In this paper, we characterize operator-theoretic properties (boundedness, compactness, and Schatten class membership) of Toeplitz operators with positive measure symbols on weighted Fock-Sobolev spaces of fractional order.

For the full range of index \(0<p\le \infty \), real weight α and real Sobolev order s, two types of weighted Fock-Sobolev spaces over \(\mathbb C^{n}\), \(F^{p}_{\alpha , s}\) and \(\widetilde F^{p}_{\alpha ,s}\), are introduced through fractional differentiation and through fractional integration, respectively. We show that they are the same with equivalent norms and, furthermore, that they are identified with the weighted Fock space \(F^{p}_{\alpha -sp,0}\) for the full range of parameters. So, the study on the weighted Fock-Sobolev spaces is reduced to that on the weighted Fock spaces. We describe explicitly the reproducing kernels for the weighted Fock spaces and then establish the boundedness of integral operators induced by the reproducing kernels. We also identify dual spaces, obtain complex interpolation result and characterize Carleson measures.

Let f and g be functions, not identically zero, in the Fock space F-alpha(2) of C-n. We show that the product TfT(g) over bar of Toeplitz operators on F-alpha(2) is bounded if and only if f(z) = e(q(z)) and g(z) = ce(-q(z)), where c is a nonzero constant and q is a linear polynomial.

We study linear combinations of composition operators acting on the Fock-Sobolev spaces of several variables. We show that such an operator is bounded only when all the composition operators in the combination are bounded individually. In other words, composition operators on the Fock-Sobolev spaces do not possess the same cancelation properties as composition operators on other well-known function spaces over the unit disk. We also show the analogues for compactness and the membership in the Schatten classes. In particular, compactness and the membership in some/all of the Schatten classes turn out to be the same.

We prove the -estimates for the -equation in the exponentially weighted -spaces on the unit disc. These estimates are extensions of the classical Hörmander weighted -estimates. We use only a family of holomorphic peak functions with precise growth conditions, which is technically simpler than using multipliers as was done by others in obtaining earlier results.

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Andersson obtained weighted Lp estimates for on analytic polyhedra. We provide an example to show why they cannot be improved.