An interpolation theorem between one-sided Hardy spaces

c 2006 by Institut Mittag-Leffler. All rights reserved

An interpolation theorem between one-sided Hardy spaces

Sheldy Ombrosi, Carlos Segovia and Ricardo Testoni

Abstract. In this paper we generalize an interpolation result due to J.-O. Str¨omberg and A. Torchinsky to the case of one-sided Hardy spaces. This generalization is important in the study of the weak type (1,1) for lateral strongly singular operators. We shall need an atomic decomposition in which for every atom there exists another atom supported contiguously at its right. In order to obtain this decomposition we have developed a rather simple technique to break up an atom into a sum of others atoms.

1. Definitions and prerequisites

Let f(x) be a Lebesgue measurable function defined on R. The one-sided Hardy–Littlewood maximal functionsM⁺f(x) andM⁻f(x) are defined as

M⁺f(x) = sup

h>0

1 h

_x+h

x |f(t)|dt and M⁻f(x) = sup

h>0

1 h

_x

x−h|f(t)|dt.

As usual, a weight ω is a measurable and non-negative function. If E⊂R is a Lebesgue measurable set, we denote itsω-measure byω(E)=

Eω(t)dt.A function f(x) belongs toL^s_ω,0<s<∞, if fL^s_ω=(_∞

−∞|f(x)|^sω(x)dx)^1/sis finite.

A weight ω belongs to the class A⁺_s, 1<s<∞,defined by E. Sawyer in [6], if there exists a constantC_ω,s such that if−∞<a<b<c<∞then

b a

ω(t)dt

c b

ω(t)^−1/(s−1)dt _s−1

≤C_ω,s(c−a)^s.

In the limit case of s=1 we say that ω belongs to the class A⁺₁ if, and only if, M⁻ω(x)≤C_ω,1ω(x) almost everywhere. We also consider the class A⁺_∞=

s≥1A⁺_s. In general a weight belonging toA⁺_∞can be equal to zero in a set with positive measure. For simplicity in this paper we shall assume all weights being positive almost everywhere.

As usual,C₀^∞(R) denotes the set of all functions with compact support having derivatives of all orders. We denote by D the space of all functions in C₀^∞(R) equipped with the usual topology and byD the space of distributions onR.

Given a positive integerγ andx∈R, we shall say that a functionψinC₀^∞(R) belongs to the class Φ_γ(x) if there exists a bounded intervalI_ψ=[x, b] containing the support ofψ such thatD^γψsatisfies

|I_ψ|^γ+1D^γψ_∞≤1.

Forf∈D we consider the one-sided maximal function f₊^∗_,γ(x) = sup{|f, ψ|:ψ∈Φ_γ(x)}.

Let ω∈A⁺_s, 0<p<∞ and that γ be a positive integer such that p(γ+1)>s.

We say that a distribution f in D belongs to H₊^p(ω) if the “p-norm” f_H^p_+(ω)= (_∞

−∞f₊^∗_,γ(x)^pω(x)dx)^1/p is finite. These spaces have been defined by L. de Rosa and C. Segovia in [4]. They also prove in [5] that the definition does not depend onγ.

LetN be an integer. A functiona(x) defined on Ris called an (∞, N)-atom if there exists an intervalI containing the support ofa(x), such that

(i)a∞≤1;

(ii) the identity

Ia(y)y^kdy=0 holds for every integerk,0≤k≤N.

The following theorem gives an atomic decomposition of theH₊^p(ω) spaces.

Theorem 1. Let ω∈A⁺_s and 0<p<∞. Then there is an integer N(p, ω)with the following property:given anyf∈H₊^p(ω)andN≥N(p, ω),we can find a sequence {λ_k}^∞_k=1 of positive coefficients and a sequence{a_k}^∞_k=1 of (∞, N)-atoms with sup- port contained in intervals {I_k}^∞_k=1 respectively, such that the sum_∞

k=1λ_ka_k con- verges unconditionally tof both in the sense of distributions and in theH₊^p(ω)-norm.

Moreover

C₁ ^∞

k=1

λ_kχ_Ik L^pω

≤ f_H+^p_(ω),

holds with C₁>0 not depending onf.

