Linear and Non-Linear Waves http://hdl.handle.net/20.500.11824/4 2019-08-22T01:24:38Z Hypocoercivity of linear kinetic equations via Harris's Theorem http://hdl.handle.net/20.500.11824/1003 Hypocoercivity 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. 2019-02-27T00:00:00Z Bilinear identities involving the $k$-plane transform and Fourier extension operators http://hdl.handle.net/20.500.11824/996 Bilinear 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 $\mathbb{R}^2$. 2019-01-01T00:00:00Z Endpoint Sobolev continuity of the fractional maximal function in higher dimensions http://hdl.handle.net/20.500.11824/995 Endpoint 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. 2019-01-01T00:00:00Z A Hardy-type inequality and some spectral characterizations for the Dirac–Coulomb operator http://hdl.handle.net/20.500.11824/991 A 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