arXiv subjects

Troels Roussau Johansen

Publications and source records attributed to Troels Roussau Johansen.

8 recordsLinked to original sources

arXiv2016

Weighted inequalities and uncertainty principles for the $\boldsymbol{(k,a)}$-generalized Fourier transform

We obtain several versions of the Hausdorff-Young and Hardy-Littlewood inequalities for the $(k,a)$-generalized Fourier transform recently investigated at length by Ben Saï d, Kobayashi, and Ø rsted. We also obtain a number of weighted inequalities - in particular Pitt's inequality - that have application to uncertainty principles. Specifically we obtain several analogs of the Heisenberg-Pauli-Weyl principle for $L^p$-functions, local Cowling-Price-type inequalities, Donoho-Stark-type inequalities and qualitative extensions. We finally use the Hausdorff-Young inequality as a means to obtain entropic uncertainty inequalities.

math.CA↗

arXiv2015

Remarks on the inverse Cherednik-Opdam transform on the real line

We obtain Hausdorff-Young inequalities for the one-dimensional Cherednik--Opdam transform and its inverse, and we establish a real Paley-Wiener theorem for its inverse that generalizes an analogous result by N. B. Andersen for the Jacobi transform.

math.CA↗

arXiv2015

Hardy--Littlewood inequalities for the Heckman--Opdam transform

We establish Hardy--Littlewood inequalities for the Heckman--Opdam transform associated to a general root datum $(\mathfrak{a},Σ,m)$ that generalizes an analogous result for the spherical Fourier transform on a Riemannian symmetric space of the non-compact type due to Eguchi and Kumahara. In particular we obtain a more precise Hausdorff--Young inequality that generalizes a recent result due to Narayanan, Pasquale, and Pusti.

math.CA↗

arXiv2011

On a Class of Non-Integrable Multipliers for the Jacobi Transform

We show that a bounded function $m$ on $\R$ not necessarily integrable at infinity may still yield $L^p$-bounded convolution operators for the Jacobi transform if the nontangential boundary values of $ωm$ along the edges of a certain strip in $\C$ yield Euclidean Fourier multipliers, for $ω$ suitably defined. This partially generalizes similar results by Giulini, Mauceri, and Meda (on rank one symmetric spaces) and Astengo (on Damek--Ricci spaces).

math.CA↗

arXiv2011

$L^p$-results for fractional integration and multipliers for the Jacobi transform

We use precise asymptotic expansions for Jacobi functions $ϕ^{(α,β)}_λ$ parameters $α$, $β$ satisfying $α>1/2$, $α>β>-1/2$, to generalizing classical Hörmander-type multiplier theorem for the spherical transform on a rank one Riemannian symmetric space (by Clerc/Stein and Stanton/Tomas) to the framework of Jacobi analysis. In particular, multiplier results for the spherical transform on Damek--Ricci spaces are subsumed by this approach, and it yields multiplier results for the hypergeometric `Heckman--Opdam transform' associated with a rank one root system. We obtain near-optimal $L^p-L^q$ estimates for the integral operator associated with the convolution kernel $m_a:λ\mapsto(λ^2+ρ^2)^{-a/2}$, $a>0$.

math.CA↗

arXiv2011

Almost Everywhere Convergence of the inverse Jacobi Transform and Endpoint Results for a Disc Multiplier

We introduce a maximal operator for the natural analogue of the disc multiplier in the framework of Jacobi analysis and determine the mapping properties of said maximal operator, including weak type endpoint results. In particular we obtain sharp information on the almost everywhere convergence of the inverse Jacobi transform.

math.CA↗

arXiv2009

An index theorem for Wiener--Hopf operators

We study multivariate generalisations of the classical Wiener--Hopf algebra, which is the C$^*$-algebra generated by the Wiener--Hopf operators, given by the convolutions restricted to convex cones. By the work of Muhly and Renault, this C$^*$-algebra is known to be isomorphic to the reduced C$^*$-algebra of a certain restricted action groupoid. In a previous paper, we have determined a composition series of this C$^*$-algebra, and compute the $K$-theory homomorphisms induced by the `symbol' maps given by the subquotients of the composition series in terms of the analytical index of a continuous family of Fredholm operators. In this paper, we obtain a topological expression for these index maps in terms of geometric-topological data naturally associated to the underlying convex cone. The resulting index formula is expressed in the framework of Kasparov's bivariant $KK$-theory. Our proof relies heavily on groupoid methods.

math.OA↗

arXiv2009

Spectrum and Analytical Indices of the C*-Algebra of Wiener-Hopf Operators

We study multivariate generalisations of the classical Wiener--Hopf algebra, which is the C$^*$-algebra generated by the Wiener--Hopf operators, given by the convolutions restricted to convex cones. By the work of Muhly and Renault, this C$^*$-algebra is known to be isomorphic to the reduced C$^*$-algebra of a certain restricted action groupoid, given by the action of Euclidean space on a certain compactification. Using groupoid methods, we construct composition series for the Wiener--Hopf C$^*$-algebra by a detailed study of this compactification. We compute the spectrum, and express homomorphisms in $K$-theory induced by the symbol maps which arise by the subquotients of the composition series in analytical terms. Namely, these symbols maps turn out to be given by an analytical family index of a continuous family of Fredholm operators. In a subsequent paper, we also obtain a topological expression of these indices.

math.OA↗