Fractional Besov Trace/Extension Type Inequalities via the Caffarelli-Silvestre extension

TL;DR

This paper establishes fractional trace and extension inequalities for the Caffarelli-Silvestre extension, linking fractional Sobolev spaces, Fourier transform estimates, and measure embeddings, advancing understanding of fractional PDEs and function space embeddings.

Contribution

It introduces new fractional trace inequalities via the Caffarelli-Silvestre extension and characterizes measure embeddings in different parameter regimes.

Findings

Fractional Sobolev and Hardy trace inequalities are established.

Control of the $ ext{H}^{-eta/2}$ norm by affine energy and derivatives.

Characterization of measure embeddings using capacity and isocapacitary inequalities.

Abstract

Let $u(\cdot,\cdot)$ be the Caffarelli-Silvestre extension of $f.$ The first goal of this article is to establish the fractional trace type inequalities involving the Caffarelli-Silvestre extension $u(\cdot,\cdot)$ of $f.$ In doing so, firstly, we establish the fractional Sobolev/ logarithmic Sobolev/ Hardy trace inequalities in terms of $\nabla_{(x,t)}u(x,t).$ Then, we prove the fractional anisotropic Sobolev/ logarithmic Sobolev/ Hardy trace inequalities in terms of $ {\partial_{t} u(x,t)}$ or $(-\Delta)^{-\gamma/2}u(x,t)$ only. Moreover, based on an estimate of the Fourier transform of the Caffarelli-Silvestre extension kernel and the sharp affine weighted $L^p$ Sobolev inequality, we prove that the $\dot{H}^{-\beta/2}(\mathbb{R}^n)$ norm of $f(x)$ can be controlled by the product of the weighted $L^p-$affine energy and the weighted $L^p-$norm of ${\partial_{t} u(x,t)}.$ The second goal of this article is to characterize non-negative measures $\mu$ on $\mathbb{R}^{n+1}_+$ such that the embeddings $$\|u(\cdot,\cdot)\|_{L^{q_0,p_0}_{\mu}(\mathbb{R}^{n+1})}\lesssim \|f\|_{\dot{\Lambda}^{p,q}_\beta(\mathbb{R}^n)}$$ hold for some $p_0$ and $q_0$ depending on $p$ and $q$ which are classified in three different cases: (1). $p=q\in (n/(n+\beta),1];$ (2) $(p,q)\in (1,n/\beta)\times (1,\infty);$ (3). $(p,q)\in (1,n/\beta)\times\{\infty\}.$ For case (1), the embeddings can be characterized in terms of an analytic condition of the variational capacity minimizing function, the iso-capacitary inequality of open balls, and other weak type inequalities. For cases (2) and (3), the embeddings are characterized by the iso-capacitary inequality for fractonal Besov capacity of open sets.