Dan Grubb

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Università di Bologna

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Universidad Nacional del Sur

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Papers by Dan Grubb

A variant of Hörmander's conditionfor singular integrals

Colloquium Mathematicum, 1997

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Irreducible partitions and the construction of quasi-measures

Transactions of the American Mathematical Society, 2001

A quasi-measure is a non-subadditive measure defined on only open or closed subsets of a compact ... more A quasi-measure is a non-subadditive measure defined on only open or closed subsets of a compact Hausdorf space. We investigate the nature of irreducible partitions as defined by Aarnes and use the results to construct quasi-measures when g(X) = 1. The cohomology ring is an important tool for this investigation.

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Signed quasi-measures and dimension theory

Proceedings of the American Mathematical Society, 1999

A quasi-linear functional on C ( X ) C(X) is a real-valued function that is linear on each closed... more A quasi-linear functional on C ( X ) C(X) is a real-valued function that is linear on each closed, singly generated subalgebra and is norm bounded. We show that if the covering dimension dim ⁡ X ≤ 1 \dim X\leq 1 , then every quasi-linear functional on C ( X ) C(X) is, in fact, linear. We do this by considering an associated set function, called a quasi-measure, and ask when such a set function can be extended to be a measure.

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Products of quasi-measures

Proceedings of the American Mathematical Society, 1996

A quasi-state is a positive functional on C ( X ) C(X) that is only assumed to be linear on singl... more A quasi-state is a positive functional on C ( X ) C(X) that is only assumed to be linear on singly-generated subalgebras. We consider the “iterated integral” of two quasi-states and determine when this gives a quasi-state on the product space. We also provide explicit formulas for the corresponding quasi-measures in case it does. Finally, we show the general failure of Fubini’s Theorem for quasi-states.

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Additivity of quasi-measures

Proceedings of the American Mathematical Society, 1998

We prove that quasi-measures on compact Hausdorff spaces are countably additive. Contained in thi... more We prove that quasi-measures on compact Hausdorff spaces are countably additive. Contained in this result is a proof that every quasi-measure decomposes uniquely into a measure and a quasi-measure that has no smaller measure beneath it. We also show that it is consistent with the usual axioms of set-theory that quasi-measures on compact Hausdorff spaces are ℵ 1 \aleph _1 -additive. Finally, we construct an example that places strong restrictions on other forms of additivity.

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Signed Quasi-Measures

Transactions of the American Mathematical Society, 1997

Let X X be a compact Hausdorff space and let A \mathcal {A} denote the subsets of X X which are e... more Let X X be a compact Hausdorff space and let A \mathcal {A} denote the subsets of X X which are either open or closed. A quasi-linear functional is a map ρ : C ( X ) → R \rho :C(X)\rightarrow \mathbb {R} which is linear on singly generated subalgebras and such that | ρ ( f ) | ≤ M ‖ f ‖ |\rho (f)|\leq M\|f\| for some M > ∞ M>\infty . There is a one-to-one correspondence between the quasi-linear functional on C ( X ) C(X) and the set functions μ : A → R \mu :\mathcal {A}\rightarrow \mathbb {R} such that i) μ ( ∅ ) = 0 \mu (\emptyset )=0 , ii) If A , B , A ∪ B ∈ A A,B,A\cup B\in \mathcal {A} with A A and B B disjoint, then μ ( A ∪ B ) = μ ( A ) + μ ( B ) \mu (A\cup B)=\mu (A)+\mu (B) , iii) There is an M > ∞ M>\infty such that whenever { U α } \{U_\alpha \} are disjoint open sets, ∑ | μ ( U α ) | ≤ M \sum |\mu (U_\alpha )|\leq M , and iv) if U U is open and ε > 0 \varepsilon >0 , there is a compact K ⊆ U K\subseteq U such that whenever V ⊆ U ∖ K V\subseteq U\setminus...

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Certain lacunary cosine series are recurrent

Studia Mathematica, 1994

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Sets of uniqueness in compact, $0$-dimensional metric groups

Transactions of the American Mathematical Society, 1987

An investigation is made of sets of uniqueness in a compact 0 0 -dimensional space. Such sets are... more An investigation is made of sets of uniqueness in a compact 0 0 -dimensional space. Such sets are defined by pointwise convergence of sequences of functions that generalize partial sums of trigonometric series on Vilenkin groups. Several analogs of classical uniqueness theorems are proved, including a version of N. Bary’s theorem on countable unions of closed sets of uniqueness.

