Toeplitz Operators with Special Symbols on Segal–Bargmann Spaces

We study the boundedness of Toeplitz operators on Segal–Bargmann spaces in various contexts. Using Gutzmer’s formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal–Bargmann spaces associated to Riemannian symmetric spaces of compact type.     

Segal–Bargmann and Weyl transforms on compact Lie groups

We present an elementary derivation of the reproducing kernel for invariant Fock spaces associated with compact Lie groups which, as Ólafsson and Ørsted showed in (Lie Theory and its Applicaitons in Physics. World Scientific, 1996), yields a simple proof of the unitarity of Hall’s Segal–Bargmann transform for compact Lie groups K. Further, we prove certain Hermite and character expansions for the heat and reproducing kernels on K and \({K_{\mathbb C}}\) . Finally, we introduce a Toeplitz (or Wick) calculus as an attempt to have a quantization of the functions on \({K_{\mathbb C}}\) as operators on the Hilbert space L ²(K).     

Segal–Bargmann and Weyl transforms on compact Lie groups

We present an elementary derivation of the reproducing kernel for invariant Fock spaces associated with compact Lie groups which, as ólafsson and ?rsted showed in (Lie Theory and its Applicaitons in Physics. World Scientific, 1996), yields a simple proof of the unitarity of Hall’s Segal–Bargmann transform for compact Lie groups K. Further, we prove certain Hermite and character expansions for the heat and reproducing kernels on K and K_{\mathbb C}{K_{\mathbb C}} . Finally, we introduce a Toeplitz (or Wick) calculus as an attempt to have a quantization of the functions on K_{\mathbb C}{K_{\mathbb C}} as operators on the Hilbert space L ²(K).     

Inclusions of Waterman–Shiba spaces into generalized Wiener classes

The characterization of the inclusion of Waterman–Shiba spaces ΛBV^(p)$\Lambda {\mathit{BV}}^{(p)}$ into generalized Wiener classes of functions BV(q;δ)$\mathit{BV}(q;\delta )$ is given. It uses a new and shorter proof and extends an earlier result of U. Goginava.     

Toeplitz Operators and Carleson Measures on Generalized Bargmann–Fock Spaces

We characterize bounded and compact Toeplitz operators defined on generalized Bargmann–Fock spaces.     

Weighted Shifts on Directed Semi-Trees: an Application to Creation Operators on Segal–Bargmann Spaces

We extend the notion of a weighted shift on a directed tree to the case of a more general graph which we call a directed semi-tree. Some basic properties of such operators are investigated. It is shown that a generalized creation operator on the Segal–Bargman space is unitarily isomorphic to a weighted shift on a directed semi-tree of a particular form.     

Connes–Dixmier Versus Dixmier Traces

This paper is a continuation of Pietsch (Math. Nachr. 285, 1999–2028, 2012) and (Stud. Math. 214, 37–66, 2013). Now we are able to bring in the harvest. Different sets of traces on the Marcinkiewicz operator ideal $$ \mathfrak M (H):= \Bigg\{T \in \mathfrak L (H) \colon \sup_{1 \le m < \infty} \tfrac 1{\log m +1}\sum_{n=1}^m a_n(T) < \infty \Bigg\} $$ are compared with each other. Their size is measured by means of the density character. In particular, it is shown that the set of Dixmier traces is properly larger than the set of Connes–Dixmier traces, which answers a question posed in Carey–Sukochev (Russ. Math. Surv. 61, 1039–1099, 2006, p. 1062). The proofs are based on considerations about shift-invariant functionals on a suitably chosen sequence space.     

Commutators of fractional maximal operator on generalized Orlicz–Morrey spaces

In the present paper, we shall give necessary and sufficient conditions for the Spanne and Adams type boundedness of the commutators of fractional maximal operator on generalized Orlicz–Morrey spaces, respectively. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators.     

Factorization in generalized Calogero–Moser spaces

Using a recent construction of Bezrukavnikov and Etingof, [R. Bezrukavnikov, P. Etingof, Induction and restriction functors for rational Cherednik algebras, arXiv: 0803.3639], we prove that there is a factorization of the Etingof–Ginzburg sheaf on the generalized Calogero–Moser space associated to a complex reflection group. In the case $W=S_n$, this confirms a conjecture of Etingof and Ginzburg, [P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphisms, Invent. Math 147 (2002) 243–348].     