Conversely, if we have a sequence {λ_k}^∞_k=1 of positive coefficients and a se- quence {a_k}^∞_k=1 of (∞, N)-atoms with support contained in intervals {I_k}^∞_k=1, re- spectively, such that _∞

k=1λ_kχ_IkL^p_ω<∞, then _∞

k=1λ_ka_k converges uncondi- tionally both in the sense of distributions and in the H₊^p(ω)-norm to an element

f∈H₊^p(ω)and

f_H+^p_(ω)≤C₂ ^∞

k=1

λ_kχ_Ik L^pω

,

holds with C₂ not depending on f. Moreover, in casef∈L¹_loc(R)then

^∞

k=1

λ^r_kχ_Ik(x)

≤C_rf₊^∗_,N(x)^r, (1)

holds for every r>0 withC_r=c^r2^r/(2^r−1), wherec is an absolute constant.

For 0<p≤1,this result is essentially proved in [4], and it can be generalized to the casep>1 following the ideas of Theorem 1 of Chapter VII in [7].

Through this paper the lettersc andCwill mean positive finite constants not necessarily the same at each occurrence.

2. Statement of the main results

We shall denote by Ω the strip in the complex plane{z∈C:0≤Re(z)≤1}. For 0<p₀, p₁<∞and z∈Ω we definep(z) as

1

p(z)=1−z p₀ +z

p₁.

Letωandνbe weights inA⁺_∞such thatω(x)>0 andν(x)>0 for almostx∈R. Given 0≤u≤1 we define

µ(u) =u p(u) p₁ ,

and, using the same notation,

µ(u)(x) =ω(x)^1−µ(u)ν(x)^µ(u). (2)

It will be proved in Lemma 4 thatµ(u)∈A⁺_∞.

In the same way, we defineq(z) and ˜µ(u), for 0<q₀, q₁<∞,ωand ˜ν weights in A⁺_∞.

Letsbe a fixed real number, 0<s<1, we shall putp=p(s),q=q(s),µ=µ(s),

˜ µ= ˜µ(s).

With this notation we have the following result.

Theorem 2. LetT_z: H₊^p0(ω)+H₊^p1(ν)!H₊^q0(ω)+H ₊^q1(˜ν)be a family of linear operators forz∈Ωsuch that iff∈H₊^p0(ω)+H₊^p1(ν)thenT_zf∗φ_t(x)is uniformly con- tinuous and bounded forz∈Ωand(x, t)in any compact subset of{(x, t):x∈Randt>0} and analytic for z in the interior ofΩ.

If, in addition, for some constantsk, β >0, T_itf_H^q0

+(ω)≤ke^β|t|f_H^p0

+(ω) and T_1+itf_H^q1

+(˜ν)≤ke^β|t|f_H^p1 +(ν), then there exists a constantC such that

T_sf_H+^q_(˜µ)≤Cf_H+^p_(µ)

for all f∈H₊^p(µ).

This type of result in complex interpolation theory was introduced by A. P.

Calder´on in [1]. By standard arguments, the previous theorem is a consequence of the following result.

Theorem 3. Letφ∈C₀^∞(R)with supp(φ)⊂(−∞,0]and

Rφ=0.

(a) Let f(z) be a function of the complex variable z∈Ω which takes values in H₊^p0(ω)+H₊^p1(ν), such that F(x, t, z)=f(z)∗φ_t(x) is uniformly continuous and bounded forz∈Ωand(x, t)in any compact subset of{(x, t):x∈Randt>0}and ana- lytic forzin the interior ofΩ. Iff(it)∈H₊^p0(ω),sup_tf(it)_H^p0

+(ω)<∞,f(1+it)∈H₊^p1(ν) andsup_tf(1+it)_H^p₊¹_(ν)<∞, thenf(s)∈H₊^p(µ)and

f(s)_H+^p_(µ)≤c

supt f(it)_H+^p0_(ω)1−s

supt f(1+it)_H^p₊¹_(ν)s .

(b)If f∈H₊^p(µ), there exists a function f(z)such that F(x, t, z)has the prop- erties stated in the first part, f(s)=f and

f(u+it)^p(u)

H^p(u)₊ (µ(u))≤cf^p_Hp +(µ)

(3)

holds forz=u+it∈Ωandc does not depend onf.

