# Bochner's Subordionation and Fractional Caloric Smoothing in Besov and Triebel--Lizorkin Spaces

## TL;DR

This paper employs Bochner's subordination to derive new caloric smoothing estimates in Besov and Triebel--Lizorkin spaces, extending classical results to more general semigroups and function spaces.

## Contribution

It introduces novel smoothing estimates using Bochner's subordination in advanced function spaces, broadening the scope of classical Gaussian-based results.

## Key findings

- Extended smoothing results to Besov and Triebel--Lizorkin spaces.
- Applicable to a wider class of semigroups beyond Gaussian.
- Provides a framework for further extensions to other function spaces.

## Abstract

We use Bochner's subordination technique to obtain caloric smoothing estimates in Besov- and Triebel--Lizorkin spaces. Our new estimates extend known smoothing results for the Gau{\ss}--Weierstra{\ss}, Cauchy--Poisson and higher-order generalized Gau{\ss}--Weierstra{\ss} semigroups. Extensions to other function spaces (homogeneous, hybrid) and more general semigroups are sketched.