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Conditional Fourier-Feynman transform given infinite dimensional conditioning function on abstract Wiener space

Jae Gil Choi, Sang Kil Shim (2023)

Czechoslovak Mathematical Journal

We study a conditional Fourier-Feynman transform (CFFT) of functionals on an abstract Wiener space ( H , B , ν ) . An infinite dimensional conditioning function is used to define the CFFT. To do this, we first present a short survey of the conditional Wiener integral concerning the topic of this paper. We then establish evaluation formulas for the conditional Wiener integral on the abstract Wiener space B . Using the evaluation formula, we next provide explicit formulas for CFFTs of functionals in the Kallianpur...

Construction de p-multiplicateurs

Francisco González Vieli (1993)

Studia Mathematica

Using characteristic functions of polyhedra, we construct radial p-multipliers which are continuous over n but not continuously differentiable through S n - 1 and give a p-multiplier criterion for homogeneous functions over 2 . We also exhibit fractal p-multipliers over the real line.

Continuity of halo functions associated to homothecy invariant density bases

Oleksandra Beznosova, Paul Hagelstein (2014)

Colloquium Mathematicae

Let be a collection of bounded open sets in ℝⁿ such that, for any x ∈ ℝⁿ, there exists a set U ∈ of arbitrarily small diameter containing x. The collection is said to be a density basis provided that, given a measurable set A ⊂ ℝⁿ, for a.e. x ∈ ℝⁿ we have l i m k 1 / | R k | R k χ A = χ A ( x ) for any sequence R k of sets in containing x whose diameters tend to 0. The geometric maximal operator M associated to is defined on L¹(ℝⁿ) by M f ( x ) = s u p x R 1 / | R | R | f | . The halo function ϕ of is defined on (1,∞) by ϕ ( u ) = s u p 1 / | A | | x : M χ A ( x ) > 1 / u | : 0 < | A | < and on [0,1] by ϕ(u) = u. It is shown that the halo...

Continuous wavelet transform on semisimple Lie groups and inversion of the Abel transform and its dual.

K. Trimèche (1996)

Collectanea Mathematica

In this work we define and study wavelets and continuous wavelet transform on semisimple Lie groups G of real rank l. We prove for this transform Plancherel and inversion formulas. Next using the Abel transform A on G and its dual A*, we give relations between the continuous wavelet transform on G and the classical continuous wavelet transform on Rl, and we deduce the formulas which give the inverse operators of the operators A and A*.

Control for Schrödinger operators on 2-tori: rough potentials

Jean Bourgain, Nicolas Burq, Maciej Zworski (2013)

Journal of the European Mathematical Society

For the Schrödinger equation, ( i t + ) u = 0 on a torus, an arbitrary non-empty open set Ω provides control and observability of the solution: u t = 0 L 2 ( 𝕋 2 ) K T u L 2 ( [ 0 , T ] × Ω ) . We show that the same result remains true for ( i t + - V ) u = 0 where V L 2 ( 𝕋 2 ) , and 𝕋 2 is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for V C ( 𝕋 2 ) and conjectured for V L ( 𝕋 2 ) . The higher dimensional generalization remains open for V L ( 𝕋 n ) .

Convergence a.e. of spherical partial Fourier integrals on weighted spaces for radial functions: endpoint estimates

María J. Carro, Elena Prestini (2009)

Studia Mathematica

We prove some extrapolation results for operators bounded on radial L p functions with p ∈ (p₀,p₁) and deduce some endpoint estimates. We apply our results to prove the almost everywhere convergence of the spherical partial Fourier integrals and to obtain estimates on maximal Bochner-Riesz type operators acting on radial functions in several weighted spaces.

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