S. Chanillo in [2] proved that strongly singular operators associated with kernels of the typee^i|x|−b/|x|for 0<b<∞satisfy a weighted weak type (1,1) inequality for weights in Muckenhoupt’s class A₁. Chanillo solves a key point of his proof with a version of Theorem 3 for weights in the class A_∞, due to J.-O. Str¨omberg and

A. Torchinsky ([7]). To obtain Chanillo’s result for one-sided strongly singular operators and for weights in the classA⁺₁ that key point is solved with Theorem 3.

3. Some preliminary results

In the following lemmas we state some results we shall need to prove Theorem 3.

They correspond to Lemma 7 and Lemma 8 of Chapter XII of [7].

Lemma 4. Let ω∈A⁺_p,ν∈A⁺_q,0<δ <1 andη >0.

Ifs=p(1−δ)+qδthen

ω(x)^1−δν(x)^δ∈A⁺_s, and there exists a constantC=C(η, δ, ω, ν)such that

1

|I⁻|

I⁻

ω(x)dx _1−δ

1

|I⁻|

I⁻

ν(x)dx _δ

≤C 1

|I⁺|

I⁺

ω(x)^1−δν(x)^δdx

holds for every pair of intervals I⁻=(a, b)andI⁺=(b, c)withb−a=η(c−b).

Proof. Assume that p>1 and q >1. By applying H¨older’s inequality with exponents α=(s−1)/(p−1)(1−δ) and β=(s−1)/(q−1)δ, it is easy to see that ω(x)^1−δν(x)^δ∈A⁺_s with constantC_ω,p^1−δC_ν,q^δ .

Let us prove the rest of the lemma. Since ω∈A⁺_p, ν∈A⁺_q and by H¨older’s inequality with exponentsαandβ, we have

I⁻

ω(x)dx _1−δ

I⁻

ν(x)dx _δ

≤C_ω,p^1−δC_ν,q^δ (c−a)^s

I⁺

ω(x)^−1/(p−1)dx

_{(p−1)(1−δ)}

I⁺

ν(x)^−1/(q−1)dx

_(q−1)δ−1

≤C_ω,p^1−δC_ν,q^δ (c−a)^s

I⁺

(ω(x)^1−δν(x)^δ)^−1/(s−1)dx

_{(s−1)(−1)}

≤C_ω,p^1−δC_ν,q^δ c−a

|I⁺| _s

I⁺

ω(x)^1−δν(x)^δdx.

Ifb−a=η(c−b) then 1

|I⁻|

I⁻

ω(x)dx _1−δ

1

|I⁻|

I⁻

ν(x)dx _δ

≤C 1

|I⁺|

I⁺

ω(x)^1−δν(x)^δdx,

whereC=C_ω,p^1−δC_ν,q^δ (1+η)^s/η.

Lemma 5. Given 0<p<∞andη >0 there exists a constantC=C(p, η) such that ifδ is a weight on the real line then

^∞

k=1

λ_kχ_Ik

L^p_δ≤C ^∞

k=1

λ_kχ_Ek L^p_δ

holds for every λ_k>0and for all intervalsI_k and allδ-measurableE_k,E_k⊂I_k with δ(E_k)≥ηδ(I_k).

Proof. The main ideas of the proof can be found on p. 116 of [7]. For the sake of completeness we give a proof here.

For the case 1≤p<∞, letg∈L^p_δ,wherepp=p+p. We assume that g≥0 and that g_Lp

δ =1. SinceM_δ(g)(y)=sup_Iy(1/δ(I))

Ig(t)δ(t)dt then we have

R

χ_Ik(y)g(y)δ(y)dy≤δ(I_k) inf

y∈Ek

M_δ(g)(y)

≤δ(E_k) η inf

y∈Ek

M_δ(g)(y)≤1 η

R

χ_Ek(y)M_δ(g)(y)δ(y)dy.

Thus,

R

_∞

k=1

λ_kχ_Ik(y)

g(y)δ(y)dy≤1 η

R

_∞

k=1

λ_kχ_Ek(y)

M_δ(g)(y)δ(y)dy

≤1 η

^∞

k=1

λ_kχ_Ek L^p_δ

M_δ(g)_Lp δ . Now, since we are working on the real line, we have thatM_δ(g)_Lp

δ ≤C_pg_Lp

holds for every weightδ,withC_p^p=2^p+1pand therefore δ

R

∞ k=1

λ_kχ_Ik(y)

g(y)δ(y)dy≤C_p1 η

^∞

k=1

λ_kχ_Ek L^p_δ

. From this inequality we obtain immediately the conclusion for p≥1.

