Toeplitz Operators whose Symbols Are Borel Measures

In this paper, we are concerned with Toeplitz operators whose symbols are complex Borel measures. When a complex Borel measure μ on the unit circle is given, we give a formal definition of a Toeplitz operator T μ with symbol μ , as an unbounded linear operator on the Hardy space. We then study various properties of T μ . Among them, there is a theorem that the domain of T μ is represented by a trichotomy. Also, it was shown that if the domain of T μ contains at least one polynomial, then T μ is densely defined. In addition, we give evidence for the conjecture that T μ with a singular measure μ reduces to a trivial linear operator.

Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F(x)=exp{∫0Tθ(t,x(t))dt} successfully exist under the certain condition, where θ(t,u)=∫Rexp{iuv}dσt(v) is a Fourier–Stieltjes transform of a complex Borel measure σt∈M(R) and M(R) is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of F(x) sucessfully holds on the Wiener space.

AN INTEGRAL REPRESENTATION FOR BESOV AND LIPSCHITZ SPACES

It is well known that functions in the analytic Besov space $B_{1}$ on the unit disk $\mathbb{D}$ admit an integral representation $$\begin{eqnarray}f(z)=\int _{\mathbb{D}}\frac{z-w}{1-z\overline{w}}\,d{\it\mu}(w),\end{eqnarray}$$ where ${\it\mu}$ is a complex Borel measure with $|{\it\mu}|(\mathbb{D})<\infty$. We generalize this result to all Besov spaces $B_{p}$ with $0<p\leq 1$ and all Lipschitz spaces ${\rm\Lambda}_{t}$ with $t>1$. We also obtain a version for Bergman and Fock spaces.

FAMILIES OF FRACTIONAL FANTAPPIÈ TRANSFORMS

Let Bn denote the unit ball in ℂn, n≥1. Given an α>0, let ℱα(n) denote the class of functions defined for z∈Bn by integrating the kernel (1−〈z,w〉)−α against a complex Borel measure dμ(w), w∈Bn. The family ℱ0(n) corresponds to the logarithmic kernel log (1/(1−〈z,w〉)). Various properties of the spaces ℱα(n), α≥0, are obtained. In particular, pointwise multiplies for ℱα(n) are investigated.

Absolutely continuous measures and compact composition operator on spaces of Cauchy transforms

The analytic self-map of the unit diskD,φis said to induce a composition operatorCφfrom the Banach spaceXto the Banach spaceYifCφ(f)=f∘φ∈Yfor allf∈X. Forz∈Dandα>0, the families of weighted Cauchy transformsFαare defined byf(z)=∫TKxα(z)dμ(x), whereμ(x)is complex Borel measure,xbelongs to the unit circleT, and the kernelKx(z)=(1−x¯z)−1. In this paper, we will explore the relationship between the compactness of the composition operatorCφacting onFαand the complex Borel measuresμ(x).

Multipliers of Fractional Cauchy Transforms and Smoothness Conditions

This paper studies conditions on an analytic function that imply it belongs to Mα, the set of multipliers of the family of functions given by where μ is a complex Borel measure on the unit circle and α > 0. There are two main theorems. The first asserts that if 0 < α < 1 and sup. The second asserts that if 0 < α < 1, ƒ ∈ H∞ and supt. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

Mean values and classes of harmonic functions

Let μ be a finite complex Borel measure supported on the unit circle.In this paper, we are concerned with the characterization of the sets of functions satisfying the generalized mean value equation of the form.and for all ξ ∈ , | ξ | = R for some fixed R > 0.