arXiv subjects

Yuanhao Yan

Publications and source records attributed to Yuanhao Yan.

3 recordsLinked to original sources

arXiv2026

Some Properties of Products of Linear Fractional Composition Operators on Weighted Dirichlet Spaces

We study the Schatten class membership and essential norms of the operators $C_φC_ψ^*$ and $C_ψ^*C_φ$, where $C_φ$ and $C_ψ$ are composition operators with nonconstant linear fractional symbols, acting on weighted Dirichlet spaces $\mathcal{D}_α$ with $α>-1$. We obtain complete characterizations of their Schatten class membership in terms of the boundary behavior of $φ$ and $ψ$. In particular, the corresponding geometric conditions are independent of the Schatten exponent, and each product belongs to every Schatten class whenever its boundary condition is satisfied. We also investigate the essential norms of these products. For $α=0$ and $α>0$, we obtain exact formulas in terms of boundary contact and the derivatives of the symbols, while for $-1<α<0$ we obtain two-sided estimates by reducing the problem to positive Toeplitz operators and local averages of generalized Nevanlinna counting functions.

math.FA↗

arXiv2026

Closed Range and Essential Norms of Composition Operators on Weighted Dirichlet Spaces

We study closed range and essential norms of bounded composition operators $C_φ$ on weighted Dirichlet spaces $\mathcal{D}_α$. For $-1<α<0$, we first consider maps whose images are obtained by removing a compact subset from a simply connected subdomain of $\mathbb{D}$. In this setting, closed range is equivalent to the Reverse Carleson property and the Reverse Carleson Disk Condition for the counting measure $μ_{φ,α}$, uniform weighted area density of the image, $\mathbb{T}\subset\overline{φ(\mathbb{D})}$, and the inclusion of an outer annulus in the image. We also obtain a closed range characterization under the Uniform Tail Condition. For all $α>-1$, we establish two-sided essential norm estimates in terms of local averages of the generalized Nevanlinna counting function and an exact formula using boundary tail integrals. If the normalized counting density has vanishing oscillation at the boundary, we obtain exact formulas in terms of its boundary limsup, Berezin transform, and local averages.

math.FA↗

arXiv2025

Polynomial Approximation in Higher-Order Weighted Dirichlet Spaces

Fejér's theorem guarantees norm convergence of Cesàro means of Taylor partial sums in the Hardy space, whereas such convergence generally fails in weighted Dirichlet-type spaces, especially in the higher-order setting. In this paper, we investigate summability problems in higher-order weighted Dirichlet spaces $\widehat{\mathcal{H}}_{μ,m}$ and show that Taylor partial sums are not uniformly bounded in these spaces and may therefore diverge in norm. To restore convergence, we introduce a family of modified polynomials whose coefficients are adjusted by a suitable weight array. Under mild boundedness and variation assumptions on the weights, we establish norm convergence of the modified sums via a coefficient correspondence principle and a Local Douglas formula. As an application, when the weight measure $μ$ is a finite sum of Dirac point masses, explicit formulas for the modified coefficients are obtained, yielding a Fejér-type summability theorem for higher-order weighted Dirichlet spaces.

math.FA↗