde{C}_S(α) := \sup_{\substack{E \subset \mathbb{Z}^n \\ 0 < \#E < \infty} } \frac{1}{\#E}\#\{m \in \mathbb{Z}^n:\, \tilde{M}_Sχ_E(m) > α\}, \] is Hölder continuous of order $1/n$. Moreover, letting $U_1, \ldots, U_n$ denote a non-periodic collection of commuting invertible transformations on the non-atomic probability space $(Ω, Σ, μ)$ we define the associated maximal operator $M_S^\ast$ by \[ M^\ast_{S}f(ω) := \sup_{0 \in R \subset \mathbb{R}^n}\frac{1}{\#(R \cap \mathbb{Z}^n)}\sum_{(j_1, \ldots, j_n)\in R}|f(U_1^{j_1}\cdots U_n^{j_n}ω)|,\qquad ω\inΩ. \] Then the corresponding Tauberian constant $C^\ast_S(α)$, defined by \[ C^\ast_S(α) := \sup_{\substack{E \subset Ω\\ μ(E) > 0}} \frac{1}{μ(E)}μ(\{ω\in Ω:\, M^\ast_Sχ_E(ω) > α\}), \] also satisfies $C^\ast_S \in C^{1/n}(0,1).$ We will also see that, in the case $n=1$, that is in the case of a single invertible, measure preserving transformation, the smoothness of the corresponding Tauberian constant is characterized by the operator enabling arbitrarily long orbits of sets of positive measure.

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arXiv2016

Sharp inequalities for one-sided Muckenhoupt weights

Let $A_\infty ^+$ denote the class of one-sided Muckenhoupt weights, namely all the weights $w$ for which $\mathsf M^+:L^p(w)\to L^{p,\infty}(w)$ for some $p>1$, where $\mathsf M^+$ is the forward Hardy-Littlewood maximal operator. We show that $w\in A_\infty ^+$ if and only if there exist numerical constants $γ\in(0,1)$ and $c>0$ such that $$ w(\{x \in \mathbb{R} : \, \mathsf M ^+\mathbf 1_E (x)>γ\})\leq c w(E) $$ for all measurable sets $E\subset \mathbb R$. Furthermore, letting $$ \mathsf C_w ^+(α):= \sup_{0 α\}) $$ we show that for all $w\in A_\infty ^+$ we have the asymptotic estimate $\mathsf C_w ^+ (α)-1\lesssim (1-α)^\frac{1}{c[w]_{A_\infty ^+}}$ for $α$ sufficiently close to $1$ and $c>0$ a numerical constant, and that this estimate is best possible. We also show that the reverse Hölder inequality for one-sided Muckenhoupt weights, previously proved by Martín-Reyes and de la Torre, is sharp, thus providing a quantitative equivalent definition of $A_\infty ^+$. Our methods also allow us to show that a weight $w\in A_\infty ^+$ satisfies $w\in A_p ^+$ for all $p>e^{c[w]_{A_\infty ^+}}$.

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arXiv2015

Solyanik estimates and local Hölder continuity of halo functions of geometric maximal operators

Let $\mathcal{B}$ be a homothecy invariant basis consisting of convex sets in $\mathbb{R}^n$, and define the associated geometric maximal operator $M_{\mathcal{B}}$ by $$ M_{\mathcal{B}} f(x) :=\sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_R |f| $$ and the halo function $ϕ_{\mathcal{B}}(α)$ on $(1,\infty)$ by $$ϕ_{\mathcal B}(α) :=\sup_{E \subset \mathbb{R}^n :\, 0 < |E| < \infty}\frac{1}{|E|}|\{x\in \mathbb{R}^n : M_{\mathcal{B}} χ_E (x) >1/α\}|. $$ It is shown that if $ϕ_{\mathcal{B}}(α)$ satisfies the Solyanik estimate $ϕ_{\mathcal B}(α) - 1 \leq C (1 - \frac{1}α)^p$ for $α\in(1,\infty)$ sufficiently close to 1 then $ϕ_{\mathcal{B}}$ lies in the Hölder class $ C^p(1,\infty)$. As a consequence we obtain that the halo functions associated with the Hardy-Littlewood maximal operator and the strong maximal operator on $\mathbb{R}^n$ lie in the Hölder class $C^{1/n}(1,\infty)$.

