Not long ago we discussed one of the main direct applications of the LittlewoodPaley theory, namely the Marcinkiewicz multiplier theorem. Recall that the singlevariable version of this theorem can be formulated as follows:
Theorem 1 [Marcinkiewicz multiplier theorem]: Let be a function on such that

 for every LittlewoodPaley dyadic interval with
where denotes the total variation of over the interval .
Then for any the multiplier defined by for functions extends to an bounded operator,
You should also recall that the total variation above is defined as
and if is absolutely continuous then exists as a measurable function and the total variation over interval is given equivalently by . We have seen that the “dyadic total variation condition” 2.) above is to be seen as a generalisation of the pointwise condition , which in dimension 1 happens to coincide with the classical differential Hörmander condition (in higher dimensions the pointwise Marcinkiewicz conditions are of product type, while the pointwise Hörmander(Mihklin) conditions are of radial type; see the relevant post). Thus the Marcinkiewicz multiplier theorem in dimension 1 can deal with multipliers whose symbol is somewhat rougher than being differentiable. It is an interesting question to wonder how much rougher the symbols can get while still preserving their mapping properties (or maybe giving up some range – recall though that the range of boundedness for multipliers must be symmetric around 2 because multipliers are selfadjoint).
Coifman, Rubio de Francia and Semmes came up with an answer to this question that is very interesting. They generalise the Marcinkiewicz multiplier theorem (in dimension 1) to multipliers that have bounded variation with > 1. Let us define this quantity rigorously.
Definition: Let and let be an interval. Given a function , its variation over the interval is
Notice that, with respect to the notation above, we have . From the fact that when we see that we have always , and therefore the higher the the less stringent the condition of having bounded variation becomes (this is linked to the Hölder regularity of the function getting worse). In particular, if we wanted to weaken hypothesis 2.) in the Marcinkiewicz multiplier theorem above, we could simply replace it with the condition that for any LittlewoodPaley dyadic interval we have instead . This is indeed what Coifman, Rubio de Francia and Semmes do, and they were able to show the following:
Theorem 2 [CoifmanRubio de FranciaSemmes, ’88]: Let and let be a function on such that

 for every LittlewoodPaley dyadic interval with
Then for any such that the multiplier defined by extends to an bounded operator,
The statement is essentially the same as before, except that now we are imposing control of the variation instead and as a consequence we have the restriction that our Lebesgue exponent satisfy . Taking a closer look at this condition, we see that when the variation parameter is the condition is empty, that is there is no restriction on the range of boundedness of : it is still the full range < < , and as grows larger and larger the range of boundedness restricts itself to be smaller and smaller around the exponent (for which the multiplier is always necessarily bounded, by Plancherel). This is a very interesting behaviour, which points to the fact that there is a certain dichotomy between variation in the range below 2 and the range above 2, with variation being the critical case. This is not an isolated case: for example, the Variation Norm Carleson theorem is false for variation with ; similarly, the Lépingle inequality is false for 2variation and below (and this is related to the properties of Brownian motion).
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