In the last post we given a proof of the following theorem.

**Theorem 1 (Mikhlin-Hormander theorem)** *Let and . If for we have*

*then is a Fourier multiplier on , . In particular, is a Fourier multiplier on if , that is, if*

In this post, we would like to extend this theorem for a class of Morrey-type spaces. In other words

**Theorem 2** *Let and . If , the Fourier multiplier on can be extended to , that is, there exists such that*

*for all .*

**Proof.** A proof can be found in [Taylor] (see also [Kozono1, Kozono2]. For reader convenience we give some details. This key theorem was important in my recent article in join with Ferreira, Lucas C.F.

We have that is a convolution operator with kernel . As it follows from [p.26, Stein] that

Based in (4) and Minlin-Hormander theorem we obtain a proof of the Theorem 2. Indeed, firstly one splits as

where

Defining and easily gets by Hölder inequality and estimate (4) the following

where denotes the conjugate exponent of . Using Tonelli’s theorem and once more estimate (4) we have

Now we ready to proof the theorem. By Mikhlin-Hormander theorem and the last estimate one has

which yields (3), because the series above is convergent.

-10.933922
-37.065357

### Like this:

Like Loading...