Fractional integration and fractional differentiation for 𝑑-dimensional Jacobi expansions

Abstract

In this paper we consider an alternative orthogonal decomposition of the space L 2 associated to the d-dimensional Jacobi measure in order to obtain an analogous result to P.A. Meyer's Multipliers Theorem for d-dimensional Jacobi expansions. Then we define and study the Fractional Integral, the Fractional Derivative and the Bessel potentials induced by the Jacobi operator. We also obtain a characterization of the Sobolev or potential spaces and a version of Calderón's reproduction formula for the d-dimensional Jacobi measure. R ÉSUM É. Dans cet article nous considérons une décomposition orthogonale alternative de l'espace L 2 associée à la mesure de Jacobi d-dimensionelle afin d'obtenir de résultat analogues au Théorème des Multiplicateurs de P.A. Meyer pour les développements d-dimensionnels de Jacobi. Nous définissons et étudions l'integral Fractionnaire, la dérivée Fractionnaire et les potentiels de Bessel induits par l'operateur de Jacobi. Nous obtenons ègalement une charactérisation des espaces

On potential spaces related to Jacobi expansions

“…We believe that our results enrich the line of research concerning Sobolev and potential spaces related to classical discrete and continuous orthogonal expansions, see in particular [3,4,6,7,9,11,20]; see also [1,2] where some results on Jacobi potential spaces can be found, though in a different Jacobi setting and with a different approach from ours. We point out that intimately connected to potential spaces are potential operators, and in the above-mentioned contexts they were studied intensively and thoroughly in the recent past.…”

“…Moreover, H α,β t f (θ) is always a smooth function of (t, θ) ∈ (0, ∞) × (0, π). All this can be verified with the aid of the bounds, see (2) and [14,Section 2],…”

Operators associated with the Jacobi semigroup

“…Also, we give very simple applications of Riesz-Jacobi transforms defined in [14]. Some similar results were obtained in [3], by use of a modified Wiener decomposition. However, we get our results in a direct way, using the classical Jacobi-Wiener decomposition.…”

Variable exponent Sobolev spaces associated with Jacobi expansions

“…We define B(Ω) as the subset of P(Ω) that consists of all those measurable functions p such that the maximal operator M is bounded from L p(·) (Ω) into itself. Diening [16,Theorem 3.5] proved that if Ω is a bounded subset of R n , p ∈ P(Ω) and there exists C > 0 such that (1) |p(x) − p(y)| ≤ C − log |x − y| , x, y ∈ Ω, |x − y| ≤ 1/2, then p ∈ B(Ω). Many classical operators in harmonic analysis (maximal operator, singular integrals, Fourier multipliers, commutators, fractional integrals, ...) have been studied in variable L p(·) -spaces (see, for instance, [15], [17], [18] and [39]).…”