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Register a new userWeighted estimates for diffeomorphic extensions of homeomorphisms
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Haiqing Xu
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Let $Omega subset mbr^2$ be an internal chord-arc domain and $varphi : mbs^1 rightarrow partial Omega$ be a homeomorphism. Then there is a diffeomorphic extension $h : mbd rightarrow Omega$ of $varphi .$ We study the relationship between weighted integrability of the derivatives of $h$ and double integrals of $varphi$ and of $varphi^{-1} .$
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Let $Omega$ be an internal chord-arc Jordan domain and $varphi:mathbb SrightarrowpartialOmega$ be a homeomorphism. We show that $varphi$ has finite dyadic energy if and only if $varphi$ has a diffeomorphic extension $h: mathbb Drightarrow Omega$ which has finite energy.
Let $mathbb{X}$ be a Jordan domain satisfying hyperbolic growth conditions. Assume that $varphi$ is a homeomorphism from the boundary $partial mathbb{X}$ of $mathbb{X}$ onto the unit circle. Denote by $h$ the harmonic diffeomorphic extension of $varphi $ from $mathbb{X}$ onto the unit disk. We establish the optimal Orlicz-Sobolev regularity and weighted Sobolev estimate of $h.$ These generalize the Sobolev regularity of $h$ by Koski-Onninen [21, Theorem 3.1].
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.
We establish a weighted inequality for the Bergman projection with matrix weights for a class of pseudoconvex domains. We extend a result of Aleman-Constantin and obtain the following estimate for the weighted norm of $P$: [|P|_{L^2(Omega,W)}leq C(mathcal B_2(W))^{{2}}.] Here $mathcal B_2(W)$ is the Bekolle-Bonami constant for the matrix weight $W$ and $C$ is a constant that is independent of the weight $W$ but depends upon the dimension and the domain.
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