Now, let us prove the case 0<p<1. By denoting Ψ=_∞

k=1λ_kχ_Ek and Φ=

_∞

k=1λ_kχ_Ik, for a fixedt>0, we define E={x∈R: Ψ(x)> t} and O=

x∈R:M_δχ_E(x)>η 2

.

Since we are working on the real line, M_δ is of weak type (1,1) with constant 2 with respect to the weightδ. So, we have

δ(O)≤4 ηδ(E).

(4)

IfO^c∩I_k=∅then δ(E ∩E_k)≤δ(E ∩I_k)≤¹₂ηδ(I_k)≤¹₂δ(E_k) and consequently, δ(E_k)≤¹₂δ(E_k)+δ(E^c∩E_k).

Therefore

ηδ(E^c∩I_k)≤δ(E_k)≤2δ(E^c∩E_k), ifO^c∩I_k=∅. (5)

Takingr>1 and using thatO^c⊆E^c we obtain

O^c

Φ(x)^rδ(x)dx≤

E^c O^c∩Ik=∅

λ_kχ_Ik(x) _r

δ(x)dx.

Since (5) holds, we can apply, with the weightδ(x)χ_Ec(x), the case r>1 we have just proved, to estimate the last term by

C_r η

E^c O^c∩Ik=∅

λ_kχ_Ek(x) _r

δ(x)dx.

So, we have

O^c

Φ(x)^rδ(x)dx≤C_r η

E^c

Ψ(x)^rδ(x)dx.

(6)

Fromδ({x:Φ(x)>t})≤δ(O)+δ(O^c∩{x:Φ(x)>t}) and by (4) we have δ({x: Φ(x)> t})≤4

ηδ(E)+ 1 t^r

O^c

Φ(x)^rδ(x)dx.

By (6) we obtain

δ({x: Φ(x)> t})≤4

ηδ(E)+C_r ηt^r

E^c

Ψ(x)^rδ(x)dx.

From the estimation above, we get ^∞

k=1

λ_kχ_Ik ^p

L^p_δ

= _∞

0

pt^p−1δ({x: Φ(x)> t})dt

≤4 η

_∞

0

pt^p−1δ(E)dt+C_r η

_∞

0

pt^p−1−r

E^c

Ψ(x)^rδ(x)dx dt.

So, the lemma follows since for 0<p<1 the last term equals 4

η+C_r η

p r−p

^∞

k=1

λ_kχ_Ek ^p

L^p_δ

.

The next lemma is contained in Theorem 1 of [3].

Lemma 6. Letδ∈A⁺_∞.

(a)There existsβ >0such that the following implication holds:givenλ>0 and an interval (a, b) such thatλ≤δ(a, x)/(x−a)for allx∈(a, b), then

|{x∈(a, b) :δ(x)> βλ}|>¹₂(b−a).

(b)There existsγ >0such that the following implication holds:givenλ>0 and an interval (a, b) such thatλ≥δ(x, b)/(b−x)for allx∈(a, b), then

δ({x∈(a, b) :δ(x)< γλ})>¹₂δ(a, b).

4. An appropriate atomic decomposition

In this section we give an atomic decomposition of a distribution f∈H₊^p(ω) with additional properties that we shall need.

We shall say that an intervalJ follows the intervalIifI=[c, d] andJ=[d, e].

Our goal is to prove that givenf∈H₊^p(ω) there is an atomic decomposition as stated in Theorem 1 such that for every atoma_k supported in an intervalI_k there is another atoma_j supported in an intervalI_j followingI_k.

First we shall need a couple of lemmas in order to ‘break up’ an atom.

Lemma 7. Letr>0. There exists a sequence{η_j}^∞_j=−∞ ofC₀^∞ functions such that

(a) 0≤η_j≤1 and _∞

j=1η_j(x)=χ_(−∞,r)(x).

(b) supp(η_j)⊂I_j=[r−2^−jr, r−2^−j−2r].

(c) If we letr_j=r/2^j andx∈I_j then ¹₄r_j≤r−x≤r_j. (d)Each xbelongs to at most three intervals I_j.

(e) For every non-negative integer m there exists a positive constantc_m such that |D^mη_j(x)|≤c_mr⁻_j^m.

