Weak-Type Boundedness of the Fourier Transform on Rearrangement Invariant Function Spaces

Abstract We study several questions about the weak-type boundedness of the Fourier transform ℱ on rearrangement invariant spaces. In particular, we characterize the action of ℱ as a bounded operator from the minimal Lorentz space Λ(X) into the Marcinkiewicz maximal space M(X), both associated with a rearrangement invariant space X. Finally, we also prove some results establishing that the weak-type boundedness of ℱ, in certain weighted Lorentz spaces, is equivalent to the corresponding strong-type estimates.


Introduction
The Fourier transform of a function f in R n , defined as is a bounded operator from L 1 to L ∞ . It can also be extended continuously on L 2 where, in fact, it defines an isometry. In particular, the Fourier transform maps L 1 + L 2 into L 2 + L ∞ .
One of the main results included in the previous work [4] concerns the boundedness of the Fourier transform in rearrangement invariant Banach function spaces (r.i. spaces, see [3] for standard definitions and notations) and asserts that the largest r.i. space which is mapped by the Fourier transform into a space of locally integrable functions is, in fact, L 1 + L 2 . It is also shown that L 2 is the only r.i. space X on R n , on which the Fourier transform is bounded: F : X → X if and only if X = L 2 . (1)

c 2018 The Edinburgh Mathematical Society
In this work, we want to study the same kind of question, but relaxing the hypothesis, assuming now that F is of restricted weak type on X. Let us first briefly recall the necessary definitions to describe this result. For a given r.i. space X, the fundamental function ϕ X is defined as Note that on a r.i. space, the expression above is independent of the set A, so ϕ X is a well-defined quasi-concave function; that is, an increasing function such that ϕ X (t)/t is non-increasing and ϕ X (t) = 0, for t > 0. Let us consider the minimal Lorentz space Λ(X) = Λ ϕX and the maximal Lorentz space (or Marcinkiewicz space) M (X) = M ϕX defined, respectively, as It is well known [3,Theorem II.5.13] that for every r.i. space X we have Regarding (1), and in view of what is known in the case of the strong-type estimates [4], it is now natural to ask about weaker conditions on the r.i. space X, in terms of its fundamental function ϕ X , to ensure that the Fourier transform F maps the minimal Lorentz space Λ(X) into the Marcinkiewicz space M (X) (i.e. it is of restricted weak-type on X). This question is completely answered in § 2 (see Theorem 2.1).
Also, motivated by the results in [2,13], we consider the boundedness of F between different weighted Lorentz spaces. If p ∈ (0, ∞) and v is a non-negative weight, the Lorentz space Λ p (v) is the set of functions f : R n −→ R for which In the particular case of v(t) = t p/q−1 , we obtain the classical Lorentz spaces L q,p , 0 < p, q < ∞. Similarly, the space Γ p (v) is the collection of those f for which is finite. Analogously, we define the corresponding weak-type space Γ p,∞ (v), which is the set of all measurable functions f such that It is easy to see that we always have the following collection of embeddings: For a general review on normability properties for these weighted Lorentz spaces, see [5].
In [2], Fourier mapping theorems in weighted Lorentz spaces were obtained in the Banach space setting. In particular, the authors proved that for 1 < p ≤ q and q ≥ 2, provided that u is a non-increasing weight, and v is in the B p class of Ariño and Muckenhoupt [1], which is given by the following inequality and also satisfies Conversely, if the inequality (2) holds for some weight functions u and v on (0, ∞) and 1 < p, q < ∞, then u and v satisfy (4). In fact, (4) is the condition obtained when (2) is tested on radial characteristic functions in R n .
An important technique to study the action of the Fourier transform on r.i. spaces was proved by Jodeit and Torchinsky [7], who showed that an operator T is of type (1, ∞) and (2,2) if and only if there is a constant C such that In [2], and also in [13], weighted extensions of this result were obtained as part of the study of the boundedness of the operator T between weighted Lorentz spaces, namely: To interpret (6) as a Lorentz space inequality we take w(t) = t p−2 v(1/t) and change the variable t to 1/t. Then, (6) is equivalent to the inequality In the case 0 < p ≤ 2 = q, one of the main results included in [13] yields a simple condition on weights u and v which is necessary and sufficient for (6) when T is the Fourier transform (see Theorem 3.3). We observe that (5) is a particular case of (6), just by taking p = q = 2 and u = v = χ (0,z) . As in our previous consideration for general r.i. spaces, in § 3 we are interested in the weak-type boundedness of the Fourier transform F on Lorentz spaces (see Theorem 3.6), and we show that the necessary and sufficient conditions exhibited in Theorem 3.3, proving the boundedness of F from Γ p (w) into Λ 2 (u), 0 < p ≤ 2, also work for characterizing the corresponding weak-type boundedness of F from Γ p (w) into Λ 2,∞ (u) or Γ 2,∞ (u), in the same range of p.

