Norm inequalities in some subspaces of Morrey space

We give norm inequalities for some classical operators in amalgam spaces and in some subspaces of Morrey space. Inégalités en norme dans certains sous-espaces d’espaces de Morrey Résumé Nous établissons des inégalités en norme pour certains opérateurs classiques dans les amalgames et certains sous-espaces d’espaces de Morrey.


Introduction
For 1 ≤ p, q ≤ ∞, the amalgam of L q and L p is the space (L q , p ) of functions f on the d-dimensional euclidean space R d which are locally in L q and such that the sequence f χ Q k q k∈Z d belongs to p (Z d ), where , χ Q k denoting the characteristic function of Q k and · q the usual Lebesgue norm in L q .
Amalgams arise naturally in harmonic analysis and were introduced by N. Wiener in 1926. But its systematic study goes back to the work of Holland [18]. We refer the reader to the survey paper of Fournier and Stewart [14] for more information about these spaces. We list here some of their basic properties.
• (L q , p ) is a Banach space when equipped with the norm if we identify functions that differ only on null subset of R d .
In this definition of amalgam spaces, we can replace the cubes Q k of side length 1 by cubes Q r k = d i=1 [rk i , r(k i + 1)) of side length r or by balls, and we can also consider continuous summation instead of discrete. More precisely, for r > 0, we put where B(y, r) is the ball centered at y with radius r. It is easy to see that for any r > 0, and f ∈ (L q , p ), there exists a constant C r > 0 depending only on r such that We can also consider on amalgam spaces, the continuous norm described in Dobler's master thesis [6] (see also [9] in the case of Wiener algebra).
Many classical results established in Fourier analysis on Lebesgue spaces have extensions in amalgams. For example, Hölder and Young inequalities are just a consequence of the analog in Lebesgue space [14,3]. The Hardy-Littlewood-Sobolev inequality for fractional integrals has been generalized 1 Hereafter we propose the following abbreviation A ≈ B for the inequalities C −1 A ≤ B ≤ CA, where C is a positive constant independent of the main parameters.
to amalgam spaces by Cowling et al in [5]. In fact, they proved a more general result which can be formulated as follows: let Q = [−1, 1) d and K : R d → C be a measurable function such that where 0 < γ, β < d. Then the operator I β γ defined by An immediate consequence of the above result is the boundedness of the Riesz potential I γ from (L q , p ) to (L q * , p * ). We recall that I γ f is defined by when the integral exists.
There are classical properties of Lebesgue spaces which are not fulfilled in amalgam spaces. For example, when q < p the translation operators τ x : f → f (· − x) for x ∈ R d which are isometric in Lebesgue spaces are just uniformly bounded in amalgam spaces equipped with the norm · q,p (it is isometric when one uses the continuous norm of Dobler). Dilation operators δ q r : f → r d q f (r·) also behave differently in these spaces. In fact, there is no real number α > 0 for which we have Fofana in [13] (see also [11,12]) considered normed spaces denoted (L q , p ) α which are subspaces of (L q , p ), satisfying property (1.4), and named these spaces integrable fractional mean function spaces. For 1 ≤ q < α fixed and p going from α to ∞, these spaces form a chain of Banach spaces beginning with Lebesgue space L α and ending by the classical Morrey's space We will see in the next paragraph that the spaces in the chain are distinct.
In this paper we give extensions of norm inequalities in Lebesgue or Morrey spaces to the setting of (L q , p ) α with 1 ≤ q ≤ α ≤ p ≤ ∞. This is often done by using the relation (1.4) and known results in amalgam spaces.
The remaining of this paper is organized as follows:

23
In the second paragraph, we recall the definition of (L q , p ) α spaces and some of their basic properties. Paragraph three is devoted to some norm inequalities in amalgams of Lebesgue and Lorentz spaces which we do not see in the literature, while in paragraph four we establish norm inequalities for some classical operators in the context of our spaces.
Throughout the paper, the letter C is used for non-negative constants that may change from one occurrence to another. Constants with subscript, such as C 0 , do not change in different occurrences. If E is a measurable subset of R d , then |E| stands for its Lebesgue measure. The notation A B will always mean that the ratio A/B is bounded away from zero by a constant independent of the relevant variables in A and B.
Acknowledgement. The author would like to thank Aline Bonami for many helpful suggestions and discussions. He also thanks the referee for his careful and meticulous reading of the manuscript.

