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Sonmez Sahutoglu

Publications and source records attributed to Sonmez Sahutoglu.

At least 37 records · Page 2Linked to original sources
arXiv2017

A local weighted Axler-Zheng theorem in $\mathbb{C}^n$

The well-known Axler-Zheng theorem characterizes compactness of finite sums of finite products of Toeplitz operators on the unit disk in terms of the Berezin transform of these operators. Subsequently this theorem was generalized to other domains and appeared in different forms, including domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-Neumann operator $N$ is compact. In this work we remove the assumption on $N$, and we study weighted Bergman spaces on smooth bounded pseudoconvex domains. We prove a local version of the Axler-Zheng theorem characterizing compactness of Toeplitz operators in the algebra generated by symbols continuous up to the boundary in terms of the behavior of the Berezin transform at strongly pseudoconvex points. We employ a Forelli-Rudin type inflation method to handle the weights.

math.CV↗
arXiv2017

Schatten class Hankel and $\overline{\partial}$-Neumann operators on pseudoconvex domains in $\mathbb{C}^n$

Let $Ω$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $φ$ be a holomorphic function on $Ω$ that is $C^2$-smooth on the closure of $Ω$. We prove that if $H_{\overlineφ}$ is in Schatten $p$-class for $p\leq 2n$ then $φ$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $Ω$ is not Hilbert-Schmidt.

math.CV↗
arXiv2017

Compactness of Hankel operators with continuous symbols

Let $Ω$ be a bounded convex Reinhardt domain in $\mathbb{C}^2$ and $ϕ\in C(\barΩ)$. We show that the Hankel operator $H_ϕ$ is compact if and only if $ϕ$ is holomorphic along every non-trivial analytic disc in the boundary of $Ω$.

math.CV↗
arXiv2013

Compactness of the dbar-Neumann operator and commutators of the Bergman projection with continuous functions

Let D be a bounded pseudoconvex domain in $C^n, n\geq 2, 0\leq p\leq n,$ and $1\leq q\leq n-1.$ We show that compactness of the dbar-Neumann operator, $N_{p,q+1},$ on square integrable (p,q+1)-forms is equivalent to compactness of the commutators $[P_{p,q}, \bar{z}_j]$ on square integrable dbar-closed (p,q)-forms for $1\leq j\leq n$ where $P_{p,q}$ is the Bergman projection on (p,q)-forms. We also show that compactness of the commutator of the Bergman projection with functions continuous on the closure percolates up in the dbar-complex on dbar-closed forms and square integrable holomorphic forms.

math.CV↗
arXiv2012

Continuity of Plurisubharmonic Envelopes in $\mathbb{C}^2$

We show that in $\mathbb{C}^2$ if the set of strongly regular points are closed in the boundary of a smooth bounded pseudoconvex domain, then the domain is c-regular, that is, the plurisubharmonic upper envelopes of functions continuous up to the boundary are continuous on the closure of the domain. Using this result we prove that smooth bounded pseudoconvex Reinhardt domains in $\mathbb{C}^{2}$ are $c$-regular.

math.CV↗
arXiv2012

Compactness of products of Hankel operators on convex Reinhardt domains in C^2

Let D be a piecewise smooth bounded convex Reinhardt domain in C^2. Assume that the symbols f and g are continuous on the closure of D and harmonic on the disks in the boundary of D. We show that if the product of Hankel operators H^*_f H_g is compact on the Bergman space of D, then on any disk in the boundary of D, either f or g is holomorphic.

math.CV↗
arXiv2011

Localization of compactness of Hankel operators on pseudoconvex domains

We prove the following localization for compactness of Hankel operators on Bergman spaces. Assume that D is a bounded pseudoconvex domain in C^n, p is a boundary point of D and B(p,r) is a ball centered at p with radius r so that U=D\cap B(p,r) is connected. We show that if the Hankel operator H^D_f is compact on A^2(D) (the symbols f is C^1 on the closure of D) then H^U_f is compact on A^2(U) where A^2(D) and A^2(U) denote the Bergman spaces on D and U, respectively.

