Riesz potential on the Heisenberg group and modified Morrey spaces

Riesz potential on the Heisenberg group and modified Morrey spaces

Vagif S. Guliyev and Yagub Y. Mammadov

Abstract

In this paper we study the fractional maximal operatorMα, 0≤α <

Qand the Riesz potential operatorIα, 0 < α < Qon the Heisenberg group in the modified Morrey spaces Lep,λ(Hn), where Q = 2n+ 2 is the homogeneous dimension on Hn. We prove that the operators Mα

and Iα are bounded from the modified Morrey spaceLe1,λ(Hn) to the weak modified Morrey spaceWLeq,λ(Hn) if and only if,α/Q≤1−1/q≤ α/(Q−λ) and fromLep,λ(Hn) toLeq,λ(Hn) if and only if, α/Q≤1/p− 1/q≤α/(Q−λ).

In the limiting case ^Q−λ_α ≤p≤^Q_α we prove that the operatorMα is bounded fromLep,λ(Hn) toL_∞(Hn) and the modified fractional integral operatorIeα is bounded fromLep,λ(Hn) toBM O(Hn).

As applications of the properties of the fundamental solution of sub- LaplacianLonHn, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of Iα from the Besov-modified Morrey spacesBLe^s_pθ,λ(Hn) toBLe^s_qθ,λ(Hn).

1 Introduction

Heisenberg group appear in quantum physics and many parts of mathematics, including Fourier analysis, several complex variables, geometry and topology.

Key Words: Heisenberg group; Riesz potential; fractional maximal function; fractional integral; modified Morrey space; BMO space

2010 Mathematics Subject Classification: 42B35, 43A15, 43A80, 47H50 Received: December, 2010.

Revised: January, 2012.

Accepted: February, 2012.

189

Analysis on the groups is also motivated by their role as the simplest and the most important model in the general theory of vector fields satisfying Hormander’s condition. In the present paper we will prove the boundedness of the Riesz potential on the Heisenberg group in modified Morrey spaces.

We state some basic results about Heisenberg group. More detailed infor- mation can be found in [9, 11, 30] and the references therein. LetHn be the 2n+ 1-dimensional Heisenberg group. That is, Hn =Cⁿ×R, with multipli- cation

(z, t)·(w, s) = (z+w, t+s+ 2Im(z·w)),¯ wherez·w¯=∑n

j=1z_jw¯_j. The inverse element ofu= (z, t) isu⁻¹= (−z,−t) and we write the identity of Hn as 0 = (0,0). The Heisenberg group is a connected, simply connected nilpotent Lie group. We define one-parameter dilations on Hn, forr > 0, by δr(z, t) = (rz, r²t). These dilations are group automorphisms and the Jacobian determinant is r^Q, where Q = 2n+ 2 is the homogeneous dimension of Hn. A homogeneous norm on Hn is given by |(z, t)| = (|z|²+|t|)^1/2. With this norm, we define the Heisenberg ball centered at u = (z, t) with radius r by B(u, r) = {v ∈ Hn : |u⁻¹v| < r},

{B(u, r) =Hn\B(u, r), and we denote byBr=B(0, r) ={v∈Hn : |v|< r} the open ball centered at 0, the identity element of Hn, with radius r. The volume of the ball B(u, r) is CQr^Q, where CQ is the volume of the unit ball B1.

Using coordinatesu= (z, t) = (x+iy, t) for points inHn, the left-invariant vector fields Xj, Yj and T onHn equal to _∂x^∂

j , _∂y^∂

j and _∂t^∂ at the origin are given by

Xj= ∂

∂x_j + 2yj

∂

∂t, Yj= ∂

∂y_j −2xj

∂

∂t, T = ∂

∂t,

respectively. These 2n+ 1 vector fields form a basis for the Lie algebra ofHn

with commutation relations

[Yj, Xj] = 4T

forj= 1, ..., n, and all other commutators equal to 0.

Given a function f which is integrable on any ball B(u, r) ⊂ Hn, the fractional maximal functionM_αf, 0≤α < Qoff is defined by

M_αf(u) = sup

r>0|B(u, r)|^−1+Qα

∫

B(u,r)

|f(v)|dV(v).

