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A.e. convergence of anisotropic partial Fourier integrals on Euclidean spaces and Heisenberg groups

D. Müller, E. Prestini (2010)

Colloquium Mathematicae

We define partial spectral integrals S R on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by l o g ( 2 + L α ) f L ² , where L α is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that S R f ( x ) converges a.e. to f(x) as R → ∞.

A.e. convergence of spectral sums on Lie groups

Christopher Meaney, Detlef Müller, Elena Prestini (2007)

Annales de l’institut Fourier

Let be a right-invariant sub-Laplacian on a connected Lie group G , and let S R f : = 0 R d E λ f , R 0 , denote the associated “spherical partial sums,” where = 0 λ d E λ is the spectral resolution of . We prove that S R f ( x ) converges a.e. to f ( x ) as R under the assumption log ( 2 + ) f L 2 ( G ) .

Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier

Troels Roussau Johansen (2011)

Studia Mathematica

The maximal operator S⁎ for the spherical summation operator (or disc multiplier) S R associated with the Jacobi transform through the defining relation S R f ^ ( λ ) = 1 | λ | R f ̂ ( t ) for a function f on ℝ is shown to be bounded from L p ( , d μ ) into L p ( , d μ ) + L ² ( , d μ ) for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from L p , 1 ( , d μ ) into L p , ( , d μ ) + L ² ( , d μ ) . In particular S R f ( t ) R > 0 converges almost everywhere towards f, for f L p ( , d μ ) , whenever (4α + 4)/(2α + 3) < p ≤ 2.

Functional calculus in weighted group algebras.

Jacek Dziubanski, Jean Ludwig, Carine Molitor-Braun (2004)

Revista Matemática Complutense

Let G be a compactly generated, locally compact group with polynomial growth and let ω be a weight on G. We look for general conditions on the weight which allow us to develop a functional calculus on a total part of L1(G,ω). This functional calculus is then used to study harmonic analysis properties of L1(G,ω), such as the Wiener property and Domar's theorem.

Multiplicative Systems on Ultra-Metric Spaces

Memic, Nacima (2010)

Mathematica Balkanica New Series

AMS Subj. Classification: MSC2010: 42C10, 43A50, 43A75We perform analysis of certain aspects of approximation in multiplicative systems that appear as duals of ultrametric structures, e.g. in cases of local fields, totally disconnected Abelian groups satisfying the second axiom of countability or more general ultrametric spaces that do not necessarily possess a group structure. Using the fact that the unit sphere of a local field is a Vilenkin group, we introduce a new concept of differentiation in...

On convolution squares of singular measures

Sanjiv K. Gupta, Kathryn E. Hare (2004)

Colloquium Mathematicae

We prove that for every compact, connected group G there is a singular measure μ such that the Fourier series of μ*μ converges uniformly on G. Our results extend the earlier results of Saeki and Dooley-Gupta.

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