Norm inequalities in some subspaces of Morrey space

BLAISE PASCAL

Justin Feuto

Norm inequalities in some subspaces of Morrey space Volume 21, n^o2 (2014), p. 21-37.

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Norm inequalities in some subspaces of Morrey space

Justin Feuto

Abstract

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.

1. Introduction

For 1 ≤ p, q ≤ ∞, the amalgam of L^q and L^p is the space (L^q, `^p) of functions f on thed-dimensional euclidean spaceR^d which are locally in L^q and such that the sequenceⁿkf χ_Qkk_q^o

k∈Z^d

belongs to `^p(Z^d), where Q_k =^Qd_i=1[k_i, k_i+ 1),χ_Qk denoting the characteristic function ofQ_kand k·k_q the usual Lebesgue norm inL^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.

Let 1≤p, q≤ ∞.

• (L^q, `^q) =L^q

Keywords: Amalgams spaces, fractional maximal operator, Riesz potential, Hilbert transform.

Math. classification:42B35, 42B20, 42B25.

• L^q∪L^p ⊂(L^q, `^p) if q≤p,

• (L^q, `^p)⊂L^q∩L^p ifp≤q,

• (L^q, `^p) is a Banach space when equipped with the norm kfk_q,p=

nkf χ_Qkk_q^o

k∈Z^d

_`p

if we identify functions that differ only on null subset of R^d.

• For 1≤p, q <∞, the dual space of (L^q, `<sup>p</sup>) is (L<sup>q0</sup>, `^p0).

In this definition of amalgam spaces, we can replace the cubes Qk of side length 1 by cubes Q^r_k = ^Qd_i=1[rki, r(ki+ 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

rkfk_q,p:=

f χQ^r_k

_q

k∈Z^d

`^p

and

rkfgk_q,p =











R

R^d

f χ_B(y,r)^p

qdy ¹_p

if p <∞ ess sup_y∈Rd

f χ_B(y,r)

q if p=∞

,

whereB(y, r) is the ball centered aty with radiusr. It is easy to see that for any r >0, andf ∈(L^q, `^p), there exists a constant C_r>0 depending only on r such that

C_r⁻¹kfk_q,p ≤ _rkfk_q,p ≤Crkfk_q,p, while

rkfkg_q,p≈r^dpkfk_q,p.¹ (1.1) 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

1Hereafter we propose the following abbreviationA≈Bfor the inequalitiesC⁻¹A≤ B≤CA, whereC 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→Cbe a measurable function such that

|K(x)| ≤D|x|^γ−dχ_Q(x) +D|x|^β−dχ_Rd\Q(x) (1.2) where 0 < γ, β < d. Then the operator I_γ^β defined by

I_γ^βf(x) = Z

R^d

K(x−y)f(y)dy

is bounded from (L^q, `<sup>p</sup>) to (L<sup>q∗</sup>, `^p∗), with 0< _q¹∗ = ¹_q −^γ_d and 0< _p¹∗ =

1

p −^β_d. An immediate consequence of the above result is the boundedness of the Riesz potential I_γ from (L^q, `<sup>p</sup>) to (L<sup>q∗</sup>, `^p∗). We recall that I_γf is defined by

I_γf(x) = Z

R^d

f(y)

|x−y|^d−γdy (1.3)

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 7→ f(· −x) for x ∈ R^d which are isometric in Lebesgue spaces are just uniformly bounded in amalgam spaces equipped with the norm k·k_q,p (it is isometric when one uses the continuous norm of Dobler). Dilation operators δ^q_r :f 7→r^dqf(r·) also behave differently in these spaces. In fact, there is no real number α >0 for which we have

kδ_r^qfk_q,p≈ kfk_q,p r >0. (1.4) Fofana in [13] (see also [11, 12]) considered normed spaces denoted (L^q, `<sup>p</sup>)<sup>α</sup> which are subspaces of (L<sup>q</sup>, `^p), satisfying property (1.4), and named these spaces integrable fractional mean function spaces. For 1≤q < αfixed and p going from α to ∞, these spaces form a chain of Banach spaces begin- ning with Lebesgue space L^α and ending by the classical Morrey’s space L^q,d(1−αq)= (L^q, `^∞)^α. 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:

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 sub- script, such asC0, do not change in different occurrences. IfEis a measur- able subset ofR^d, then |E|stands for its Lebesgue measure. The notation A 4B 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.

