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

## Introduction

In general, the space is hard to study in many sense. First, if we consider Hölder’s inequality, then the natural pair of is . In the case of bounded domain, for any So we can think 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 -estimates. But these techniques cannot be applicable to -type estimate. Third, the dual of is , but 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 , 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 . 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 with .

For any , we have

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 with .

For any , we have

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

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

Theorem.Let and . Then there exists a constant such that

for all such that , where is the vector Riesz transform.

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

then we obtain the following result:

Theorem.Let and . Then there exists a constant such that

for all .

For additional information, see the following pdf file.