Boundary values of holomorphic functions and heat kernel method in translation-invariant distribution spaces

TL;DR

This paper investigates boundary values of holomorphic functions in translation-invariant distribution spaces, introduces new edge of the wedge theorems, and applies heat kernel methods for element representation, unifying several classical cases.

Contribution

It develops new edge of the wedge theorems and heat kernel representations in translation-invariant distribution spaces, extending classical results to broader functional frameworks.

Findings

Established new edge of the wedge theorems.

Represented distribution space elements via heat kernel methods.

Unified boundary value and analytic representation frameworks.

Abstract

We study boundary values of holomorphic functions in translation-invariant distribution spaces of type $\mathcal{D}'_{E'_{\ast}}$. New edge of the wedge theorems are obtained. The results are then applied to represent $\mathcal{D}'_{E'_{\ast}}$ as a quotient space of holomorphic functions. We also give representations of elements of $\mathcal{D}'_{E'_{\ast}}$ via the heat kernel method. Our results cover as particular instances the cases of boundary values, analytic representations, and heat kernel representations in the context of the Schwartz spaces $\mathcal{D}'_{L^{p}}$, $\mathcal{B}'$, and their weighted versions.