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arXiv2015

Solyanik estimates in ergodic theory

Let $U_1, \ldots, U_n$ be a collection of commuting measure preserving transformations on a probability space $(Ω, Σ, μ)$. Associated with these measure preserving transformations is the ergodic strong maximal operator $\mathsf M ^\ast _{\mathsf S}$ given by \[ \mathsf M ^\ast _{\mathsf S} f(ω) := \sup_{0 \in R \subset \mathbb{R}^n}\frac{1}{\#(R \cap \mathbb{Z}^n)}\sum_{(j_1, \ldots, j_n) \in R\cap \mathbb{Z}^n}\big|f(U_1^{j_1}\cdots U_n^{j_n}ω)\big|, \] where the supremum is taken over all open rectangles in $\mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. For $0 < α< 1$ we define the sharp Tauberian constant of $\mathsf M ^\ast _{\mathsf S}$ with respect to $α$ by \[ \mathsf C ^\ast _{\mathsf S} (α) := \sup_{\substack{E \subset Ω\\ μ(E) > 0}}\frac{1}{μ(E)}μ(\{ω\in Ω: \mathsf M ^\ast _{\mathsf S} χ_E (ω) > α\}). \] Motivated by previous work of A. A. Solyanik and the authors regarding Solyanik estimates for the geometric strong maximal operator in harmonic analysis, we show that the Solyanik estimate \[ \lim_{α\rightarrow 1}\mathsf C ^\ast _{\mathsf S}(α) = 1 \] holds, and that in particular we have \[\mathsf C ^\ast _{\mathsf S}(α) - 1 \lesssim_n (1 - \frac{1}α)^{1/n}\] provided that $α$ is sufficiently close to $1$. Solyanik estimates for centered and uncentered ergodic Hardy-Littlewood maximal operators associated with $U_1, \ldots, U_n$ are shown to hold as well. Further directions for research in the field of ergodic Solyanik estimates are also discussed.

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arXiv2015

A note on local Hölder continuity of weighted Tauberian functions

Let $\mathsf M$ and $\mathsf M _{\mathsf S}$ respectively denote the Hardy-Littlewood maximal operator with respect to cubes and the strong maximal operator on $\mathbb{R}^n$, and let $w$ be a nonnegative locally integrable function on $\mathbb{R}^n$. We define the associated Tauberian functions $\mathsf{C}_{\mathsf{HL},w}(α)$ and $\mathsf{C}_{\mathsf{S},w}(α)$ on $(0,1)$ by \[ \mathsf{C}_{\mathsf{HL},w}(α) :=\sup_{\substack{E \subset \mathbb{R}^n \\ 0 < w(E) < \infty}} \frac{1}{w(E)}w(\{x \in \mathbb{R}^n : \mathsf M χ_E(x) > α\}) \] and \[ \mathsf{C}_{\mathsf{S},w}(α) := \sup_{\substack{E \subset \mathbb{R}^n \\ 0 < w(E) < \infty}} \frac{1}{w(E)}w(\{x \in \mathbb{R}^n : \mathsf M _{\mathsf S}χ_E(x) > α\}). \] Utilizing weighted Solyanik estimates for $\mathsf M$ and $\mathsf M_{\mathsf S}$, we show that the function $\mathsf{C}_{\mathsf{HL},w} $ lies in the local Hölder class $C^{(c_n[w]_{A_{\infty}})^{-1}}(0,1)$ and $\mathsf{C}_{\mathsf{S},w} $ lies in the local Hölder class $C^{(c_n[w]_{A_{\infty}^\ast})^{-1}}(0,1)$, where the constant $c_n>1$ depends only on the dimension $n$.

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arXiv2015

Weighted Solyanik Estimates for the Hardy-Littlewood maximal operator and embedding of $A_\infty$ into $A_p$

Let $w$ denote a weight in $\mathbb{R}^n$ which belongs to the Muckenhoupt class $A_\infty$ and let $\mathsf{M}_w$ denote the uncentered Hardy-Littlewood maximal operator defined with respect to the measure $w(x)dx$. The \emph{sharp Tauberian constant} of $\mathsf M_w$ with respect to $α$, denoted by $\mathsf{C}_w (α)$, is defined by \[ \mathsf{C}_w (α) := \sup_{E:\, 0 < w(E) < \infty}w(E)^{-1}w\big(\big\{x \in \mathbb{R}^n:\, \mathsf{M}_w χ_E (x) > α\big\}\big). \] In this paper, we show that the Solyanik estimate \[ \lim_{α\rightarrow 1^-}\mathsf{C}_w(α) = 1 \] holds. 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 $\mathsf M$ and a weight $w$: \[ \mathsf C ^w (α) := \sup_{E:\, 0 < w(E) < \infty} w(E)^{-1} w\big(\big\{x \in \mathbb R^n:\, \mathsf{M} χ_E (x) > α\big\}\big). \] We show that we have $\lim_{α\to 1^{-}}\mathsf{C}^w(α)=1$ if and only if $w\in A_\infty$. As a corollary of our methods we obtain a quantitative embedding of $A_\infty$ into $A_p$.