The fractional maximal functionMαf coincides forα= 0 with the Hardy- Littlewood maximal functionM f ≡M0f (see [9, 30]) and is intimately related

to the fractional integral Iαf(u) =

∫

Hn

|v⁻¹u|^α−Qf(v)dV(v), 0< α < Q.

The operatorsM_αandI_αplay important role in real and harmonic analysis (see, for example [8, 9, 30]).

The classical Riesz potential is an important technical tool in harmonic analysis, theory of functions and partial differential equations. The classical Riesz potential Iα is defined onRⁿ by

Iαf = (−∆)^−α/2f, 0< α < n,

where ∆ is the Laplacian operator. It is known, thatIαf(x) =γ(α)⁻¹∫

Rⁿ|x− y|^α−nf(y)dy≡Iαf(x).

The potential and related topics on the Heisenberg group we consider the sub-Laplacian defined by

L=−

∑n j=1

(X_j²+Y_j²) .

The Riesz potential on the Heisenberg group is defined in terms of the sub-LaplacianL.

Definition 1.1. For0< α < Q, the Riesz potentialIα is defined, initially on the Schwartz spaceS(Hn), by

Iαf(z, t) =L^−α2f(z, t).

In [33] the relation between Riesz potential and heat kernel on the Heisen- berg group is studied. The following theorem give expression of Iα, which provides a bridge to discuss the boundedness of the Riesz potential (see [33], Theorem 1).

Theorem A.Let qs(z, t) be the heat kernel on Hn. For 0≤ α < Q, we have for f ∈S(Hn)

Iαf(z, t) =

∫∞ 0

Γ(α 2

)₋1

s^α2−1qs(·)ds∗f(z, t).

The Riesz potentialIα satisfies the following estimate (see [33], Theorem 2)

|Iαf(z, t)|.Iαf(z, t). (1)

Inequality (1) gives a suitable estimate for the Riesz potential on the Heisen- berg group.

In the theory of partial differential equations, together with weightedL_p,w(Rⁿ) spaces, Morrey spacesLp,λ(Rⁿ) play an important role. Morrey spaces were introduced by C. B. Morrey in 1938 in connection with certain problems in el- liptic partial differential equations and calculus of variations (see [22]). Later, Morrey spaces found important applications to Navier-Stokes ([20], [32]) and Schr¨odinger ([23], [24], [25], [27], [28]) equations, elliptic problems with dis- continuous coefficients ([3], [6]), and potential theory ([1], [2]). An exposition of the Morrey spaces can be found in the book [19].

Definition 1.2. Let 1≤p <∞,0≤λ≤Q, [t]₁ = min{1, t}. We denote by Lp,λ(Hn) the Morrey space, and by Lep,λ(Hn) the modified Morrey space, the set of locally integrable functionsf(u),u∈Hn, with the finite norms

∥f∥L_p,λ= sup

u∈Hn, t>0

( t^−λ

∫

B(u,t)

|f(y)|^pdV(v) )1/p

,

∥f∥_Lep,λ = sup

u∈Hn, t>0

( [t]⁻₁^λ

∫

B(u,t)

|f(y)|^pdV(v) )1/p

respectively.

Note that

Lep,0(Hn) =Lp,0(Hn) =Lp(Hn),

Lep,λ(Hn)⊂≻ Lp,λ(Hn)∩Lp(Hn) and max{∥f∥L_p,λ,∥f∥L_p} ≤ ∥f∥Lep,λ (2) and ifλ <0 orλ > Q, then L_p,λ(Hn) =Le_p,λ(Hn) = Θ, where Θ is the set of all functions equivalent to 0 onHn.

Definition 1.3. [15] Let 1≤p <∞,0≤λ≤Q. We denote by W Lp,λ(Hn) the weak Morrey space and byWLe_p,λ(Hn)the modified weak Morrey space the set of locally integrable functionsf(u),u∈Hn with finite norms

∥f∥_{W Lp,λ}= sup

r>0

r sup

u∈Hn, t>0

(t^−λ |{v∈B(u, t) : |f(v)|> r}|)1/p

,

∥f∥_WLep,λ = sup

r>0

r sup

u∈Hn, t>0

([t]⁻₁^λ |{v∈B(u, t) : |f(v)|> r}|)1/p

respectively.