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Quasi-measures and image transformations on completely regular spaces

Topology and its Applications, 2004

A map ρ : C(X) → C(Y) is called quasi-linear if it is linear on each singly generated subalgebra ... more A map ρ : C(X) → C(Y) is called quasi-linear if it is linear on each singly generated subalgebra of C(X). We prove the representation theorem for positive quasi-linear functionals in terms of integration with respect to quasi-measures. We then consider positive quasi-linear maps in more generality and show that they can be described in terms of transformations of sets in the underlying spaces. We call these image transformations. It is shown how smoothness properties of the quasilinear map are reflected in the corresponding image transformation and the simplifications that occur when the underlying spaces are compact.

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Smoothness Properties of Quasi-Measures

Rocky Mountain Journal of Mathematics, 2008

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Growth Conditions for Thin Sets in Vilenkin Groups of Bounded Order

Proceedings of the American Mathematical Society, 1993

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Quasi-linear maps

Fundamenta Mathematicae, 2008

A quasi-linear map from a continuous function space C(X) is one which is linear on each singly ge... more A quasi-linear map from a continuous function space C(X) is one which is linear on each singly generated subalgebra. We show that the collection of quasi-linear functionals has a Banach space pre-dual with a natural order. We then investigate quasilinear maps between two continuous function spaces, classifying them in terms of generalized image transformations.

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Covering spaces and irreducible partitions

Fundamenta Mathematicae, 2011

An irreducible partition of a space is a partition of that space into solid sets with a certain m... more An irreducible partition of a space is a partition of that space into solid sets with a certain minimality property. Previously, these partitions were studied using the cup product in cohomology. This paper obtains similar results using the fundamental group instead. This allows the use of covering spaces to obtain information about irreducible partitions. This is then used to generalize Knudsen's construction of topological measures on the torus. We give examples of such measures that are invariant under Hamiltonian flows on certain symplectic manifolds.

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$U$-sets in compact, $0$-dimensional, metric groups

Canadian Mathematical Bulletin, 1989

This paper studies a pointwise definition of sets of uniqueness on compact, 0-dimensional, metric... more This paper studies a pointwise definition of sets of uniqueness on compact, 0-dimensional, metric groups. It is shown that this definition is equivalent for closed sets to one based on supports of pseudo functions. An analog of Rajchman's theorem is given leading to examples of sets of uniqueness.

Spaces of Quasi-Measures

Canadian Mathematical Bulletin, 1999

We give a direct proof that the space of Baire quasi-measures on a completely regular space (or t... more We give a direct proof that the space of Baire quasi-measures on a completely regular space (or the space of Borel quasi-measures on a normal space) is compact Hausdorff. We show that it is possible for the space of Borel quasi-measures on a non-normal space to be non-compact. This result also provides an example of a Baire quasi-measure that has no extension to a Borel quasi-measure. Finally, we give a concise proof of theWheeler-Shakmatov theorem, which states that if X is normal and dim(X) ≤ 1, then every quasi-measure on X extends to a measure.

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Dirichlet sets in Vilenkin groups

Acta Mathematica Hungarica, 1993

For the purposes of this paper, G win denote a compact, totally disconnected, abe]Jan, metric gro... more For the purposes of this paper, G win denote a compact, totally disconnected, abe]Jan, metric group, i.e. a Vilenkin group, and F will denote the Pontryagin dual of G. We set A(G) to be the algebra of absolutely convergent Fourier series on G and PM(G) the Banach space dual of A(G). Elements of PM(G) are called pseudomeasures. If S is a pseudomeasure, write S(7)= = (S, 7) where ( , } denotes the dual pairing of PM(G) and A(G). We call S a pseudofunction if S(7) ~ 0 as 7 ~ co in F. The collection of all pseudofunctions is called PF(G)o A pseudomeasure is said to be supported on a closed set E if it is supported as a distribution with test functions from A(G). A closed set is called a U-set if it supports no nontrivial pseudofunction. The set E is called a strong U-set if ]Jm sup )S(7)I = sup 1~'(7)1

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