2. Definition and basic properties of (L^q, `^p)^α spaces

For 1≤q, p, α≤ ∞, the space (L^q, `<sup>p</sup>)<sup>α</sup>:= (L<sup>q</sup>, `^p)^α(R^d) consists of those elements of (L^q, `^p) such that

kfk_q,p,α:= sup

r>0

kδ_r^αfk_q,p<∞,

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]).

(1) ((L^q, `^p)^α,k·k_q,p,α) is a complex Banach space.

(2) If 1≤q₁ ≤q₂≤α≤p₁ ≤p₂ ≤ ∞, then we have kfk_q

1,p2,α≤ kfk_q

2,p1,α≤Ckfk_α. Notice that

rkfk_q,p,α:=kδ_r^αfk_q,p=r^d(1α−1q)_rkfk_q,p, (2.1) so that

kfk_q,p,α= sup

r>0

r^d(α1−1q)rkfk_q,p≈sup

r>0

r^{d(1α−1p−1q)}rkfgk_q,p. (2.2)

Proposition 2.2. Let 1≤q ≤α≤p≤ ∞.

(1) For f ∈(L^q, `^p)^α, we have

kδ_r^αfk_q,p,α=kfk_q,p,α.

(2) The space(L^q, `<sup>p</sup>)<sup>α</sup> is the biggest norm space which is continuously included in (L<sup>q</sup>, `^p) and for which sup_r>0kδ^α_rfk<∞.

Proof. The first assertion is an immediate consequence of the definition of the norm k·k_q,p,α. For the second, let (E,k·k) be a norm space for which there exists C >0 such that we have

kfk_q,p ≤Ckfk and sup

r>0

kδ_r^αfk<∞, (2.3) for all f ∈E. Then for f ∈E, we have

sup

r>0

kδ_r^αfk_q,p≤Csup

r>0

kδ^α_rfk<∞,

so that f ∈(L^q, `^p)^α by definition.

As we say in the introduction, the family {(L^q, `<sup>p</sup>)<sup>α</sup>}<sub>α≤p≤∞</sub> consists of distinct spaces. To see this on the real line, we let q = 1. A positive function f onR belongs to (L<sup>1</sup>, `^p)^α if and only if there exists a constant C <∞ such that

r^α1 kE_rfk_`p≤C, r >0 (2.4) where E_rf is the sequence defined by

(E_rf)_k= 1 r

Z

r(k+I)

f(y)dy,

where I = [0,1). Notice that it is enough to have condition (2.4) just for r = 2^m, m∈Z. Let 1≤p1 < p2 <∞, anda = (a_n)_n∈N be a sequence of positive reals numbers which belongs to `<sup>p2</sup> without being in `^p1. Put for all k∈Z

λk=

0 if k6= 2ⁿ for all n∈N

a_n if k= 2ⁿ ,

and let us consider the functionf defined byf(x) =λ_kifx∈k+I,k∈Z. We claim that f ∈(L¹, `<sup>p2</sup>)<sup>α</sup>\(L<sup>1</sup>, `^p1)^α. The function does not belong to (L¹, `<sup>p1</sup>), since the sequencea∈/ `^p1. Let us prove now thatf ∈(L¹, `^p2)^α. Fix r= 2^m, withm∈Z.

• If m ≤0 then (rk, r(k+ 1))∩Z=∅, so that f(x) = λ_brkc for all x∈I_k^r. Thus

1 r f χ_I^r

k

₁ = 1

r

Z r(k+1) rk

f(x)dx=λ_brkc, and

r^α1

1 r

f χI_k^r

₁

k∈Z

`^p2

= r^α1

nλ_brkc^o

k∈Z

_`p2

= r^α1 kak_`p2

≤ kak_`p2 .

• Ifm >0 then [rk, r(k+ 1)) =∪^r(k+1)−1_j=rk [j, j+ 1), such that r^α1

1 r f χ_I^r

k

₁

k∈Z

`^p2

= r^α1−1







r(k+1)−1

X

j=rk

λj





k∈Z

`^p2

= r^α1−1kak_`p2

≤ kak_`p2. The assumption follows.

