On the distribution kernels of Toeplitz operators on CR manifolds

We study the distribution kernel of a Toeplitz operator associated with a classical pseudodifferential operator on a compact, embeddable, strictly pseudoconvex CR manifold. The main result consists of a formula for the values at the diagonal of the second coefficient in the expansion of the symbol of the kernel. We also establish asymptotic expansions for Toeplitz operators on the positive part of a compact not necessary strictly pseudoconvex CR orbifold under certain natural assumptions.

💡 Research Summary

The paper investigates the distribution kernels of Toeplitz operators on compact, embeddable, strictly pseudoconvex CR manifolds and on a broader class of CR orbifolds. Starting from the Szegő projection Π, which projects L²‑functions onto the kernel of the tangential Cauchy–Riemann operator (\bar\partial_b), the authors consider a classical pseudodifferential operator (E) of order (m) and form the Toeplitz operator (T_E = \Pi E \Pi). The central goal is to describe the kernel (T_E(x,y)) as an oscillatory integral with a globally defined phase function and a classical symbol, and to compute explicitly the diagonal values of the first two coefficients in the symbol expansion.

Section 2 sets up the necessary notation, recalls Hörmander symbol classes (S^m_{1,0}) and the subclass of classical symbols (S^m_{\mathrm{cl}}), and reviews the geometry of a CR manifold ((X,T^{1,0}X)). A contact form (\omega_0) and its Reeb vector field (T) are fixed; the Levi form is non‑degenerate, and the Tanaka–Webster connection provides a decomposition of the cotangent bundle into horizontal and vertical parts. The Szegő kernel (\Pi(x,y)) is expressed (Theorem 1.1) as (\int_0^\infty e^{i t\varphi(x,y)} a(x,y,t),dt) with a phase (\varphi) satisfying (\operatorname{Im}\varphi\ge0), (\varphi(x,y)=0) iff (x=y), and a symbol (a) whose leading term (a_0(x,x)= (2\pi)^{-(n+1)}). By applying the Malgrange preparation theorem, the authors replace (\varphi) with an equivalent phase (\hat\varphi) that satisfies the additional condition (T_y^2\hat\varphi(x,x)=0).

The main result (Theorem 1.3) states that for any classical pseudodifferential operator (E) with principal symbol (e_0) and subprincipal symbol (e_{\mathrm{sub}}), the kernel of (T_E) admits the representation
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