Template iterations and maximal cofinitary groups

Vera Fischer; Asger Törnquist

Displaying similar documents to “Template iterations and maximal cofinitary groups”

Comparing the closed almost disjointness and dominating numbers

Dilip Raghavan, Saharon Shelah (2012)

Fundamenta Mathematicae

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We prove that if there is a dominating family of size ℵ₁, then there are ℵ₁ many compact subsets of $ω^{ω}$ whose union is a maximal almost disjoint family of functions that is also maximal with respect to infinite partial functions.

A MAD Q-set

Arnold W. Miller (2003)

Fundamenta Mathematicae

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A MAD (maximal almost disjoint) family is an infinite subset of the infinite subsets of ω = 0,1,2,... such that any two elements of intersect in a finite set and every infinite subset of ω meets some element of in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative $G_{δ}$ -set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology it inherits as a subset of $P (ω) = 2^{ω}$ .

A note on rare maximal functions

Paul Alton Hagelstein (2003)

Colloquium Mathematicae

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A necessary and sufficient condition is given on the basis of a rare maximal function $M_{l}$ such that $M_{l} f \in L ¹ ([0, 1])$ implies f ∈ L log L([0,1]).

Ordinal remainders of classical ψ-spaces

Alan Dow, Jerry E. Vaughan (2012)

Fundamenta Mathematicae

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Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain $T_{α} : α < λ$ of infinite subsets of ω, there exists $ℳ \subset {[ω]}^{ω}$ , an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ∖ψ of the associated ψ-space, ψ = ψ(ω,ℳ ), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < ⁺, where is the tower number, there exists a mod-finite ascending chain $T_{α} : α < λ$ , hence a ψ-space with...

Local integrability of strong and iterated maximal functions

Paul Alton Hagelstein (2001)

Studia Mathematica

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Let $M_{S}$ denote the strong maximal operator. Let $M_{x}$ and $M_{y}$ denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that $\int_{Q} M_{y} M_{x} h < \infty$ but $\int_{Q} M_{x} M_{y} h = \infty$ . It is shown that if f is a function supported on Q such that $\int_{Q} M_{y} M_{x} f < \infty$ but $\int_{Q} M_{x} M_{y} f = \infty$ , then there exists a set A of finite measure in ℝ² such that $\int_{A} M_{S} f = \infty$ .

Problems on averages and lacunary maximal functions

Andreas Seeger, James Wright (2011)

Banach Center Publications

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We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H¹ to $L^{1, \infty}$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^{p}$ regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L² boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^{p}$ ...

Radial maximal function characterizations for Hardy spaces on RD-spaces

Loukas Grafakos, Liguang Liu, Dachun Yang (2009)

Bulletin de la Société Mathématique de France

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An RD-space $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type $𝒳$ having “dimension” $n$ , there exists a $p_{0} \in (n / (n + 1), 1)$ such that for certain classes of distributions, the $L^{p} (𝒳)$ quasi-norms of their radial maximal functions and grand maximal functions are equivalent when $p \in (p_{0}, \infty]$ . This result yields a radial maximal function characterization for Hardy spaces on $𝒳$ . ...

Weak-type inequalities for maximal operators acting on Lorentz spaces

Adam Osękowski (2014)

Banach Center Publications

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We prove sharp a priori estimates for the distribution function of the dyadic maximal function ℳ ϕ, when ϕ belongs to the Lorentz space $L^{p, q}$ , 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p,q as above and r ∈ [1,p], we determine the best constant $C_{p, q, r}$ such that for any $ϕ \in L^{p, q}$ , ${| | ℳ ϕ | |}_{r, \infty} \leq C_{p, q, r} {| | ϕ | |}_{p, q}$ .

MAD families and $P$ -points

Salvador García-Ferreira, Paul J. Szeptycki (2007)

Commentationes Mathematicae Universitatis Carolinae

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The Katětov ordering of two maximal almost disjoint (MAD) families $𝒜$ and $ℬ$ is defined as follows: We say that $𝒜 \leq_{K} ℬ$ if there is a function $f : ω \to ω$ such that $f^{- 1} (A) \in ℐ (ℬ)$ for every $A \in ℐ (𝒜)$ . In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called $K$ -uniform if for every $X \in ℐ {(𝒜)}^{+}$ , we have that ${𝒜 |}_{X} \leq_{K} 𝒜$ . We prove that CH implies that for every $K$ -uniform MAD family $𝒜$ there is a $P$ -point $p$ of $ω^{*}$ such that the set of all Rudin-Keisler predecessors of $p$ is dense...

Asymmetric tie-points and almost clopen subsets of $ℕ^{*}$

Alan S. Dow, Saharon Shelah (2018)

Commentationes Mathematicae Universitatis Carolinae

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A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of $ℕ^{*}$ . One especially important application, due to Veličković, was to the existence of nontrivial involutions on $ℕ^{*}$ . A tie-point of $ℕ^{*}$ has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost...

The minimal operator and the geometric maximal operator in ℝⁿ

David Cruz-Uribe, SFO (2001)

Studia Mathematica

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We prove two-weight norm inequalities in ℝⁿ for the minimal operator $f (x) = i n f_{Q ∋ x} 1 / | Q | \int_{Q} | f | d y$ , extending to higher dimensions results obtained by Cruz-Uribe, Neugebauer and Olesen [8] on the real line. As an application we extend to ℝⁿ weighted norm inequalities for the geometric maximal operator $M ₀ f (x) = {s u p}_{Q ∋ x} e x p (1 / | Q | \int_{Q} l o g | f | d x)$ , proved by Yin and Muckenhoupt [27]. We also give norm inequalities for the centered minimal operator, study powers of doubling weights and give sufficient conditions for the geometric maximal operator to be equal...