For q < α < p, the weak Lebesgue space L^α,∞ is a subset of (L^q, `^p)^α. Moreover,

kfk_q,p,α≤Ckfk^∗_α,∞, where the quasi-norm k·k^∗_p,q is defined by

kfk^∗_p,q=





 p

q

R∞ 0

t^1pf^∗(t)^{q dt}_t ¹_q

if 1≤p, q <∞

sup_t>0t^1pf^∗(t) if 1≤p≤ ∞andq =∞ ,

and f^∗ being the non increasing function rearrangement off on R^d, i.e., f^∗(t) = infⁿα >0 :ⁿx∈R^d:|f(x)|> α^o≤t^o, t >0.

It is well known that sup_t>0t^1pf^∗(t) = sup_α>0αⁿx∈R^d:|f(x)|> α^o

1 p.

3. Some new results in amalgams

It is a classical result (see [3, 11, 14]) that iff ∈(L^q1, `<sup>p1</sup>) andg∈(L<sup>q2</sup>, `^p2) with _p¹

1+_p¹

2 ≥1 and _q¹

1+_q¹

2 ≥1, thenf∗g∈(L^q, `^p), where ¹_p = _p¹

1+_p¹

2−1 and ¹_q = _q¹

1 +_q¹

2 −1 . Moreover,

kf∗gk_q,p≤Ckfk_q

1,p1kgk_q

2,p2.

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.

Theorem 3.1 (Theorem 2.10.1 [20]). Let 1 ≤ p1, p2, q1, q2 ≤ ∞ with

1 p1 +_p¹

2 >1. If f ∈L^p1,q1 andg∈L^p2,q2 then f∗g∈L^p,q, where 1

p₁ + 1

p₂ −1 = 1 p and q ≥1 such that

1 q₁ + 1

q₂ ≥ 1 q. Moreover, we have

kf∗gk^∗_p,q≤Ckfk^∗_p

1,q1kgk^∗_p

2,q2.

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 (kf χ_Qkk^∗_p,q)_k∈

Z^d belongs to `<sup>t,s</sup>. We put kfk<sub>(L</sub>q,p,`^t,s) =(kf χ_Qkk^∗_p,q)_k∈

Z^d

∗

t,s. The following result is established as the classical one in amalgams, just by replacing the Young inequality by that of Theorem 3.1.

Theorem 3.2. Let

1≤ p1, p2, q1, q2 ≤ ∞ 1≤ r₁, r₂, s₁, s₂ ≤ ∞ with

( ₁

p1 +_p¹

2 > 1

1 r1 +_r¹

2 > 1 ,

f ∈(L^p1,q1, `<sup>r1,s1</sup>) and g∈(L<sup>p2,q2</sup>, `^r2,s2). Then f∗g∈(L^p,q, `^r,s), where ( ₁

p = _p¹

1 + _p¹

2 −1

1

r = _r¹

1 +_r¹

2 −1 and

( ₁

q1 +_q¹

2 ≥ ¹_q

1 s1 + _s¹

2 ≥ ¹_s .

Moreover, there exists a constant C >0 such that

kf∗gk_(Lp,q,`<sup>r,s</sup>) ≤Ckfk<sub>(L</sub>p1,q1,`^r1,s1)kgk_(Lp2,q2,`^r2,s2).

This result can be seen as one realization of the general result (convo- lution triples) stated in Theorem 3 of [10].

The next result follows immediately using the fact that L^p,p =L^p. Corollary 3.3. Let 1≤p₁, p₂≤ ∞ with _p¹

1 +_p¹

2 >1 and 1< q₁, q₂ <∞ with _q¹

1 + _q¹

2 > 1. If f ∈ (L^q1, `<sup>p1</sup>) and g ∈ (L<sup>q2,∞</sup>, `^p2,∞) then f ∗g ∈ (L^s, `^r), where ¹_r = _p¹

1+_p¹

2−1and ¹_s = _q¹

1+_q¹

2−1. Moreover, there exists C >0 such that

kf∗gk_s,r≤Ckfk_q

1,p1kgk_(Lq2,∞,`^p2,∞).

