A Marcinkiewicz maximal-multiplier theorem

For r<2, we prove the boundedness of a maximal operator formed by applying all multipliers m with $\|m\|_{V^r} \leq 1$ to a given function.


Introduction
Given an exponent r and a function f defined on R, consider the r-variation norm where the supremum is over all strictly increasing finite length sequences of real numbers. The classical Marcinkiewicz multiplier theorem states that if r = 1 and a function m is of bounded r-variation uniformly on dyadic shells, then m is an L p multiplier for 1 < p < ∞ and ] andˆ,ˇdenote the Fourier-transform and its inverse. Later, Coifman, Rubio de Francia, and Semmes [2], see also [8], showed that the requirement of bounded 1-variation can be relaxed to allow for functions of bounded 2-variation, and in fact (1) holds whenever r ≥ 2 and | 1 p − 1 2 | < 1 r . The estimate [2] does not discriminate between multipliers of bounded 2-variation and those of bounded r-variation where r < 2, and so one might ask whether there is anything to be gained by controlling the variation norm of multipliers in the latter range of exponents.
Defining the maximal-multiplier operator where the supremum is over all functions in the V r unit ball, we have The case r = 1 was observed independently by Lacey [4]. Note that, in the definition of M r , each m is required to have finite r-variation on all of R rather than simply on each dyadic shell as in (1). This is necessary for boundedness, as can be seen from the counterexamples of Christ, Grafakos, Honzík and Seeger [1].
Although the maximal operator (2) would seem to be fairly strong, we do not yet know of an application for the bound above. We will, however, quickly illustrate a strategy for its use that falls an (important) ǫ short of success. Let Ψ be (say) a Schwartz function, and for each ξ, x ∈ R and k ∈ Z consider the 2 k -truncated partial Fourier integral It was proven by Demeter, Lacey, Tao, and Thiele [3] that for q = 2 and 1 < p < ∞ If we had the bound ) ≤ C p,r f L p for some r < 2, then an application of Theorem 1.1 would give (4) for q > r by rather different means than [3]. In fact, one can see by applying the method in Appendix D of [6] that (5) holds for r > 2 and p > r ′ . Unfortunately, it does fail for r ≤ 2.

Proof of Theorem 1.1
The following lemma was proven in [2], see also [5].
Lemma 2.1. Let m be a compactly supported function on R of bounded r-variation for some 1 ≤ r < ∞. Then for each integer j ≥ 0, one can find a collection Υ j of pairwise disjoint subintervals of R and coefficients {b υ } υ∈Υj ⊂ R so that |Υ j | ≤ 2 j , |b υ | ≤ 2 −j/r m Vr , and where the sum in j converges uniformly.
The lemma above was applied in concert with Rubio de Francia's square function estimate [7] to obtain (1). Here, we will argue similarly, exploiting the analogy between the Rubio de Francia square function estimate and the variation-norm Carleson theorem.
It was proven in [7] that for p ≥ 2 where the supremum above is over all collections of pairwise disjoint subintervals of R. Consider the partial Fourier integral It was proven in [6] that for s > 2 and p > s ′ sup Note that, by standarding limiting arguments, taking the supremum in (2) to be over all compactly supported m such that m V r ≤ 1 does not change the definition of M r . Applying the decomposition (6), we see that for any compactly supported m with m V r ≤ 1 we have where, for the last inequality, we require s < r ′ . Provided that r < 2 and p > r we can choose an s < r ′ with s > 2 and p > s ′ , giving (7) and hence (3).
The argument of Lacey [4] for r = 1 follows a similar pattern, except with Marcinkiewicz's method in place of [2] and the standard Carleson-Hunt theorem in place of [6].