On the Rademacher maximal function

Mikko Kemppainen (2011)

Studia Mathematica

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This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The $L^{p}$ -boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σ-finite measure spaces with filtrations and the $L^{p}$ -boundedness is shown not to depend on the underlying measure space or the filtration. Martingale...

Maximal almost disjoint families of functions

Dilip Raghavan (2009)

Fundamenta Mathematicae

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We study maximal almost disjoint (MAD) families of functions in $ω^{ω}$ that satisfy certain strong combinatorial properties. In particular, we study the notions of strongly and very MAD families of functions. We introduce and study a hierarchy of combinatorial properties lying between strong MADness and very MADness. Proving a conjecture of Brendle, we show that if $c o v (ℳ) <_{}$ , then there no very MAD families. We answer a question of Kastermans by constructing a strongly MAD family from = . Next, we...

Another ⋄-like principle

Michael Hrušák (2001)

Fundamenta Mathematicae

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A new ⋄-like principle $⋄_{}$ consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that $\neg ⋄_{}$ is consistent with CH and that in many models of = ω₁ the principle $⋄_{}$ holds. As $⋄_{}$ implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that $⋄_{}$ holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁...

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

Kristóf Szarvas, Ferenc Weisz (2016)

Czechoslovak Mathematical Journal

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The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces $L_{p} (ℝ^{d})$ (in the case $p > 1$ ), but (in the case when $1 / p (\cdot)$ is log-Hölder continuous and $p_{-} = inf {p (x) : x \in ℝ^{d}} > 1$ ) on the variable Lebesgue spaces $L_{p (\cdot)} (ℝ^{d})$ , too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1, 1)$ . In the present note we generalize Besicovitch’s covering theorem for the so-called $γ$ -rectangles. We introduce a general maximal operator $M_{s}^{γ, δ}$ and with the help of generalized $Φ$ -functions, the strong-...

Sums of commuting operators with maximal regularity

Christian Le Merdy, Arnaud Simard (2001)

Studia Mathematica

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Let Y be a Banach space and let $S \subset L_{p}$ be a subspace of an $L_{p}$ space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to $S (Y) \subset L_{p} (Y)$ . We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and $e^{- t B}$ is a positive contraction...

Transference and restriction of maximal multiplier operators on Hardy spaces

Zhixin Liu, Shanzhen Lu (1993)

Studia Mathematica

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The aim of this paper is to establish transference and restriction theorems for maximal operators defined by multipliers on the Hardy spaces $H^{p} (ℝ^{n})$ and $H^{p} (^{n})$ , 0 < p ≤ 1, which generalize the results of Kenig-Tomas for the case p > 1. We prove that under a mild regulation condition, an $L^{\infty} (ℝ^{n})$ function m is a maximal multiplier on $H^{p} (ℝ^{n})$ if and only if it is a maximal multiplier on $H^{p} (^{n})$ . As an application, the restriction of maximal multipliers to lower dimensional Hardy spaces is considered. ...

Special sets of reals and weak forms of normality on Isbell--Mrówka spaces

Vinicius de Oliveira Rodrigues, Victor dos Santos Ronchim, Paul J. Szeptycki (2023)

Commentationes Mathematicae Universitatis Carolinae

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We recall some classical results relating normality and some natural weakenings of normality in $Ψ$ -spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$ -sets, $λ$ -sets and $σ$ -sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being $ℵ_{0}$ -separated. This new class fits between $λ$ -sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $𝒜$ being...

Maximal regularity of discrete and continuous time evolution equations

Sönke Blunck (2001)

Studia Mathematica

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We consider the maximal regularity problem for the discrete time evolution equation $u_{n + 1} - T u ₙ = f ₙ$ for all n ∈ ℕ₀, u₀ = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup ${(T ⁿ)}_{n \in ℕ ₀}$ and of the resolvent R(λ,T), secondly by the maximal regularity of the continuous time evolution equation u’(t) - Au(t) = f(t) for all t > 0, u(0) = 0, where A:= T - I. By recent...

An observation on spaces with a zeroset diagonal

Wei-Feng Xuan (2020)

Mathematica Bohemica

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We say that a space $X$ has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. A space $X$ has a zeroset diagonal if there is a continuous mapping $f : X^{2} \to [0, 1]$ with $Δ_{X} = f^{- 1} (0)$ , where $Δ_{X} = {(x, x) : x \in X}$ . In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most $𝔠$ .

Maximal non $λ$ -subrings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with unity. The notion of maximal non $λ$ -subrings is introduced and studied. A ring $R$ is called a maximal non $λ$ -subring of a ring $T$ if $R \subset T$ is not a $λ$ -extension, and for any ring $S$ such that $R \subset S \subseteq T$ , $S \subseteq T$ is a $λ$ -extension. We show that a maximal non $λ$ -subring $R$ of a field has at most two maximal ideals, and exactly two if $R$ is integrally closed in the given field. A determination of when the classical $D + M$ construction is a maximal non $λ$ -domain is given. A necessary condition...

A solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita (2005)

Fundamenta Mathematicae

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In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $α \leq 2^{}$ , a topological group G such that $G^{γ}$ is countably compact for all cardinals γ < α, but $G^{α}$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M A_{c o u n t a b l e}$ . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M A_{c o u n t a b l e}$ . However, the question has...