Notice that a functionKwhich satisfies (1.2) belongs to (L^d−dγ,∞, `<sup>d−dβ,∞</sup>) so that the boundedness of the operator I<sub>γ</sub><sup>β</sup> from (L<sup>q</sup>, `^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 cased= 1, the Hilbert transform of a function f is the function Hf defined by

Hf(x) = 1 πv.p

Z +∞

−∞

f(y) x−ydy, where v.p denotes the Cauchy principal value.

Proposition 3.4. Let d= 1 and1< q, p <∞. The Hilbert transform H is bounded on (L^q, `^p).

Proof. To simplify the formulas, we adopt the abbreviation f_k forf χ_Qk. Let m∈Z. We have

(Hf)_m = ^X

n−m∈Q˜

Hf_nχ_[m,m+1)+ ^X

n−m /∈Q˜

Hf_nχ_[m,m+1) =F_m+G_m where ˜Q= (−2,2). Since the Hilbert transform is bounded inL^qit follows that

kF_mk_q ≤C ^X

n−m∈Q˜

kf_nk_q

so that

X

m

kF_mk^p_q

!_p¹

≤C ^X

n

kf_nk^p_q

!¹_p .

For x∈Qm andn−m /∈Q, we have˜ Hfn(x) = 1

π 1

m−n Z

R

fn(y)dy+ Z

R

( 1

x−y − 1

m−n)fn(y)dy

, with

1

x−y − 1 m−n

≤ 2

(|m−n| −1)|m−n|, fory∈Q_n. So, if u is the sequence defined byu_n=^R_nⁿ⁺¹f(y)dy, then we have

kHuk_{`</sub>p ≤Ckuk<sub>`}p =Ckfk_q,p. Besides,





 X

m∈Z



 Z m+1

m

X

n−m /∈Q˜

Z

R

( 1

x−y − 1

m−n)f_n(y)dy

q

dx





p q





1 p

≤





 X

m∈Z



 X

n−m /∈Q˜

2

(|m−n| −1)|m−n|

Z

R

|f_n(y)|dy





p





1 p

.

(3.1)

Thus, if we consider the sequencesv andwdefined respectively byv0 = 0 and v_n= _n¹₂ forn6= 0, andw_n=^R

R|f_n(y)|dy, we havev∈`<sup>1</sup> andw∈`^p with

kwk_`p ≤ kfk_q,p.

Since the second member of (3.1) is the `<sup>p</sup> norm of v∗w, it is less than kvk<sub>`</sub>1kfk_q,p.It follows that



 X

m∈Z

kG_mk^p_q





1 p

≤Ckfk_q,p,

which ends the proof.

4. Norm inequalities involved in (L^q, `^p)^α spaces

The Riesz potential Iγf (0< γ < d) of a function f as defined by (1.3) is closely related to the fractional maximal function M_γf defined by

M_γf(x) = sup

r>0

|B(x, r)|^γd−1 Z

B(x,r)

|f(y)|dy.

It follows from the definitions that

M_γf ≤I_γ|f|. (4.1)

Let r >0 and α≥1. We have

I_γ(δ^α_rf) =δ_r^α∗(I_γf) andM_γ(δ^α_rf) =δ_r^α(M_γf) (4.2) with _α¹∗ = _α¹ −^γ_d and, on the real line,

H(δ_r^αf) =δ_r^α(Hf), (4.3) so that the next proposition follows from the definition of (L^q, `<sup>p</sup>)<sup>α</sup>spaces, the boundedness of Iγ from (L<sup>q</sup>, `^p) to (L^q∗, `<sup>p∗</sup>) and the boundedness of Hilbert transform on (L<sup>q</sup>, `^p) in the case d= 1.

Proposition 4.1. Let 1< q≤α≤p <∞ and ^γ_d ≤ ¹_p.

(1) The Riesz potential Iγ and the associated fractional maximal op- erator M_γ are bounded from (L^q, `<sup>p</sup>)<sup>α</sup> to(L<sup>q∗</sup>, `^p∗)^α∗.

(2) When d= 1, the Hilbert transform is bounded on(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 M₀. We recall that for a locally integrable function f, the (centered) Hardy-Littlewood maximal function M₀f is defined by

M₀f(x) = sup

r>0

|B(x, r)|⁻¹ Z

B(x,r)

|f(z)|dz.