Note that

W Lp(Hn) =W Lp,0(Hn) =WLep,0(Hn), L_p,λ(Hn)⊂W L_p,λ(Hn) and ∥f∥W L_p,λ ≤ ∥f∥L_p,λ, Le_p,λ(Hn)⊂WLe_p,λ(Hn) and ∥f∥_WLep,λ ≤ ∥f∥Lep,λ.

The classical result by Hardy-Littlewood-Sobolev states that if 1< p < q <

∞, thenIα is bounded fromLp(Hn) to Lq(Hn) if and only ifα=Q (1

p−¹_q) and for p= 1< q <∞, Iα is bounded fromL1(Hn) to W Lq(Hn) if and only ifα=Q

( 1−¹_q)

.

Spanne (see [29]) and Adams [1] studied boundedness ofIαonRⁿin Morrey spacesLp,λ(Rⁿ). Later on Chiarenza and Frasca [5] was reproved boundedness of Iα in these spacesLp,λ(Rⁿ). By more general results of Guliyev [12] (see, also [13, 14, 15]) one can obtain the following generalization of the results in [1, 5, 29] to the Heisenberg group case (see, also [21]).

Theorem B.Let0< α < Q and0≤λ < Q−α,1≤p < ^Q−_α^λ.

1) If1< p < ^Q_α^−λ, then condition ¹_p−¹_q = _n−^α_λ is necessary and sufficient for the boundedness of the operator Iα fromLp,λ(Hn)toLq,λ(Hn).

2) Ifp= 1, then condition1−¹_q = _Q^α_−λ is necessary and sufficient for the boundedness of the operator Iα from L1,λ(Hn)toW Lq,λ(Hn).

Ifα= ^Q_p −^Q_q,thenλ= 0 and the statement of Theorem B reduces to the aforementioned result by Hardy-Littlewood-Sobolev.

Recall that, for 0< α < Q, M_αf(u)≤v

α Q−1

n I_α(|f|)(u), (3)

hence Theorem B also implies the boundedness of the fractional maximal operator M_α, wherev_n=|B(e,1|is the volume of the unit ball in Hn.

We defineBM O(Hn), the set of locally integrable functionsf with finite norms

∥f∥∗= sup

r>0, u∈Hn

|B_r|⁻¹

∫

B_r

|f(v⁻¹u)−f_Br(u)|dV(v)<∞, where f_Br(u) =|B_r|⁻¹∫

Brf(v⁻¹u)dV(v).

In this paper we study the fractional maximal operatorM_αand the Riesz potential operator Iα on the Heisenberg group in the modified Morrey space.

In the case p= 1 we prove that the operators Mα andIα are bounded from Le1,λ(Hn) to WLeq,λ(Hn) if and only if, α/Q≤1−1/q ≤α/(Q−λ). In the case 1< p <(Q−λ)/αwe prove that the operatorsMα andIαare bounded from Lep,λ(Hn) toLeq,λ(Hn) if and only if,α/Q≤1/p−1/q≤α/(Q−λ).

In the limiting case ^Q_α^−λ ≤ p ≤ ^Q_α we prove that the operator Mα is bounded fromLep,λ(Hn) toL_∞(Hn) and the modified fractional integral oper- atorIe_α is bounded fromLe_p,λ(Hn) toBM O(Hn).

As an application of the properties of the fundamental solution of sub- Laplacian LonHn, in Theorem 3.3 we prove the following modified Morrey version of Sobolev inequality onHn:

∥u∥Leq,λ≤C∥∇Lu∥Lep,λ, for every u∈C₀^∞(Hn), where 0≤λ < Q, 1< p < Q−λand _Q¹ ≤ ¹_p−¹_q ≤_Q−¹_λ.

In Theorem 7 we obtain boundedness of the operator Iα from the Besov- modified Morrey spaces BLe^s_pθ,λ(Hn) toBLe^s_qθ,λ(Hn), 1< p < q <∞,α/Q≤ 1/p−1/q≤α/(Q−λ), 1≤θ≤ ∞and 0< s <1.

