Linear and Non-Linear Waveshttp://hdl.handle.net/20.500.11824/42019-08-22T00:35:08Z2019-08-22T00:35:08ZHypocoercivity of linear kinetic equations via Harris's TheoremCañizo J. A.Cao C.Evans J.Yoldaş H.http://hdl.handle.net/20.500.11824/10032019-08-09T01:00:10Z2019-02-27T00:00:00ZHypocoercivity of linear kinetic equations via Harris's Theorem
Cañizo J. A.; Cao C.; Evans J.; Yoldaş H.
We study convergence to equilibrium of the linear relaxation
Boltzmann (also known as linear BGK) and the linear Boltzmann
equations either on the torus
$(x,v) \in \mathbb{T}^d \times \mathbb{R}^d$ or on the whole space
$(x,v) \in \mathbb{R}^d \times \mathbb{R}^d$ with a confining
potential. We present explicit convergence results in total
variation or weighted total variation norms (alternatively $L^1$ or
weighted $L^1$ norms). The convergence rates are exponential when
the equations are posed on the torus, or with a confining potential
growing at least quadratically at infinity. Moreover, we give
algebraic convergence rates when subquadratic potentials
considered. We use a method from the theory of Markov processes
known as Harris's Theorem.
2019-02-27T00:00:00ZBilinear identities involving the $k$-plane transform and Fourier extension operatorsBeltran D.Vega L.http://hdl.handle.net/20.500.11824/9962019-07-27T01:00:12Z2019-01-01T00:00:00ZBilinear identities involving the $k$-plane transform and Fourier extension operators
Beltran D.; Vega L.
We prove certain $L^2(\mathbb{R}^n)$ bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the $k$-plane transform. As the estimates are $L^2$-based, they follow from bilinear identities: in particular, these are the analogues of a known identity for paraboloids, and may be seen as higher-dimensional versions of the classical $L^2(\mathbb{R}^2)$-bilinear identity for Fourier extension operators associated to curves in $\mathbb{R}^2$.
2019-01-01T00:00:00ZEndpoint Sobolev continuity of the fractional maximal function in higher dimensionsBeltran D.Madrid J.http://hdl.handle.net/20.500.11824/9952019-07-27T01:00:10Z2019-01-01T00:00:00ZEndpoint Sobolev continuity of the fractional maximal function in higher dimensions
Beltran D.; Madrid J.
We establish continuity mapping properties of the non-centered fractional maximal operator $M_{\beta}$ in the endpoint input space $W^{1,1}(\mathbb{R}^d)$ for $d \geq 2$ in the cases for which its boundedness is known. More precisely, we prove that for $q=d/(d-\beta)$ the map $f \mapsto |\nabla M_\beta f|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$ for $ 0 < \beta < 1$ if $f$ is radial and for $1 \leq \beta < d$ for general $f$. The results for $1\leq \beta < d$ extend to the centered counterpart $M_\beta^c$. Moreover, if $d=1$, we show that the conjectured boundedness of that map for $M_\beta^c$ implies its continuity.
2019-01-01T00:00:00ZA Hardy-type inequality and some spectral characterizations for the Dirac–Coulomb operatorCassano B.Pizzichillo F.Vega L.http://hdl.handle.net/20.500.11824/9912019-07-04T01:00:09Z2019-06-01T00:00:00ZA Hardy-type inequality and some spectral characterizations for the Dirac–Coulomb operator
Cassano B.; Pizzichillo F.; Vega L.
We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials $\mathbf V$ of Coulomb type: we characterise its eigenvalues in terms of the Birman—Schwinger principle and we bound its discrete spectrum from below, showing that the ground-state energy is reached if and only if $\mathbf V$ verifies some rigidity conditions. In the particular case of an electrostatic potential, these imply that $\mathbf V$ is the Coulomb potential.
2019-06-01T00:00:00Z