Toeplitz quantization on Fock space

For Toeplitz operators $T_f^{(t)}$ acting on the weighted Fock space $H_t^2$, we consider the semi-commutator $T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}$, where $t>0$ is a certain weight parameter that may be interpreted as Planck's constant ? in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit

($?$)$\underset{t\to 0}{\lim }?{6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t.$

It is well-known that ${6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t$ tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to $f,g\in \mathrm{BUC}({\mathbb{C}}^n)$ by Bauer and Coburn. We now further generalize (?) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra $\mathrm{VMO}\cap L^{\infty }$ of bounded functions having vanishing mean oscillation on ${\mathbb{C}}^n$. Our approach is based on the algebraic identity $T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}=?{(H_{\overline{f}}^{(t)})}^{?}{H}_g^{(t)}$, where $H_g^{(t)}$ denotes the Hankel operator corresponding to the symbol g, and norm estimates in terms of the (weighted) heat transform. As a consequence, only f (or likewise only g) has to be contained in one of the above classes for (?) to vanish. For g we only have to impose ${\lim \sup }_{t\to 0}{6H_g^{(t)}6}_t<\infty$, e.g. $g\in L^{\infty }({\mathbb{C}}^n)$. We prove that the set of all symbols $f\in L^{\infty }({\mathbb{C}}^n)$ with the property that ${\lim }_{t\to 0}?{6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t={\lim }_{t\to 0}?{6T_g^{(t)}{T}_f^{(t)}?T_{gf}^{(t)}6}_t=0$ for all $g\in L^{\infty }({\mathbb{C}}^n)$ coincides with $\mathrm{VMO}\cap L^{\infty }$. Additionally, we show that $\underset{t\to 0}{\lim }?{6T_f^{(t)}6}_t={6f6}_{\infty }$ holds for all $f\in L^{\infty }({\mathbb{C}}^n)$. Finally, we present new examples, including bounded smooth functions, where (?) does not vanish.     

Regularity in Lp Sobolev spaces of solutions to fractional heat equations

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations with a nonlocal operator P:

(*)$Pu+{?}_{t}u=f(x,t),\text{ for }x\in \mathrm{\Omega }?{\mathbb{R}}^n,t\in I?\mathbb{R}.$

1) For strongly elliptic pseudodifferential operators (ψdo's) P on ${\mathbb{R}}^n$ of order $d\in {\mathbb{R}}_{+}$, a symbol calculus on ${\mathbb{R}}^{n+1}$ is introduced that allows showing optimal regularity results, globally over ${\mathbb{R}}^{n+1}$ and locally over $\mathrm{\Omega }\times I$:

$f\in H_{p,\mathrm{loc}}^{(s,s/d)}(\mathrm{\Omega }\times I)?u\in H_{p,\mathrm{loc}}^{(s+d,s/d+1)}(\mathrm{\Omega }\times I),$

for $s\in \mathbb{R}$, $1<p<\infty$. The $H_p^{(s,s/d)}$ are anisotropic Sobolev spaces of Bessel-potential type, and there is a similar result for Besov spaces.2) Let Ω be smooth bounded, and let P equal ${(?\mathrm{\Delta })}^a$ ($0<a<1$), or its generalizations to singular integral operators with regular kernels, generating stable Lévy processes. With the Dirichlet condition $\mathrm{supp}u?\stackrel{￣}{\mathrm{\Omega }}$, the initial condition $u{|}_{t=0}=0$, and $f\in L_p(\mathrm{\Omega }\times I)$, (*) has a unique solution $u\in L_p(I;H_p^{a(2a)}(\stackrel{￣}{\mathrm{\Omega }}))$ with ${?}_{t}u\in L_p(\mathrm{\Omega }\times I)$. Here $H_p^{a(2a)}(\stackrel{￣}{\mathrm{\Omega }})={\dot{H}}_p^{2a}(\stackrel{￣}{\mathrm{\Omega }})$ if $a<1/p$, and is contained in ${\dot{H}}_p^{2a?\epsilon }(\stackrel{￣}{\mathrm{\Omega }})$ if $a=1/p$, but contains nontrivial elements from $d^a{\stackrel{￣}{H}}_p^a(\mathrm{\Omega })$ if $a>1/p$ (where $d(x)=\mathrm{dist}(x,?\mathrm{\Omega })$). The interior regularity of u is lifted when f is more smooth.     

Formality theorem for g-manifolds

With any $\mathfrak{g}$-manifold M are associated two dglas $\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{T}_{\mathrm{poly}}^{?}(M))$ and $\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{D}_{\mathrm{poly}}^{?}(M))$, whose cohomologies $H_{\mathrm{CE}}^{?}(\mathfrak{g},T_{\mathrm{poly}}^{?}(M)\stackrel{0}{\to }{T}_{\mathrm{poly}}^{?+1}(M))$ and $H_{\mathrm{CE}}^{?}(\mathfrak{g},D_{\mathrm{poly}}^{?}(M)\stackrel{d_H}{\to }{D}_{\mathrm{poly}}^{?+1}(M))$ are Gerstenhaber algebras. We establish a formality theorem for $\mathfrak{g}$-manifolds: there exists an $L_{\infty }$ quasi-isomorphism $\mathrm{\Phi }:\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{T}_{\mathrm{poly}}^{?}(M))\to \mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{D}_{\mathrm{poly}}^{?}(M))$ whose first ‘Taylor coefficient’ (1) is equal to the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd cocycle of the $\mathfrak{g}$-manifold M, and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the $\mathfrak{g}$-manifold M is an isomorphism of Gerstenhaber algebras from $H_{\mathrm{CE}}^{?}(\mathfrak{g},T_{\mathrm{poly}}^{?}(M)\stackrel{0}{\to }{T}_{\mathrm{poly}}^{?+1}(M))$ to $H_{\mathrm{CE}}^{?}(\mathfrak{g},D_{\mathrm{poly}}^{?}(M)\stackrel{d_H}{\to }{D}_{\mathrm{poly}}^{?+1}(M))$.