As an another application, in Theorem 3.5 we obtain the following Sobolev- Stein embedding inequality in Besov-modified Morrey space onHn.

∥u∥BeL^s_qθ,λ ≤C∥∇Lu∥_BLe^s_pθ,λ, for every u∈C₀^∞(Hⁿ)

where 0 ≤ λ < Q−1, 1 < p < Q−λ, 1 ≤ θ ≤ ∞, 0 < s < 1 and 1/Q≤1/p−1/q≤1/(Q−λ).

By A.B we mean thatA≤CB with some positive constantC indepen- dent of appropriate quantities. IfA.BandB.A, we writeA≈Band say thatA andB are equivalent.

2 Main Results

The following Hardy-Littlewood-Sobolev inequality in modified Morrey spaces on the Heisenberg group is valid.

Theorem 2.1. Let 0< α < Q,0≤λ < Q−αand1≤p < ^Q_α^−λ.

1) If 1 < p < ^Q−_α^λ, then condition _Q^α ≤ ¹_p −¹_q ≤ _Q^α_−λ is necessary and sufficient for the boundedness of the operators Iα and Iα from Lep,λ(Hn) to Leq,λ(Hn).

2) If p = 1 < ^Q−_α^λ, then condition _Q^α ≤1−¹_q ≤ _Q^α_−λ is necessary and sufficient for the boundedness of the operators Iα and I_α from Le_1,λ(Hn) to WLeq,λ(Hn).

Corollary 2.1. Let 0< α < Q,0≤λ < Q−αand1≤p≤^Q_α^−λ.

1) If 1 < p < ^Q−_α^λ, then condition _Q^α ≤ ¹_p −¹_q ≤ _Q^α_−λ is necessary and sufficient for the boundedness of the operatorMα fromLep,λ(Hn)toLeq,λ(Hn).

2) Ifp= 1< ^Q−_α^λ, then condition _Q^α ≤1−¹_q ≤ _Q^α_−λ is necessary and suf- ficient for the boundedness of the operator Mα from Le1,λ(Hn)toWLeq,λ(Hn).

3) If ^Q_α^−λ ≤p ≤ ^Q_α, then the operator Mα is bounded from Lep,λ(Hn) to L_∞(Hn).

The examples show that the operatorIα are not defined for all functions f ∈Le_p,λ(Hn), 0≤λ < Q−α, ifp≥ ^Q−_α^λ.

We consider the modified fractional integral Ieαf(u) =

∫

Hn

(|uv⁻¹|^α−Q− |v|^α−Qχ{B(e,1)(v) )

f(v)dV(v),

whereχ_{

B(e,1)is the characteristic function of the set ^{B(e,1) =Hn\B(e,1).

Note that in the limiting case ^Q−_α^λ ≤ p ≤ ^Q_α statement 1) in Theorem B does not hold. Moreover, there exists f ∈ Lep,λ(Hn) such that Iαf(u) =

∞ for all u ∈ Hn. However, as will be proved, statement 1) holds for the modified fractional integral Ieα if the space L_∞(Hn) is replaced by a wider spaceBM O(Hn).

The following theorem we obtain conditions ensuring that the operatorIeα

is bounded from the spaceLep,λ(Hn) toBM O(Hn).

Theorem 2.2. Let 0 < α < Q, 0 ≤λ < Q−α, and ^Q−_α^λ ≤ p ≤ ^Q_α, then the operator Ieα is bounded from Lep,λ(Hn) to BM O(Hn). Moreover, if the integral Iαf exists almost everywhere for f ∈Lep,λ(Hn), ^Q_α^−λ ≤p≤ ^Q_α, then Iαf ∈BM O(Hn)and the following inequality is valid

∥Iαf∥BM O≤C∥f∥Lep,λ, where C >0 is independent off.

3 Some applications

It is known that (see [4], p. 247) if|·|is a homogeneous norm onHn, then there exists a positive constant C0 such that Γ(x) = C0|x|^2−Q is the fundamental solution ofL.

