Fourier multipliers on the real Hardy spaces

We provide a variant of Hytönen’s embedding theorem, which allows us to extend and unify several sufficient conditions for a function to be a Fourier multiplier on the real Hardy spaces.


S. Król
Arch. Math. Theorem 1.1. Let d ∈ N, t ∈ (1,2], and l ∈ { n/t + 1, . . . , dn}. Let m ∈ L ∞ (R n ) with D α m ∈ L 1 loc (R n ) for all multi-indices α ∈ I l,d . Suppose that there exists a constant ρ > 0 such that Then m is a Fourier multiplier on H p (R n ) for all p ∈ (0, 2] such that In the case of lim inf |μ|→∞ χ Sμ m ∞ > 0, the condition (1) is equivalent to the Hytönen-type condition, i.e. sup μ∈Z,α∈I l,d 2 μ(|α|1−n/t) χ Sμ D α m t < ∞, studied in [5]; see also Remark 4.3. Otherwise, it presents a diverse feature. For instance, if |m(x)| ≤ C(1 + |x|) −a (x ∈ R n ) for some a > 0 and sup μ∈Z,α∈I l,d for some b ≥ 0, then the conclusion of Theorem 1.1 holds with ρ = b/a. This shows that under an additional information on the decay of a function m, a weakened Hytönen-type condition still ensures that m is a Fourier multiplier on H p (R n ) for some p < 2.
The extension of the classical Hörmander-type condition which involves an additional analytic behaviour of the multiplier function goes back to [3,9,10]. More recently, such conditions were systematically studied by Kolomoitsev [6,7].
The Kolomoitsev-type conditions differ from (1) in a few points. For the convenience of the reader we restate [6, Theorem 2] and a special case of [6,Theorem 1], which can be reformulated in our setting as follows. Set is the canonical basis of R n . Theorem A (Kolomoitsev, [6, Theorems 1 and 2]). Let d > n/2 and let m ∈ Then m is a Fourier multiplier on H p (R n ) for every p ∈ (0, 2] such that Note that the sets of multi-indices α involved in the conditions (1) and (4) differ essentially, as well as the corresponding conclusions. A close analysis shows that neither result is superior in general and, as soon as n ≥ 2, the one or the other might be better suited to a particular situation. For instance, note that in order to use Theorem A to verify whether bounded functions m, which are locally in the Sobolev spaces W d 2 (R n ), are Fourier multipliers on any desired H p (R n ) for p ∈ (0, 1), one needs that the smoothness parameter d is grater than n/2. Theorem 1.1 shows that this restriction on the parameter d can be relaxed, if we have an additional information on the behaviour of the mixed partial derivatives D α m, α ∈ I d,l , with l > n/2. Note that functions which are not locally in W n/2 +1 2 (R n ) but satisfy the condition (1) for d ≤ n/2, l ∈ { n/2 + 1, . . . , dn} and ρ > 0 small, can be easily constructed.
On the other hand, if (1) holds for d > n/2, l = d, and t = 2, then we get that the right-hand side of (2) is strictly less than that of (5). Therefore, Theorem 1.1 does not reproduce Theorem A. It should be also underlined that Kolomoitsev's result is sharp; see [6, Remark 1, p. 1565]. We do not know if one can modify Hytönen's ∞-approach, which is involved in the proof of Theorem 4.1, to remove this drawback and cover Theorem A.
We point out that the Fourier embedding result which is applied (implicitly) in the proofs of Kolomoitsev's multiplier theorems says that the Fourier transform maps the Besov space B s t,q (R n ) isomorphically onto K s t ,q (R n ) for every s ≥ 0, 0 < q ≤ ∞, and t = 2, where  (9), shows immediately that B s t,q (R n ) embedds in K s t ,q (R n ) for every s ≥ 0, t ∈ [1,2], and t ≤ q ≤ ∞. It allows us to extend and unify some results known in the literature.
In particular, combining Lemma 3.1 with techniques from [5], we extend the Strömberg-Torchinsky results, [12, Theorems 5 and 6, Chapter XI], which fall outside the scope of the treatment in [5]; see e.g. [5,Example 9.7]. We refer the reader to Theorems 3.4 and 3.7 for more details.
We conclude with some additional comments. Except for some details, we reproduce ideas which have been presented in [5] and the standard techniques from the multiplier theory developed in the above cited papers. Therefore, the presentation of the proofs of our results is restricted to providing only main supplementary observations which should be made. It requires an adaptation of the techniques developed in the corresponding papers. This allows us to keep the novelty of the paper in a more transparent way.
Moreover, the presentation is restricted to the scalar-valued case. The vector-valued counterparts of our results, with having a priori boundedness 460
on one L p space of vector-valued functions, can be easily obtained by standard arguments. The interested reader can for instance mimic the presentation given in [5].

