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We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube $I^{M} = {[0, 1)}^{M}$ , with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on $I^{M}$ ; (3) the measures have the convolution property that $μ * L^{p} \subseteq L^{p + ε}$ for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then $μ * L^{p} \subseteq L^{q}$ for any measure μ in our class.

A convolution property of the Cantor-Lebesgue measure

Daniel M. Oberlin (1982)

Colloquium Mathematicae

A convolution property of the Cantor-Lebesgue measure, II

Daniel M. Oberlin (2003)

Colloquium Mathematicae

For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from $L^{p} ()$ to $L^{q} ()$ . We also give a condition on p which is necessary if this operator maps $L^{p} ()$ into L²().

A few remarks on the results of Rosinski and Suchanecki concerning unconditional convergence and $C$ -sequences

W. A. Woyczynski (1974/1975)

Séminaire Analyse fonctionnelle (dit "Maurey-Schwartz")

A Heat Semigroup Version of Bernstein's Theorem on Lie Groups.

G.I. Gaudry, S. Meda, R. Pini (1990)

Monatshefte für Mathematik

A Large Bi-Invariant Nuclear Function Space on a Locally Compact Group.

Johan F. Aarnes (1971)

Mathematica Scandinavica

A multiplier counter-example for mixed-norm spaces

Charles McCarthy (1974)

Studia Mathematica

A new proof of the $L^{p}$ -conjecture for locally compact abelian groups

David L. Johnson (1982)

Colloquium Mathematicae

A new semilattice of function algebras and its Boolean form on a lattice of groups.

H. Sedaghat (1993)

Semigroup forum

A numerical invariant for finitely generated groups via actions on graphs.

William L. Paschke (1993)

Mathematica Scandinavica

À propos des convoluteurs d'un groupe quotient

A. Derighetti (1982)

Publications du Département de mathématiques (Lyon)

A remark on a Marcinkiewicz-Hörmander multiplier theorem for some non-differential convolution operators

Jacek Dziubański (1989)

Colloquium Mathematicae

A Remark on Continuity of Translation Invariant Functionals on L... (G) for Compact Lie Groups G.

Jaroslaw Krawczyk (1990)

Monatshefte für Mathematik

A Remark on Discontinuous Translation Invariant Functionals on L... (G) for Certain Compact Groups G.

W. Hebisch, J. Krawczyk (1992)

Monatshefte für Mathematik

A remark on the asymmetry of convolution operators

Saverio Giulini (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A convolution operator, bounded on $L^{q}(\mathbb{R}^{n})$ , is bounded on $L^{p}(\mathbb{R}^{n})$ , with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace $\mathbb{R}^{n}$ with a general non-commutative locally compact group $G$ . In this paper we give a simple construction of a convolution operator on a suitable compact group $G$ , wich is bounded on $L^{q}(G)$ for every $q\in[2,\infty)$ and is unbounded on $L^{p}(G)$ if $p\in[1,2)$ .

A semi-group of probability measures with non-smooth differentiable densities on a Lie group

Paweł Głowacki, Andrzej Hulanicki (1987)

Colloquium Mathematicae

A Simplified Constructive Proof of the Existence and Uniqueness of Haar Measure.

E.M. Alfsen (1963)

Mathematica Scandinavica

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