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We discuss some aspects of the theory of subelliptic estimates.
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We obtain uniform estimates for the canonical solution to $barpartial u=f$ on the Cartesian product of smoothly bounded planar domains, when $f$ is continuous up to the boundary. This generalizes Landuccis result for the bidisc toward higher dimensional product domains.
We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type. Specifically, we show that there exist positive constants $C_1$, $C_2$ and a polynomial correction function $Q_t$ on $G$ such that $$C_1 Q_t e^{-frac{d^2}{4t}} le p_t le C_2 Q_t e^{-frac{d^2}{4t}}$$ where $p_t$ is the heat kernel, and $d$ the Carnot-Caratheodory distance on $G$. We also obtain similar bounds on the norm of its subelliptic gradient $| abla p_t|$. Along the way, we record explicit formulas for the distance function $d$ and the subriemannian geodesics of H-type groups.
In this paper we investigate the regularity properties of weighted Bergman projections for smoothly bounded pseudo-convex domains of finite type in $mathbb{C}^{n}$. The main result is obtained for weights equal to a non negative rational power of the absolute value of a special defining function $rho$ of the domain: we prove (weighted) Sobolev-$L^{p}$ and Lipchitz estimates for domains in $mathbb{C}^{2}$ (or, more generally, for domains having a Levi form of rank $geq n-2$ and for decoupled domains) and for convex domains. In particular, for these defining functions, we generalize results obtained by A. Bonami & S. Grellier and D. C. Chang & B. Q. Li. We also obtain a general (weighted) Sobolev-$L^{2}$ estimate.
In this note we show that the weighted $L^{2}$-Sobolev estimates obtained by P. Charpentier, Y. Dupain & M. Mounkaila for the weighted Bergman projection of the Hilbert space $L^{2}left(Omega,dmu_{0}right)$ where $Omega$ is a smoothly bounded pseudoconvex domain of finite type in $mathbb{C}^{n}$ and $mu_{0}=left(-rho_{0}right)^{r}dlambda$, $lambda$ being the Lebesgue measure, $rinmathbb{Q}_{+}$ and $rho_{0}$ a special defining function of $Omega$, are still valid for the Bergman projection of $L^{2}left(Omega,dmuright)$ where $mu=left(-rhoright)^{r}dlambda$, $rho$ being any defining function of $Omega$. In fact a stronger directional Sobolev estimate is established. Moreover similar generalizations are obtained for weighted $L^{p}$-Sobolev and lipschitz estimates in the case of pseudoconvex domain of finite type in $mathbb{C}^{2}$ and for some convex domains of finite type.
Let $phi$ be a normalized convex function defined on open unit disk $mathbb{D}$. For a unified class of normalized analytic functions which satisfy the second order differential subordination $f(z)+ alpha z f(z) prec phi(z)$ for all $zin mathbb{D}$, we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.
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