Gradient Estimate for the Heat Kernel on Some Fractal-Like Cable Systems and Quasi-Riesz Transforms

TL;DR

This paper provides pointwise gradient estimates for the heat kernel on fractal-like cable systems such as Vicsek and Sierpiński, and explores applications to the boundedness of quasi-Riesz transforms in $L^p$ spaces.

Contribution

It introduces new pointwise upper estimates for the heat kernel gradient on specific fractal cable systems and applies these results to analyze quasi-Riesz transforms.

Findings

Established pointwise upper bounds for heat kernel gradients.

Derived $L^p$-boundedness results for quasi-Riesz transforms.

Extended analysis to fractal-like cable systems such as Vicsek and Sierpiński.

Abstract

We give pointwise upper estimate for the gradient of the heat kernel on some fractal-like cable systems including the Vicsek and the Sierpi\'nski cable systems. Applications to $L^p$-boundedness of quasi-Riesz transforms are derived.