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arXiv2014

Weighted Solyanik estimates for the strong maximal function

Let $\mathsf M_{\mathsf S}$ denote the strong maximal operator on $\mathbb R^n$ and let $w$ be a non-negative, locally integrable function. For $α\in(0,1)$ we define the weighted sharp Tauberian constant $\mathsf C_{\mathsf S}$ associated with $\mathsf M_{\mathsf S}$ by $$ \mathsf C_{\mathsf S} (α):= \sup_{\substack {E\subset \mathbb R^n \\ 0 α\}). $$ We show that $\lim_{α\to 1^-} \mathsf C_{\mathsf S} (α)=1$ if and only if $w\in A_\infty ^*$, that is if and only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate $\mathsf C_{\mathsf S}(α)-1\lesssim_{n} (1-α)^{(cn [w]_{A_\infty ^*})^{-1}}$ as $α\to 1^-$, where $c>0$ is a numerical constant; this estimate is sharp in the sense that the exponent $1/(cn[w]_{A_\infty ^*})$ can not be improved in terms of $[w]_{A_\infty ^*}$. As corollaries, we obtain a sharp reverse Hölder inequality for strong Muckenhoupt weights in $\mathbb R^n$ as well as a quantitative imbedding of $A_\infty^*$ into $A_{p}^*$. We also consider the strong maximal operator on $\mathbb R^n$ associated with the weight $w$ and denoted by $\mathsf M_{\mathsf S} ^w$. In this case the corresponding sharp Tauberian constant $\mathsf C_{\mathsf S} ^w$ is defined by $$ \mathsf C_{\mathsf S} ^w α) := \sup_{\substack {E\subset \mathbb R^n \\ 0 α\}).$$ We show that there exists some constant $c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that $\mathsf C_{\mathsf S} ^w (α)-1 \lesssim_{w,n} (1-α)^{c_{w,n}}$ as $α\to 1^-$ whenever $w\in A_\infty ^*$ is a strong Muckenhoupt weight.

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arXiv2014

Solyanik estimates in harmonic analysis

Let $\mathcal{B}$ denote a collection of open bounded sets in $\mathbb{R}^n$, and define the associated maximal operator $M_{\mathcal{B}}$ by $$ M_{\mathcal{B}}f(x) := \sup_{x \in R \in \mathcal{B}} \frac{1}{|R|}\int_R |f|. $$ The sharp Tauberian constant of $M_{\mathcal{B}}$ associated to $α$, denoted by $C_{\mathcal{B}}(α)$, is defined as $$ C_{\mathcal{B}}(α) := \sup_{E :\, 0 < |E| < \infty}\frac{1}{|E|}\big|\big\{x \in \mathbb{R}^n:\, M_{\mathcal{B}}χ_E (x) > α\big\}\big|.$$ Motivated by previous work of A. A. Solyanik, we show that if $M_{\mathcal{B}}$ is the uncentered Hardy-Littlewood maximal operator associated to balls, the estimate $$ \lim_{α\rightarrow 1^-}C_{\mathcal{B}}(α) = 1 $$ holds. Similar results for iterated maximal functions are obtained, and open problems in the field of Solyanik estimates are also discussed.

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arXiv2013

Recurrent Linear Operators

We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions.

math.FA↗

arXiv2013

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

We give an alternative characterization of the class of Muckenhoupt weights $A_{\infty, \mathfrak B}$ for homothecy invariant Muckenhoupt bases $\mathfrak B$ consisting of convex sets. In particular we show that $w\in A_{\infty, \mathfrak B}$ if and only if there exists a constant $c>0$ such that for all measurable sets $E\subset \mathbb R^n$ we have $$ w({x\in \mathbb R^n: M_{\mathfrak B} (\mathbf {1}_E)(x)>1/2}) < c w(E).$$ This applies for example to the collection $\mathfrak R$ of rectangles with sides parallel to the coordinate axes, giving a new characterization of strong (multiparameter) Muckenhoupt weights. We also show versions of these results under the presence of a doubling measure. Thus the strong maximal function $M_{\mathfrak R,μ}$, defined with respect to a product-doubling measure $μ$, is bounded on $L^p(μ)$ for some $p>1$ if and only if $$μ({x\in \mathbb R^n: M_{\mathfrak R,μ} (\mathbf{1}_E)(x)>1/2}) < c μ(E)$$ for all measurable sets $E\subset \mathbb R^n$. Finally we discuss applications in differentiation theory, proving among other things that Tauberian conditions as above imply that the corresponding bases differentiate $L^\infty(μ)$, with respect to the measure $μ$.