Segal－Bargmann－Hall Transform and Geometric Quantization

Using geometric methods, Hall has proved that the Segal-Bargmann transform for a con-nected Lie group K of compact type is an isometric isomorphism [H1] and is unique when Kis simply connected [H7]. Furthermore, Hall considered geometric quantization of T~*(K), K'scotangent bundle [H9]. Using the vertical polarization and a natural Khler polarization obtainedby identifying T~*(K) with the complexified group KC, Hall concluded that the pairing map be-tween the two Hilbert Spaces induced by these two polarizations coincides with the generalizedSegal-Bargmann transform C_t (up to constant).     

Newton’s method on generalized Banach spaces

We present a weaker convergence analysis of Newton’s method than in Kantorovich and Akilov (1964), Meyer (1987), Potra and Ptak (1984), Rheinboldt (1978), Traub (1964) on a generalized Banach space setting to approximate a locally unique zero of an operator. This way we extend the applicability of Newton’s method. Moreover, we obtain under the same conditions in the semilocal case weaker sufficient convergence criteria; tighter error bounds on the distances involved and an at least as precise information on the location of the solution. In the local case we obtain a larger radius of convergence and higher error estimates on the distances involved. Numerical examples illustrate the theoretical results.     

Relations among Besov-type spaces,Triebel–Lizorkin-type spaces and generalized Carleson measure spaces

In this article, the authors construct some counterexamples to show that the generalized Carleson measure space and the Triebel–Lizorkin-type space are not equivalent for certain parameters, which was claimed to be true in Lin and Wang [C.-C. Lin and K.Wang, Equivalency between the generalized Carleson measure spaces and Triebel–Lizorkin-type spaces, Taiwanese J. Math. 15 (2011), pp. 919–926]. Moreover, the authors show that for some special parameters, the generalized Carleson measure space, the Triebel–Lizorkin-type space and the Besov-type space coincide with certain Triebel–Lizorkin space, which answers a question posed in Remark 6.11(i) of Yuan et al. [W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010]. In conclusion, the Triebel–Lizorkin-type space and the Besov-type space become the classical Besov spaces, when the fourth parameter is sufficiently large.     

New Sobolev spaces via generalized Poincaré inequalities on metric measure spaces

In this paper, we introduce some new function spaces of Sobolev type on metric measure spaces. These new function spaces are defined by variants of Poincaré inequalities associated with generalized approximations of the identity, and they generalize the classical Sobolev spaces on Euclidean spaces. We then obtain two characterizations of these new Sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity. For the well-known Hajłasz–Sobolev spaces on metric measure spaces, we also establish some new characterizations related to generalized approximations of the identity. Finally, we clarify the relations between the Sobolev-type spaces introduced in this paper and the Hajłasz–Sobolev spaces on metric measure spaces.     

The Cauchy problem for the generalized Zakharov–Kuznetsov equation on modulation spaces

We consider the Cauchy problem for the generalized Zakharov–Kuznetsov equation ${?}_{t}u+{?}_{x_1}\mathrm{\Delta }u={?}_{x_1}(u^{m+1})$ on three and higher dimensions. We mainly study the local well-posedness and the small data global well-posedness in the modulation space $M_{2,1}^0({\mathbb{R}}^n)$ for $m\ge 4$ and $n\ge 3$. We also investigate the quartic case, i.e., $m=3$.     

Asymptotic Expansion of the Bergman Kernel via Perturbation of the Bargmann–Fock Model

We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated with the kth tensor powers of a positive line bundle L in a \(\frac{1}{\sqrt{k}}\)-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential \(k\varphi \) in a \(\frac{1}{\sqrt{k}}\)-neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann–Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann–Fock Bergman kernel.     

New generalized topologies on generalized topological spaces due to Cs??sz??r

We introduce new types of sets called $\bigwedge_{\mu}$ -sets and $\bigvee_{\mu}$ -sets and study some of their fundamental properties. We then investigate the topologies obtained from these sets.