Definition and basic properties of
with the usual convention that 1 ∞ = 0. As proved in [11,13] the space (L q , p ) α is non trivial if and only if q ≤ α ≤ p; thus in the remaining of the paper we will always assume that this condition is fulfilled. We have the following properties. Proposition 2.1 ([11, 12, 13]).
Notice that Norm inequalities Proof. The first assertion is an immediate consequence of the definition of the norm · q,p,α . For the second, let (E, · ) be a norm space for which there exists C > 0 such that we have As we say in the introduction, the family {(L q , p ) α } α≤p≤∞ consists of distinct spaces. To see this on the real line, we let q = 1. A positive function f on R belongs to (L 1 , p ) α if and only if there exists a constant C < ∞ such that r where E r f is the sequence defined by where I = [0, 1). Notice that it is enough to have condition (2.4) just for r = 2 m , m ∈ Z. Let 1 ≤ p 1 < p 2 < ∞, and a = (a n ) n∈N be a sequence of positive reals numbers which belongs to p 2 without being in and let us consider the function f defined by We claim that f ∈ (L 1 , p 2 ) α \ (L 1 , p 1 ) α . The function does not belong to (L 1 , p 1 ), since the sequence a / ∈ p 1 . Let us prove now that f ∈ (L 1 , p 2 ) α . Fix r = 2 m , with m ∈ Z.
For q < α < p, the weak Lebesgue space L α,∞ is a subset of (L q , p ) α . Moreover, and f * being the non increasing function rearrangement of f on R d , i.e., It is well known that sup t>0 t

Some new results in amalgams
It is a classical result (see [3,11,14]) that if f ∈ (L q 1 , p 1 ) and g ∈ (L q 2 , p 2 ) with 1 p 1 + 1 p 2 ≥ 1 and 1 The proof of this result uses the Young inequality in Lebesgue spaces and in the space of real sequences. We can weaken the second member of the inequality if instead of the Young inequality we use the following result.  [20]). Let 1 ≤ p 1 , p 2 , q 1 , q 2 ≤ ∞ with Moreover, we have f * g * p,q ≤ C f * p 1 ,q 1 g * p 2 ,q 2 . Let 1 ≤ p, q, t, s ≤ ∞. The amalgam of the Lorentz space L p,q and its discrete version t,s is the space of measurable functions f locally in L p,q and such that the sequence ( f χ Q k * p,q ) k∈Z d belongs to t,s . We put . The following result is established as the classical one in amalgams, just by replacing the Young inequality by that of Theorem 3.1.

J. Feuto
This result can be seen as one realization of the general result (convolution triples) stated in Theorem 3 of [10].
The next result follows immediately using the fact that L p,p = L p .
Notice that a function K which satisfies (1.2) so that the boundedness of the operator I β γ from (L q , p ) to (L q * , p * ) is just a consequence of Corollary 3.3.
The next result which gives the continuity of the Hilbert transform in the amalgam spaces is well known. However, since we couldn't find its proof in the literature, we give one here. We recall that in the case d = 1, the Hilbert transform of a function f is the function Hf defined by where v.p denotes the Cauchy principal value.

Norm inequalities
For x ∈ Q m and n − m / ∈Q, we have So, if u is the sequence defined by u n = n+1 n f (y)dy, then we have Besides, Thus, if we consider the sequences v and w defined respectively by v 0 = 0 and v n = 1 n 2 for n = 0, and w n = R |f n (y)| dy, we have v ∈ 1 and w ∈ p with w p ≤ f q,p .
Since the second member of (3.1) is the p norm of v * w, it is less than v 1 f q,p . It follows that which ends the proof.

Norm inequalities involved in (L q , p ) α spaces
The Riesz potential I γ f (0 < γ < d) of a function f as defined by (1.3) is closely related to the fractional maximal function M γ f defined by |f (y)| dy. 29 Let r > 0 and α ≥ 1. We have so that the next proposition follows from the definition of (L q , p ) α spaces, the boundedness of I γ from (L q , p ) to (L q * , p * ) and the boundedness of Hilbert transform on (L q , p ) in the case d = 1.
(1) The Riesz potential I γ and the associated fractional maximal operator M γ are bounded from (L q , p ) α to (L q * , p * ) α * .
We will prove that we can have a result better than the above for Riesz potential. For this purpose, we need a control of the classical Hardy-Littlewood maximal operator M 0 . We recall that for a locally integrable function f , the (centered) Hardy-Littlewood maximal function M 0 f is defined by
As mentioned in [8], Condition (4.13) can be satisfied by many operators such as Bochner-Riesz operators at the critical index, Ricci-Stein's oscillatory singular integral, C. Fefferman's singular multiplier, and some Calderón-Zygmund operators.
We define the linear commutator  Proof. If α < p = ∞ then the result is just Theorem 2.2 of [8], and when α = q or α = p, there is nothing to prove. We suppose now that 1 < q < α < p < ∞. Proceeding as in the proof of Theorem 2.2 [8], we have that for all y ∈ R d and r > 0, so that the use of John-Nirenberg theorem on BM O functions (see Theorem 7.1.6 in [17]) and the properties of BM O lead to We end the proof as above.
The next result gives a norm equivalence of the Riesz potential and associated fractional maximal operator when we deal with non negative measurable functions. Proof. In view of inequality (4.1) it suffices to prove that I γ f q,p,α M γ f q,p,α . We can assume that 1 ≤ q < α < p < ∞, since the case α ∈ {q, p} is solved in Theorem 1 of [19] where the last inequality comes from Theorem 5.2 of [16]. Taking into account this inequality and the relation (4.17), the L p -norm of both sides of the inequality leads to It follows from the above inequality and Relation (2.2) that Thus the result follows by taking the supremum over r > 0.