Control of pseudodifferential operators by maximal functions via weighted inequalities

We establish general weighted L 2 inequalities for pseudodifferential operators associated to the Hörmander symbol classes S ρ,δm . Such inequalities allow one to control these operators by fractional “non-tangential” maximal functions and subsume the optimal range of Lebesgue space bounds for pseudodifferential operators. As a corollary, several known Muckenhoupt-type bounds are recovered, and new bounds for weights lying in the intersection of the Muckenhoupt and reverse Hölder classes are obtained. The proof relies on a subdyadic decomposition of the frequency space, together with applications of the Cotlar–Stein almost orthogonality principle and a quantitative version of the symbolic calculus

Pressure-induced phase transitions in AgClO4

AgClO4 has been studied under compression by x-ray diffraction and density functional theory calculations. Experimental evidence of a structural phase transition from the tetragonal structure of AgClO4 to an orthorhombic barite-type structure has been found at 5.1 GPa. The transition is supported by total-energy calculations. In addition, a second transition to a monoclinic structure is theoretically proposed to take place beyond 17 GPa. The equation of state of the different phases is reported as well as the calculated Raman-active phonons and their pressure evolution. Finally, we provide a description of all the structures of AgClO4 and discuss their relationships. The structures are also compared with those of AgCl in order to explain the structural sequence determined for AgClO4.Comment: 38 pages, 11 figures, 4 table

Sparse bounds for pseudodifferential operators

We prove sparse bounds for pseudodifferential operators associated to H\"ormander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of weighted estimates for pseudodifferential operators. The results naturally apply to the context of oscillatory Fourier multipliers, with applications to dispersive equations and oscillatory convolution kernels

Endpoint Sobolev continuity of the fractional maximal function in higher dimensions

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

Bilinear identities involving the kk-plane transform and Fourier extension operators

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 $\mathbb{R}^2$

Regularity of fractional maximal functions through Fourier multipliers

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function maps $L^p$ into a first order Sobolev space in dimensions $n \geq 5$