Preliminaries.
We follow the notation introduced in [5]. For an arbitrary function f : R n → C, we define the translation τ h and the difference operators Δ d h and ∂ α h as follows: where e i stands for the ith standard unit vector in R n , hβ : To illustrate our general results expressed in the terms of the difference operators we restate relevant Hörmander-type conditions studied by Strömberg and Torchinsky in [12,Chapter XI].
For real numbers l ≥ 0 and t ∈ [1,2] we say that m satisfies the condition for all multi-indices α with |α| ≤ l when l is a positive integer, and when l is not an integer, in addition, ⎛ for all multi-indices α with |α| = l , where := l − l .
Following [11], given 1 ≤ r ≤ ∞, we say that a function k ∈ L 1 loc (R n \{0}) satisfies the (D r )-condition, and write k ∈ (D r ), if there exists a sequence (for r = ∞ this condition is understood in the usual way). Note that (D 1 ) is the classical Hörmander condition. Moreover, for k := m with m ∈ L ∞ we also We refer the reader to [12] for the background on weighted Hardy spaces H p w (R n ), and relevant classes of weights, i.e. the class of weights satisfying the Muckenhoupt A s -condition, the reverse Hölder RH r -condition, and the doubling D b -condition; see [12,Chapter I].
We say that a function m ∈ L ∞ (R n ) is a Fourier multiplier on H p w (R n ), and write m ∈ M(H p w ) if the operator T given by extends to a bounded operator on H p w (R n ), where D 0 stands for {f ∈ S(R n ) : suppf is compact and 0 / ∈ suppf }. Finally, recall that H p w (R n ) is identical to L p w (R n ) for every p ≥ s and w ∈ A s , where s > 1; see [12, Theorem 1, Chapter VI]. In the sequel, we omit '(R n )' in the symbols of the function spaces.

Proof. Set E(i, ρ, j)
Therefore, taking the logarithmic average over ρ i and changing a variable This completes the proof.

Then there exists a constant C such that
and . Let s denote the smallest integer at least s.   The fact that we can replace h −j in (10) with h −jγ for every γ ∈ (0, 1) follows, for instance, from [5,Lemma 10.1]. This easily leads to the desired estimates.
Then, for every multi-index β ∈ N n 0 with |β| < s − n/t, there exists a constant C such that for every R > 0 and y ∈ R n with 0 < 2|y| ≤ R we have In particular, k ∈ K(t , s − n/t), and k ∈ (D t ).
The proof follows the ideas provided in [5]. We give only the main supplementary observation which should be made, and leave the detailed verification for the reader.
satisfies the assumption of Lemma 3.2. By a straightforward computation, one can obtain the conclusion of [5,Lemma 11.3] with q = t for all multi-indices β. Consequently, following the lines of the proofs of [5,Lemmas 11.4 and 11.5] we get also their conclusions for q = t and all multi-indices β with |β| < s − n/t. This yields the desired estimates. Now, the second claim follows from the first one by a direct computation.  m ∈ B(s, t, ∞) for some s > 0 and t ∈ (1, 2] such that n/t < s ≤ n. If (i) n s < p < ∞ and w ∈ A ps/n or (ii) 1 < p < ( n s ) and w −1/(p−1) ∈ A p s/n , then m is a Fourier multiplier on L p w . Furthermore, if s < n, one may take p = n/s in (i), and p = (n/s) in (ii).
Proof. We make use of the generalisation of [8, Theorem 1] due to Rubio de Francia, Ruiz & Torrea, see [11, Theorem 1.6, Part I], which in particular says that a singular integral operator of convolution type with kernel k ∈ (D r ) (1 < r < ∞) is bounded on L p w if w ∈ A p/r and r ≤ p < ∞ or if w ∈ A r p and 1 < p ≤ r. Note that by a kernel in [11] the authors mean a function in L 1 loc (R n \{0}). However, the conclusion of [11, Theorem 1.6, Part I] still holds in our a bit more general case of k = m with m ∈ L ∞ and k ∈ (D r ). Indeed, according to the above definition, one can apply [11, Theorem 1.6, Part I] to ν μ=−ν k μ (ν ∈ N) and then the standard approximation and density arguments.
Indeed, since Δ j y [m(2 μ ·)φ] = i≤j j i (−1) i τ iy [m(2 μ ·)φ] (y ∈ R n ), the claim follows immediately from Hölder's inequality. Now, if w ∈ A ps/n , then there exists r ∈ (1, ∞) such that n/s < r < min(p, t) with w ∈ A p/r . By the above observation and Theorem 3.4, we get k ∈ (D r ), and consequently (i) holds. By the standard duality argument one can show (ii).
Since for every w ∈ A r (r ≥ 1) there exists > 1 such that w ∈ A r , and m is a multiplier on L p for every p > 1, see Remark 3.5, the last statement can be obtained by an interpolation argument with a change of measure. This completes the proof.
We conclude with further results on multipliers on weighted Hardy spaces which extend slightly the results due to Strömberg and Torchinsky in [12]. We start with preliminary observations. First, note that if m ∈ M (t, l) (t ∈ [1,2], l > 0), then m ∈ B(l, t, ∞). Indeed, this follows, for instance, from [5,Lemma 4.2] by taking as the stable set I in its formulation the set {je i : i, . . . , n, j = 0, . . . , l }, and noting that in this case |α| 1 = |α| ∞ for all α ∈ I.
We point out that the multiplier results of Strömberg and Torchinsky [12, Theorems 4, 5, and 6, Chapter XI] are formulated in the terms of the condition M (t, l), though, their proofs are based on the behaviour of m reflecting the fact that m ∈ M (t, l). In other words, [12,Theorems 4,5, and 6, Chapter XI] follow