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arXiv2013

A variation norm Carleson theorem for vector-valued Walsh-Fourier series

We prove a variation norm Carleson theorem for Walsh-Fourier series of functions with values in a UMD Banach space. Our only hypothesis on the Banach space is that it has finite tile-type, a notion introduced by Hytönen and Lacey. Given q \geq 2 we show that, if the space X has tile-type t for all t>q, then the r-variation of the Walsh-Fourier sums of any function f \in L^p ([0,1) ; X) belongs to L^p, whenever q q. For intermediate spaces, i.e. spaces X= [Y,H]_s which are complex interpolation spaces between some UMD space Y and a Hilbert space H, the tile-type is q=2/s. We show that in this case the variation norm Carleson theorem remains true for all r > q in the larger range p > (2r/q)'.

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arXiv2013

The endpoint Fefferman-Stein inequality for the strong maximal function

Let Mf 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>1 we prove the natural endpoint Fefferman-Stein inequality for M and any strong Muckenhoupt weight w: w({x \in R^n: M f (x) > t}) \lesssim_{w,n} \int_{R^n} |f|/t [1 + (log^+ |f|/t)^{n-1}] Mw. This extends the corresponding two-dimensional result of T. Mitsis.

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arXiv2012

The Vector Valued Quartile Operator

Certain vector-valued inequalities are shown to hold for a Walsh analog of the bilinear Hilbert transform. These extensions are phrased in terms of a recent notion of quartile type of a UMD (Unconditional Martingale Differences) Banach space X. Every known UMD Banach space has finite quartile type, and it was recently shown that the Walsh analog of Carleson's Theorem holds under a closely related assumption of finite tile type. For a Walsh model of the bilinear Hilbert transform however, the quartile type should be sufficiently close to that of a Hilbert space for our results to hold. A full set of inequalities is quantified in terms of quartile type.

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arXiv2012

Circle discrepancy for checkerboard measures

Consider the plane as a union of congruent unit squares in a checkerboard pattern, each square colored black or white in an arbitrary manner. The discrepancy of a curve with respect to a given coloring is the difference of its white length minus its black length, in absolute value. We show that for every radius t>1 there exists a full circle of radius either t or 2t with discrepancy greater than ct^(1/2) for some numerical constant c>0. We also show that for every t>1 there exists a circular arc of radius exactly t with discrepancy greater than ct^(1/2). Finally we investigate the corresponding problem for more general curves and their interiors. These results answer questions posed by Kolountzakis and Iosevich.

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arXiv2011

Dynamics of tuples of matrices in Jordan form

A tuple (T_1,...,T_k) of (n x n) matrices over R is called hypercyclic if for some x in R^n the set {T^{m_1} T^{m_2}...T^{m_k} x : m_1,m_2,...,m_k in N} is dense in R^n. We prove that the minimum number of (n x n) matrices in Jordan form over R which form a hypercyclic tuple is n+1. This answers a question of Costakis, Hadjiloucas and Manoussos.

math.FA↗

arXiv2011

Szemeredi's theorem, frequent hypercyclicity and multiple recurrence

Let T be a bounded linear operator acting on a complex Banach space X and (λ_n) a sequence of complex numbers. Our main result is that if |λ_n|/|λ_{n+1}| \to 1 and the sequence (λ_n T^n) is frequently universal then T is topologically multiply recurrent. To achieve such a result one has to carefully apply Szemerédi's theorem in arithmetic progressions. We show that the previous assumption on the sequence (λ_n) is optimal among sequences such that |λ_n|/|λ_{n+1}| converges in [0,+\infty]. In the case of bilateral weighted shifts and adjoints of multiplication operators we provide characterizations of topological multiple recurrence in terms of the weight sequence and the symbol of the multiplication operator respectively.

math.FA↗

arXiv2010

A Three Dimensional Signed Small Ball Inequality

The Small Ball Inequality is a conjectural lower bound on sums the L-infinity norm of sums of Haar functions supported on dyadic rectangles of a fixed volume in the unit cube. The conjecture is fundamental to questions in discrepancy theory, approximation theory and probability theory. In this article, we concentrate on a special case of the conjecture, and give the best known lower bound in dimension 3, using a conditional expectation argument.

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