Proof. Let

h(y) = _y/2

y

ρ(t)dt,

whereρis a non-negativeC₀^∞(R) function with support contained in [−2,−1] and

Rρ(t)dt=1. We define

η_j(x) =h x−r

2^−j−2r

.

It is not hard to see that{η_j}^∞_j=−∞satisfy the five conditions. For details see [4].

Lemma 8. Let a(y) be an (∞, N)-atom with support contained in an inter- val I. There exists a sequence {a_j(x)}^∞_j=1 of (∞, N)-atoms with associated inter- vals{I_j}^∞_j=1such thata(x)=c_∞

j=1a_j(x)almost everywhere, withcbeing a positive constant independent ofa(x). In addition I=_∞

j=1I_j and no point x∈I belongs to more than three intervalsI_j.

Proof. Without lost of generality, we can assume thatI=[0, r]. Sincea(y) is bounded with compact support,

A(x) = 1 N!

_∞

x

(y−x)^Na(y)dy is well defined.

The vanishing moments condition ofa(x) implies that supp(A)⊂[0, r].More- over, it is not hard to see thatD^N+1A(x)=(−1)^N+1a(x) for almost everyx.

Let{η_j}^∞_j=−∞ be the functions of Lemma 7 associated to the interval (−∞, r).

Then, by condition (a) of Lemma 7, we have A(x) =

∞ j=−1

A(x)η_j(x).

(7)

If we let A_j(x)=A(x)η_j(x) and b_j(x)=(−1)^N+1D^N+1A_j(x), we have that supp(b_j)⊂[0, r]∩[r−2^−jr, r−2^−j−2r]=I_j, and, since supp(A_j) is bounded, it can be shown by integration by parts thatb_j(x) hasN vanishing moments.

We claim thatb_j∞≤c, whereconly depends onN. By Leibniz’s formula we have

D^N+1A_j(x) =

N+1 k=0

c_k,ND^kA(x)D^N+1−kη_j(x).

(8)

Forx∈supp(η_j) andk≤N,sincea∞≤1,we obtain

|D^kA(x)| ≤c _r

x

(t−x)^N−k|a(t)|dt≤c(r−x)^N+1−k. Thus, from (8) and conditions (e) and (c) of Lemma 7, we get

|b_j(x)|=|D^N+1A_j(x)| ≤c_N

N+1 k=0

(r−x)^N+1−k|D^N+1−kη_j(x)| (9)

≤c_N

N+1 k=0

(r−x)^N+1−k

r^N_j ^+1−k ≤c_N(N+2) =c.

Furthermore, as a consequence of (7), we have, a(x) =c

∞ j=−1

a_j(x)

for a.e. x, wherea_j(x)=b_j(x)/cwithcas in (9) are (∞, N) atoms.

Remark 9. We observe thatI_j+2 followsI_j and that|I_j+2|≤|I_j|≤4|I_j+2|. Taking into account this remark and as a consequence of Theorem 1 and the previous lemma we have the following result.

Theorem 10. Let ω∈A⁺_s and 0<p<∞. Then there is an integer N(p, ω) with the following property:given anyf∈H₊^p(ω)andN≥N(p, ω), we can find a se- quence {λ_k}^∞_k=1 of positive coefficients and a sequence {a_k,j(x)}^∞_k,j=1 of (∞, N)- atoms with support contained in intervals{I_k,j}^∞_k,j=1 respectively such that the sum _∞

k,j=1λ_ka_k,j converges unconditionally tof both in the sense of distributions and in theH₊^p(ω)-norm. Moreover,

f_H+^p_(ω)∼ ^∞

k,j=1

λ_kχ_Ik,j L^pω

and, for every j andk,I_k,j+2 follows I_k,j,and |I_k,j+2|≤|I_k,j|≤4|I_k,j+2|.

5. Proof of Theorem 3

The first part of the theorem can be obtained as in the proof of Theorem 3 in Chapter XII of [7] using the maximal function M₁⁺(f, φ, x) defined on [5]. For the second part it is enough to define f(z) and to prove inequality (3).

Letf∈H₊^p(µ) then there exists an atomic decompositionf=_∞

k,j=1λ_ka_k,j as the one given in Theorem 10. With the notation introduced in Section 2 and for z∈Ω,we define

f(z) = ∞ k,j=1

λ_k,j(z)a_k,j, where

λ_k,j(z) =λ^p/p(z)_k

ω(I_k,j+2) ν(I_k,j+2)

_{(z−s)p/p0p1} .