Proposition 4.2. (1) Let 1< q≤α ≤p≤ ∞. Then

kM₀fk_q,p,α≤Ckfk_q,p,α. (4.4)

(2) For q= 1, we have

kM₀fk_(L1,∞,L^p)^α ≤Ckfk_q,p,α, (4.5) where for f ∈L^1,∞_loc , andp <∞,

kfk_(L1,∞,L^p)^α := sup

r>0

r^{d(1α−1p−1)} Z

R^d

(f χ_B(y,r)^∗

1,∞)^pdy ¹_p

.

Proof. Ifα∈ {q, p}then we recover the classical result in Lebesgue spaces.

We just have to consider the cases where 1 < q < α < p; but, if p =∞ this is nothing but Theorem 1 of [4]. We suppose now that p <∞. Using the classical inequality

Z

R^d

(M₀f)^q(x)φ(x)dx≤C Z

R^d

|f(x)|^q(M₀φ)(x)dx, (4.6) for all measurable functions f and φ > 0 given by Theorem 2.12 of [15]

with the characteristic function of a ball as φ, and proceeding as in the proof of Theorem 1 of [4], we obtain

rkM^₀fk^q_q,p≤C (

2rkfgk^q_q,p+

∞

X

k=1

1

(2^k−1)^{d 2k+1r}kf^gk^q_q,p )

,

for f ∈(L^q, `^p)^α. Let us multiply both sides by r^{dq(α1−1q−p1)}, we obtain (r^{d(α1−1q−p1)} _rkM^₀fk_q,p)^q 4 2^{−dq(α1−1q−1p)}kfk^q_q,p,α

+

∞

X

k=1

2^{(k+1)dq(1q+1p−α1)}

(2^k−1)^d kfk^q_q,p,α 4 kfk^q_q,p,α,

sincedq(¹_q+¹_p−_α¹)−d=d(_p^q−_α^q)<0. Thus, taking the supremum over r >0, we obtain

kM₀fk_q,p,α≤Ckfk_q,p,α, using Relation (2.2).

As for the caseq= 1, the proof is the same using the following inequality Z

{x∈R^d:M0f(x)>t}χ_B(y,r)(x)dx≤ C t

Z

R^d

|f(x)|(M₀χ_B(y,r))(x)dx

instead of (4.6).

We now use the Proposition 4.2 to give some estimates of the Riesz potential in (L^q, `^p)^α spaces.

Theorem 4.3. Let 1≤q≤α≤p≤ ∞and 0< ^γ_d < _α¹. Put _α¹∗ = _α¹ −^γ_d. (1) If 1< q then we have

kI_γfk_q,˜˜p,α∗ ≤Ckfk¹⁻_q,p,α^αdγkfk_q,∞,α^αdγ (4.7) with ¹_q˜ = ¹_q −^α_q^γ_d and ¹_p˜= ¹_p −^α_p^γ_d.

(2) If q = 1 then we have

kI_γfk_(L1,∞,L^p˜)^α∗ ≤Ckfk¹⁻

α dγ 1,p,α kfk

α dγ

1,∞,α. (4.8)

Proof. Let 1≤q ≤α≤p≤ ∞, 0< γ < dand f ∈(L^q, `^p)^α.

If p=∞, then the theorem is exactly Theorem 3.1 of [1]. We consider the case where 1≤q≤α≤p <∞. Since forα∈ {q, p}the space (L^q, `^p)^α is equal to the Lebesgue spaceL^α, we just have to look at the case where 1≤q < α < p <∞.

For all >0, we have Iγf(x) =

Z

|x−y|≤

f(y)

|x−y|^d−γdy+ Z

|x−y|>

f(y)

|x−y|^d−γdy. (4.9) We recall that (L^q, `^p)^αis a subspace of the Morrey spaceL^q,d(1−αq), so that M₀f is finite almost everywhere inR^d. Following the proof of Theorem 2 in [4], we can say that

Z

|x−y|≤

f(y)

|x−y|^d−γdy

≤C^γM₀f(x), and

Z

|x−y|>

f(y)

|x−y|^d−γdy

≤C^γ−dαkfk_q,∞,α. It comes that

|I_γf| ≤C^γM₀f +^γ−dαkfk_q,∞,α. Taking =

M0f kfk_q,∞,α

−^α_d

, we have

Iγf ≤C(M₀f)^−αγd+1kfk_q,∞,α^αdγ .

