Monthly Archives: July 2016

Detail calculation of An $L^{1}$ -type estimates for Riesz potentials

In this note, we fill all details of the paper “An L^1-type estimates for Riesz potentials“, written by Armin Schikorra, Daniel Spector, and Jean Van Schaftingen.

Introduction

In general, the space L^1 is hard to study in many sense. First, if we consider Hölder’s inequality, then the natural pair of L^1 is L^\infty. In the case of bounded domain, L^\infty \subset L^p for any p<\infty So we can think L^\infty has less elements than any other space.   Second, when we study the theory of partial differential equation, Calderón-Zygmund singular integral and Maximal function estimate are useful techniques. These techniques gives a good way to obtain L^p-estimates. But these techniques cannot be applicable to L^1-type estimate. Third, the dual of L^1 is L^\infty, but L^1 cannot be considered as a dual of meaningful space. So we cannot apply weak* convergence in this space. By these reasons, many mathematician study a proper subspace H^1, real Hardy space. It has fruitful structure that all disadvantages we discussed do not happen. For further information, see Stein’s harmonic analysis or Coifman-Meyer-Lions-Semes’s paper.

Currently, there is no general method for studying the space L^1. In 2004, however, Bourgain and Brezis  gives a new type estimate which is related to this type of difficulty. They proved the following inequality:

Theorem. Let f\in\Leb 1\left(\mathbb{R}^{n};\mathbb{R}^{n}\right) with \mathrm{div} f=0.
For any u\in\Sob{1}{n}\left(\mathbb{R}^{n};\mathbb{R}^{n}\right)\cap\Leb{\infty}\left(\mathbb{R}^{n};\mathbb{R}^{n}\right), we have

    \[ \left|\int_{\mathbb{R}^{n}}f\cdot udx\right|\le C \norm{f}{\Leb 1} \norm{\nabla u}{\Leb n}. \]

They proved this inequality by using Littlewood-Paley decomposition.
But there is a quite elementary proof given by Van Schaftigen in 2004. He proved similar inequality in more general setting.

Theorem. Let f\in\Leb 1\left(\mathbb{R}^{n};\mathbb{R}^{n}\right) with \mathrm{div} f\in\Leb 1.
For any u\in W^{1,n}\left(\mathbb{R}^{n};\mathbb{R}^{n}\right)\cap\Leb{\infty}\left(\mathbb{R}^{n};\mathbb{R}^{n}\right), we have

    \[ \left|\int_{\mathbb{R}^{n}}f\cdot udx\right|\le C\left(\norm{f}{\Leb 1}\norm{\nabla u}{\Leb n}+\norm{\mathrm{div}  f}{\Leb 1}\norm{u}{\Leb n}\right). \]

The method is based on dimension reduction and Morrey-Sobolev embedding.

In 2014, Schikorra, Spector and Van Schaftigen proved the following theorem.

Theorem. Let n\ge2 and 0<\alpha<n. Then there exists a constant C=\left(\alpha,n\right)>0 such that

    \[ \norm{I_{\alpha}u}{\Leb{\frac{n}{n-\alpha}}\left(\mathbb{R}^{n}\right)}\le C\norm{Ru}{\Leb 1\left(\mathbb{R}^{n};\mathbb{R}^{n}\right)} \]

for all u\in C_{0}^{\infty}\left(\mathbb{R}^{n}\right) such that Ru\in\Leb 1\left(\mathbb{R}^{n};\mathbb{R}^{n}\right), where R is the vector Riesz transform.

There is another analogue for this result. If we define the Riesz fractional gradient by

    \[ D^{\alpha}u:=DI_{1-\alpha}u, \]

then we obtain the following result:

Theorem. Let n\ge2 and 0<\alpha<1. Then there exists a constant C=\left(\alpha,n\right)>0 such that

    \[ \norm{u}{\Leb{\frac{n}{n-\alpha}}\left(\mathbb{R}^{n}\right)}\le C\norm{D^{\alpha}u}{\Leb 1\left(\mathbb{R}^{n};\mathbb{R}^{n}\right)} \]

for all u\in C_{0}^{\infty}\left(\mathbb{R}^{n}\right).

For additional information, see the following pdf file.