Fourier transform of restricted weak type in rearrangement invariant spaces
For a given r.i. space X, with rearrangement invariant norm · X , let us denote by X its associate space (see [3] for a definition and properties), which is also a r.i. space whose norm, for a function g, is defined by the expression The following result extends (1) to the a priori weaker case of restricted weak-type estimates for F. However, we can still prove that, even under these conditions, there are no other spaces where the boundedness holds, except those sharing the same fundamental function as L 2 .
Proof. The assumption that ϕ X (t) t 1/2 implies that Λ(X) = L 2,1 and M (X) = L 2,∞ ; since F maps L 2 into L 2 continuously, the assertion follows from the fact that For the converse, we observe that F : Λ(X) → M (X) is equivalent, by duality, to F : Λ(X ) → M (X ). Since L 1 + L 2 is the largest r.i. space which is mapped by the Fourier transform into a space of locally integrable functions [4, Theorem 1], it follows that or, equivalently, Taking into account the minimal and maximal properties, respectively, of Λ(X) and M (X), we obtain that Λ(X) ⊂ L 1 + L 2,1 and L 2,∞ ∩ L ∞ ⊂ M (X).
These inclusions can be written in terms of the corresponding fundamental functions of the spaces involved as follows min(t, t 1/2 ) ϕ X (t) max(1, t 1/2 ). Therefore, we obtain that ϕ X (t) t 1/2 for t ≥ 1.

Remark 2.3.
From the proof of Theorem 2.1, we observe that, given a r.i. space X, a necessary condition to obtain the boundedness of F from Λ(X) into M (X) is that both minimal Lorentz spaces Λ(X) and Λ(X ) are contained in L 1 + L 2 , but this is not sufficient.

Weak-type boundedness on weighted Lorentz spaces
The use of level function techniques introduced in [6] and [8] was an important tool to study the boundedness of the Fourier transform in the context of Lorentz spaces, as established in [13]. The following proposition gives the basic properties needed to prove our main result, which is an extension to weighted inequalities on weak-type spaces (see [12, Proposition 2.1 and Proposition 5.1] and also [11] and [10]). Let us denote by L + the set of Lebesgue measurable functions h : (0, ∞) −→ [0, ∞].

(c) If h is bounded and compactly supported then there exists a (necessarily finite or countable) collection of disjoint intervals (a j , b j ), each of finite length, such that
To use this proposition, and following [13], we introduce the class A of averaging operators A defined by where {(a j , b j )} j is a finite or countable collection of disjoint subintervals of (0, ∞) each of finite length. Proposition 3.1 implies that, for h bounded and compactly supported, there exists A h ∈ A such that h o = A h h. The next lemma relates the level function to the averaging operators A.

Lemma 3.2 (Sinnamon [13, Lemma 2.5]). If
For u and v positive measurable functions defined on (0, ∞) and u o the level function of the weight u, let us consider the constant: Using Lemma 3.2, one can prove that K 0 is equivalent to which is independent of u o . We will also need to recall the following result, whose extension to weak-type boundedness on Lorentz spaces is the main goal of Theorem 3.6.
The next lemma provides a very useful expression to calculate necessary conditions for weak-type estimates by means of the action of F on some test functions, and is the analogue (in the weak-type setting) to [13,Corollary 4.8].
Lemma 3.5. If z > 0, w z (t) = min(t −2 , z −2 ), and u ∈ L + , then Proof. For each s > 0, define A (0,s) to be the averaging operator in A based on the single interval (0, s). That is, If t ≥ z, we may take s = t to get On the one hand, each On the other hand, since w z is non-increasing, w z (t) ≤ A (0,t) w z (t); moreover, the largest average of w z on any interval that includes t is the average over (0, t). Thus, for each A ∈ A, Combining these two estimates, applying Lemma 3.2 and using that u o is decreasing, we finally get Theorem 3.6. Let 0 < p ≤ 2, and u, w be two functions in L + . Then, the following conditions are equivalent: Consequently, the strong-type conditions included in Theorem 3.3 are also equivalent to the corresponding weak-type boundedness.
Proof. First, we observe that the finiteness of condition K 1 is equivalent to condition (8) but expressed in terms of the weight v(t) = t p−2 w(1/t). So, we will first restrict to prove that F : Γ p (w) −→ Λ 2,∞ (u) if and only if K 0 < ∞ or, equivalently, K 1 < ∞.
The sufficiency of the condition follows from [13, Theorem 5.1], since K 0 < ∞ or K 1 < ∞ is equivalent to the fact that F : Γ p (w) −→ Λ 2 (u). To prove the necessity, we suppose that the boundedness holds, which, in terms of the weight v(t) = t p−2 w(1/t), by a change of variables implies that Following the same idea as in [13,Corollary 4.8], let us consider z > 0 and w z (t) = min(t −2 , z −2 ). Fix R > 0 and A ∈ A. Then, by [13,Corollary 4.7], there exists a function f such that f * ≤ χ (0,1/z) and for all y ∈ (0, R) Aw z (y) 1/2 ( f ) * (y).