The inequalities (4.7) and (4.8) follow respectively from the inequalities

(4.4) and (4.5).

The following corollary is a consequence of the fact that kfk_q,∞,α ≤ Ckfk_q,p,α for all 1≤q≤α≤p≤ ∞.

Corollary 4.4. Let 1≤q≤α≤p≤ ∞and 0< ^γ_d < _α¹. Put _α¹∗ = _α¹ −^γ_d

(1) If 1< q then we have

kI_γfk_˜q,˜p,α∗ ≤Ckfk_q,p,α (4.10) with ¹_q˜ = ¹_q −^α_q^γ_d and ¹_p˜= ¹_p −^α_p^γ_d.

(2) If q = 1 then we have

kI_γfk_(L1,∞,L^p˜)^α∗ ≤Ckfk_1,p,α. (4.11) Using a different method, Dosso et al in [7] proved that if 0< ^γ_d < _α¹−¹_p then

kI_γfk_q∗,p,α^∗ ≤Ckfk_q,p,α (4.12) where _q¹∗ = ¹_q − ^γ_d.The next result is a generalization of Theorem 2.1 of [8], and its proof is just an adaptation of that given there.

Theorem 4.5. Let1< q <∞andq≤α≤p. IfT is a sublinear operator which is bounded on L^q and satisfy

|T f(x)| ≤C Z

R^d

|f(y)|

|x−y|^ddy x /∈suppf, (4.13) for any f ∈L¹ with compact support, then T is also bounded on (L^q, `^p)^α. Proof. We assume that 1 < q < α < p < ∞, since the case p = ∞ is Theorem 2.1 of [8] and when α ∈ {q, p} we have nothing to prove. Fix y ∈R^d andr >0 we have

f =f χ_B(y,2r)+

∞

X

k=1

f χ_B(y,2^k+1_r)\B(y,2^k_r) so that taking the L^q-norm on the ball B(y, r), we obtain

T f χ_B(y,r)

q≤C f χ_B(y,2r)

q+

∞

X

k=1

(2^k)^−dpf χ_B(y,2k+1r)

_q

! , using the L^q boundedness of T and relation (4.13). Taking the L^p-norm of both sides with respect to y, it comes that

rkT f]k_q,p≤C _2rkfk^g_q,p+

∞

X

k=1

(2^k)^−dq ₂k+1rkfkg_q,p

!

. (4.14)

We multiply both sides of (4.14) by r^{d(α1−p1−1q)}. It follows that r^{d(α1−1p−1q)} rkT fk]_q,p ≤C 1 +

∞

X

k=1

(2^k)^d(1p−α1)

!

kfk_q,p,α, r >0.

Taking the supremum over r >0, Relation (2.2) yields the result.

As mentioned in [8], Condition (4.13) can be satisfied by many oper- ators 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 [b, T] by [b, T]f(x) = T(bf)(x) − b(x)T f(x), and recall that the space BM O consists of functionsb∈L¹_loc satisfying kbk_{BM O} <∞, where

kbk_{BM O}:= sup

r>0,x∈R^d

1

|B(x, r)|

Z

B(x,r)

b(y)−b_B(x,r)dy, (4.15) with b_B(x,r) denoting the average overB(x, r) of b.

Theorem 4.6. Let 1 < q < ∞, q ≤ α ≤ p ≤ ∞ and b ∈ BM O. If a linear operator T satisfies (4.13) and [b, T]is bounded on L^q, then [b, T] is also bounded on (L^q, `^p)^α.

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 andr >0,

[b, T]f χ_B(y,r)

q 4f χ_B(y,2r)

q

+

∞

X

k=1

1 (2^kr)^d

"

Z

B(y,r)

Z

B(y,2^k+1r)

|b(x)−b(z)| |f(x)|dx

!q

dz

#¹_q , so that the use of John-Nirenberg theorem on BM O functions (see The- orem 7.1.6 in [17]) and the properties of BM O lead to

[b, T]f χ_B(y,r)

q 4f χ_B(y,2r)

q+kbk_{BM O}

∞

X

k=1

(2^k)^−dq f χ_B(y,2^k+1_r)

q.

We end the proof as above.