1. Mikhlin-Hormander type symbols
Let be vector-spaces of measurable functions from to itself and let be a bounded linear operator from to . Recall that is called translation invariant if for all and . Let and with , we found that each such operator is determined by a certain tempered distribution such that for every (Schwartz space). So taking Fourier transform into we have . This motivate us to define a Fourier multiplier as a map given by
where is a tempered distribution and denotes the Fourier transform . We refer to as symbol of , sometimes one writes as to relate it with more general operators so-called pseudo-differential operators. No standard example of such symbols is, for ,
In the limit case, , the above symbol can be written as , where denotes the characteristic function of unit disk . It’s well-known that the condition and is necessary for be a Fourier multiplier on (see e.g., ). However, Fefferman (see ,) gave an intricate proof which show us that this condition is not sufficient, that is, he showed that the operator does not extend to a bounded operator on for any and . This result give us a negative answer to the famous disk conjecture which states that is bounded on for . In this post we will work with symbols more regular than (1) such as Minklin symbols .
In a few months ago, based on Littlewood-Paley theorem, we gave a proof that the operator is a Fourier multiplier from to itself (see Theorem 7) provided that and satisfies
Let us recall some important definitions. Let and , a function lies in Sobolev spaces if for every with there exists such that
and we say that is the derivative of in distribution sense (more precisely in ) and we write in to mean (5). The space equipped with the norm
is a Banach space. Also, notice that Sobolev spaces coincide with inhomogeneous fractional Sobolev spaces because of the norm equivalence
It follows that
Using Leibniz’s formula we written the term as
Now making the change of variable one has . Hence, by (3) it follows that
Hence which implies (8).