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We consider Keldysh-type operators, $ P = x_1 D_{x_1}^2 + a (x) D_{x_1} + Q (x, D_{x} ) $, $ x = ( x_1, x) $ with analytic coefficients, and with $ Q ( x, D_{x} ) $ second order, principally real and elliptic in $ D_{x} $ for $ x $ near zero. We show that if $ P u =f $, $ u in C^infty $, and $ f $ is analytic in a neighbourhood of $ 0 $ then $ u $ is analytic in a neighbourhood of $ 0 $. This is a consequence of a microlocal result valid for operators of any order with Lagrangian radial sets. Our result proves a generalized version of a conjecture made by the second author and Lebeau and has applications to scattering theory.
Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an asymptotic
Motivated by applications to stochastic differential equations, an extension of H{o}rmanders hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established using point-w
Let (X j , d j , $mu$ j), j = 0, 1,. .. , m be metric measure spaces. Given 0 < p $kappa$ $le$ $infty$ for $kappa$ = 1,. .. , m and an analytic family of multilinear operators T z : L p 1 (X 1) x $bullet$ $bullet$ $bullet$ L p m (X m) $rightarrow$ L
We consider the passive scalar equations subject to shear flow advection and fractional dissipation. The enhanced dissipation estimates are derived. For classical passive scalar equation ($gamma=1$), our result agrees with the sharp one obtained in cite{Wei18}
In this note, we study the boundedness of integral operators $I_{g}$ and $T_{g}$ on analytic Morrey spaces. Furthermore, the norm and essential norm of those operators are given.