By Theorem 1 there exists a constantC such that f(u+it)_Hp(u)

+ (µ(u))≤C ^∞

k,j=1

|λ_k,j(u+it)|χ_Ik,j L^p(u)_µ(u)

,

as long as the right-hand side is finite. Then we shall prove that ^∞

k,j=1

|λ_k,j(u+it)|χ_Ik,j^p(u)

L^p(u)_µ(u)≤Cf^p_H^p

+(µ)<∞.

First we claim that there exist β_k,j∈I_k,j+2 and a constant c such that, for every x∈I_k,j∪I_k,j+2,

µ(u)(x, β_k,j)

β_k,j−x ≤4cµ(u)(I_k,j+2)

|I_k,j+2| . (10)

In fact, sinceM⁻is of weak type (1,1) with constant 2 with respect to the Lebesgue measure, we have, for everyk andj, that

x∈I_k,j+2:M⁻(µ(u)χ_{Ik,j∪Ik,j+2})(x)>4µ(u)(I_k,j∪I_k,j+2)

|I_k,j+2|

≤|I_k,j+2| 2 . So, there existsβ_k,j∈I_k,j+2 such that

M⁻(µ(u)χ_{Ik,j∪Ik,j+2})(β_k,j)≤4cµ(u)(I_k,j+2)

|I_k,j+2| , wherec is the left doubling constant ofµ(u). This implies (10).

We denote byα_k,j the left end point ofI_k,j and E_k,j=

x∈(α_k,j, β_k,j) :µ(u)(x)<4γcµ(u)(I_k,j+2)

|I_k,j+2|

,

where γ is the constant given in Lemma 6(b). By (10) we can apply Lemma 6(b) withλ=4c(µ(u)(I_k,j+2))/|I_k,j+2| to obtain

µ(u)(E_k,j)≥µ(u)(α_k,j, β_k,j)

2 .

(11)

SinceI_k,j⊂(α_k,j, β_k,j), we have ^∞

k,j=1

|λ_k,j(u+it)|χ_Ik,j

L^p(u)_µ(u)≤ ^∞

k,j=1

|λ_k,j(u+it)|χ_{(αk,j,βk,j)} L^p(u)_µ(u)

.

From (11) we can apply Lemma 5, and estimate the last term by C

^∞

k,j=1

|λ_k,j(u+it)|χ_Ek,j L^p(u)_µ(u)

=C ^∞

k,j=1

|λ_k,j(u+it)|µ(u)(·)^1/p(u)χ_Ek,j L^p(u)

.

By the definition ofλ_k,j(u+it), taking into account that ifx∈E_k,j then µ(u)(x)<4γcµ(u)(I_k,j+2)

|I_k,j+2| and from (2), the last term is bounded by

C ^∞

k,j=1

λ^p/p(u)_k

ω(I_k,j+2) ν(I_k,j+2)

_{(u−s)p/p0p1}

× 1

|I_k,j+2|

Ik,j+2

ω(x)^1−µ(u)ν(x)^µ(u)dx _1/p(u)

χ_Ek,j L^p(u)

. Since (u−s)p/p₀p₁=(µ(u)−µ)/p(u) and using H¨older’s inequality the last ex- pression is bounded by

C

^∞

k,j=1

λ^p/p(u)_k

ω(Ik,j+2) ν(Ik,j+2)

(µ(u)−µ)/p(u)

ω(Ik,j+2)^1−µ(u)ν(Ik,j+2)^µ(u)

|Ik,j+2|

_1/p(u) χE_k,j

L^p(u)

≤C

^∞

k,j=1

λ^p/p(u)_k

ω(Ik,j+2)^1−µν(Ik,j+2)^µ

|Ik,j+2|

_1/p(u)

χI_k,j∪I_k,j+2∪I_k,j+4∪I_k,j+6 L^p(u)

.

Therefore, by Lemma 4 and Lemma 5, we get f(u+it)_H^p(u)

+ (µ(u))≤C ^∞

k,j=1

λ^p/p(u)_k

µ(I_k,j+4)

|I_k,j+4| _1/p(u)

χ_Ik,j+6 L^p(u)

≤C ^∞

k,j=1

λ^p/p(u)_k

µ(I_k,j+2)

|I_k,j+2| _1/p(u)

χ_Ik,j+4 L^p(u)

.