math.CV↗
arXiv2011

On compactness of the dbar-Neumann problem and Hankel operators

Let $\D=\D_1\setminus \Dc_2$, where $\D_1$ and $\D_2$ are two smooth bounded pseudoconvex domains in $\C^n, n\geq 3,$ such that $\Dc_2\subset \D_1.$ Assume that the $\dbar$-Neumann operator of $\D_1$ is compact and the interior of the Levi-flat points in the boundary of $\D_2$ is not empty (in the relative topology). Then we show that the Hankel operator on $\D$ with symbol $ϕ, H^{\D}_ϕ,$ is compact for every $ϕ\in C(\Dc)$ but the $\dbar$-Neumann operator on $\D$ is not compact.

math.CV↗
arXiv2011

Compactness of products of Hankel operators on the polydisk and some product domains in $\mathbb{C}^2$

Let $\mathbb{D}^n$ be the polydisk in $\mathbb{C}^n$ and the symbols $ϕ,ψ\in C(\bar{\mathbb{D}^n})$ such that $ϕ$ and $ψ$ are pluriharmonic on any $(n-1)$-dimensional polydisk in the boundary of $\mathbb{D}^{n}.$ Then $H^*_ψH_ϕ$ is compact on $A^2(\mathbb{D}^n)$ if and only if for every $1\leq j,k\leq n$ such that $j\neq k$ and any $(n-1)$-dimensional polydisk $D$, orthogonal to the $z_j$-axis in the boundary of $\mathbb{D}^n,$ either $ϕ$ or $ψ$ is holomorphic in $z_k$ on $D.$ Furthermore, we prove a different sufficient condition for compactnes of the products of Hankel operators. In $\mathbb{C}^2,$ our techniques can be used to get a necessary condition on some product domains involving annuli.

math.FA↗
arXiv2011

Strong Stein neighborhood bases

Let D be a smooth bounded pseudoconvex domain in C^n. We give several characterizations for the closure of D to have a strong Stein neighborhood basis in the sense that D has a defining function r such that {z\in C^n:r(z) 0. We also show that this condition is invariant under proper holomorphic maps that extend smoothly to the boundary.

math.CV↗
arXiv2009

Complex Singularities and the Lorenz Attractor

The Lorenz attractor is one of the best known examples of applied mathematics. However, much of what is known about it is a result of numerical calculations and not of mathematical analysis. As a step toward mathematical analysis, we allow the time variable in the three dimensional Lorenz system to be complex, hoping that solutions that have resisted analysis on the real line will give up their secrets in the complex plane. Knowledge of singularities being fundamental to any investigation in the complex plane, we build upon earlier work and give a complete and consistent formal development of complex singularities of the Lorenz system using {\it psi series}. The psi series contain two undetermined constants. In addition, the location of the singularity is undetermined as a consequence of the autonomous nature of the Lorenz system. We prove that the psi series converge, using a technique that is simpler and more powerful than that of Hille, thus implying a two parameter family of singular solutions of the Lorenz system. We pose three questions, answers to which may bring us closer to understanding the connection of complex singularities to Lorenz dynamics.

math.DS↗
arXiv2009

Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains

Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let f be a function smooth up to the boundary on a smooth bounded pseudoconvex domain D in C^n. We show that, if D is convex or the Levi form of the boundary of D is of rank at least n-2, then compactness of the Hankel operator H_f implies that f is holomorphic "along" analytic discs in the boundary. Furthermore, when D is convex in C^2 we show that the condition on f is necessary and sufficient for compactness of H_f

math.CV↗
arXiv2007

A remark on irregularity of the dbar-Neumann problem on non-smooth domains

It is an observation due to J.J. Kohn that for a smooth bounded pseudoconvex domain D in $C^n$ there exists s>0 such that the dbar-Neumann operator on D maps $W^s_{(0,1)}(D)$ (the space of $(0,1)$-forms with coefficient functions in $L^2$-Sobolev space of order s) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain D in $C^2$, smooth except at one point, whose dbar-Neumann operator is not bounded on $W^s_{(0,1)}(D)$ for any s>0.

math.CV↗