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
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.