Now we shall consider M_µ⁺g(x)=sup_h>0(1/µ(x, x+h))_x+h

x g(t)µ(t)dt. Since M_µ⁺ is of weak type (1,1) with constant 2 with respect to the weightµwe have

µ

x∈I_k,j+2:M_µ⁺(µ⁻¹χ_{Ik,j+2∪Ik,j+4})(x)>4|I_k,j+2∪I_k,j+4| µ(I_k,j+2)

≤µ(I_k,j+2)

2 ,

so we can choosec_k,j∈I_k,j+2 such that

M_µ⁺(µ⁻¹χ_{Ik,j+2∪Ik,j+4})(c_k,j)≤8 |I_k,j+2| µ(I_k,j+2). (12)

Since, for everyx∈I_k,j+2∪I_k,j+4,x>c_k,j, x−c_k,j

µ(c_k,j, x)≤M_µ⁺(µ⁻¹χ_{Ik,j+2∪Ik,j+4})(c_k,j), we have from (12) that

µ(I_k,j+2)

8|I_k,j+2|≤µ(c_k,j, x) x−c_k,j .

Then if we denote byd_k,j the right end point ofI_k,j+4 and F_k,j=

x∈(c_k,j, d_k,j) :µ(x)>βµ(I_k,j+2) 4|I_k,j+2|

,

we can apply Lemma 6(a) to obtain

|F_k,j| ≥d_k,j−c_k,j

2 .

So, by Lemma 5 and the definition ofF_k,j, ^∞

k,j=1

λ^p/p(u)_k

µ(I_k,j+2)

|I_k,j+2| _1/p(u)

χ_Ik,j+4 L^p(u)

≤C ^∞

k,j=1

λ^p/p(u)_k

µ(I_k,j+2)

|I_k,j+2| _1/p(u)

χ_Fk,j L^p(u)

≤C ^∞

k,j=1

λ^p/p(u)_k µ(·)^1/p(u)χ_Fk,j

L^p(u)≤C ^∞

k,j=1

λ^p/p(u)_k χ_Ik,j+4 L^p(u)µ

.

Therefore,

f(u+it)_Hp(u)

+ (µ(u))≤C ^∞

k,j=1

λ^p/p(u)_k χ_Ik,j L^p(u)µ

.

By a density argument such as the one given on p. 187 of [7], we can assume that f∈L¹_loc(R). Thus, using (1) in the last inequality, we have

f(u+it)^p(u)_Hp(u)

+ (µ(u))≤Cf_N^{∗p/p(u)p(u)}_Lp(u)

µ =Cf^p_H^p

+(µ), as we wanted to prove.

References

1. Calder´on, A. P., Intermediate spaces and interpolation, the complex method,Studia Math.24(1964), 113–190.

2. Chanillo, S., Weighted norm inequalities for strongly singular convolution operators, Trans. Amer. Math. Soc.281(1984), 77–107.

3. Mart´ın-Reyes, F. J.,Pick, L. andde la Torre, A.,A⁺_∞condition,Canad. J. Math.

45(1993), 1231–1244.

4. de Rosa, L. andSegovia, C., WeightedH^pspaces for one sided maximal functions, Contemp. Math.189(1995), 161–183.

5. de Rosa, L. andSegovia, C., Equivalence of norms in one-sided H^p spaces, Collect.

Math.53(2002), 1–20.

6. Sawyer, E., Weighted inequalities for the one-sided Hardy–Littlewood maximal func- tions,Trans. Amer. Math. Soc.297(1986), 53–61.

7. Str¨omberg, J.-O. and Torchinsky, A., Weighted Hardy Spaces, Lecture Notes in Math.1381, Springer, Berlin, 1989.

Sheldy Ombrosi

Departamento de Matem´atica Universidad Nacional del Sur Avenida Alem 1253

(8000) Bah´ıa Blanca, Buenos Aires Argentina

[email protected]

Carlos Segovia

Instituto Argentino de Matem´atica CONICET

Saavedra 15, 3^o piso

(1083) Ciudad de Buenos Aires Argentina

[email protected] Ricardo Testoni

Departamento de Matem´atica FCEyN, Universidad de Buenos Aires Pabell´on 1, Ciudad Universitaria (1428) Ciudad de Buenos Aires Argentina

[email protected]

Received August 17, 2005

published online November 24, 2006