From Theorem 2.1, one easily obtains an inequality extending the classical Sobolev embedding theorem to the Heisenberg groups.

Theorem 3.3. (Sobolev-Stein embedding on modified Morrey spaces) Let 0 ≤λ < Q,1 < p < Q−λand _Q¹ ≤ ¹_p −¹_q ≤ _Q¹_−λ. Then there exists a positive constantC such that

∥u∥_Leq,λ ≤C∥∇_Lu∥_Lep,λ, for every u∈C₀^∞(Hn).

Proof. Letu∈C₀^∞(Hn). By using the integral representation formula for the fundamental solution (see [4], p. 237), we have

u(x) =

∫

Hn

Γ(x⁻¹y)Lu(y)dy (4)

Keeping in mind that L= ∑n

i=1(X_i²+Y_i²) and X_i^∗ =−Xi, Y_i^∗ =−Yi, by integrating by parts at the right-hand side (4), we obtain

u(x) =

∫

Hn

(∇LΓ)(x⁻¹y)∇Lu(y)dy. (5) On the other hand, out of the origin, we have

∇LΓ(x) =C₀∇L(

|x|^2−Q)

= (2−Q)C₀|x|^1−Q∇L|x|,

so that, since∇L|·|is smooth inHn\{0}andδλ-homogeneous of degree zero,

∇_LΓ(x)≤C|x|^1−Q,

for a suitable constant C >0 depending only onL. Using this inequality in (5), we get

|u(x)| ≤C

∫

Hn

|∇_Lu(y)||x|^1−Qdy=CI1(|∇_Lu|)(x). (6) Then, by Theorem 2.1,

∥u∥Leq,λ ≤C∥I1(|∇Lu|)∥Leq,λ ≤C∥∇Lu∥Lep,λ.

In the following theorem we prove the boundedness of Iα in the Besov- modified Morrey spaces onHn

BLe^s_pθ,λ(Hn) = {

f :∥f∥_BLes pθ,λ

=∥f∥_Lep,λ+ ( ∫

Hn

∥f(x·)−f(·)∥^θ_Le

p,λ

|x|^Q+sθ dx )¹_θ

<∞} (7) where 1≤p, θ≤ ∞and 0< s <1.

Besov spaces B_pθ^s (Hn) in the setting Lie groups were studied by many authors (see, for example [8, 10, 16, 26, 31]).

Theorem 3.4. Let 0 < α < Q, 0 ≤ λ < Q−α and 1 ≤ p < ^Q−_α^λ. If

α

Q ≤ ¹_p − ¹_q ≤ _Q^α_−λ, 1 ≤ θ ≤ ∞ and 0 < s < 1, then the operator Iα is bounded from the spaces BLe^s_pθ,λ(Hn)toBLe^s_qθ,λ(Hn). More precisely, there is a constant C >0such that

∥Iαf∥_BeLs

qθ,λ≤C∥f∥_BeLs pθ,λ

holds for all f ∈BLe^s_pθ,λ(Hn).

Proof. By the definition of the Besov-modified Morrey spaces onHnit suffices to show that

∥τyIαf−Iαf∥_Leq,λ ≤C∥τyf −f∥_Lep,λ, where τyf(x) =f(yx).

It is easy to see thatτ_yf commutes with Iα, i.e.,τ_yIαf =Iα(τ_yf). Hence we obtain

|τ_yIαf −Iαf|=|Iα(τ_yf)−Iαf| ≤Iα(|τ_yf−f|).

Taking Le_p,λ-norm on both sides of the last inequality, we obtain the desired result by using the boundedness ofIαfrom Le_p,λ(Hn) toLe_q,λ(Hn).

Thus the proof of the Theorem 3.4 is completed.

From Theorem 3.4 we obtain the following Sobolev-Stein embedding in- equality on Besov-modified Morrey space.

Theorem 3.5. (Sobolev-Stein embedding on Besov-modified Morrey space) Let 0≤λ < Q−1,1< p < Q−λ,1≤θ≤ ∞ and0< s <1. Then there exists a positive constant C such that

∥u∥_BeL^s

qθ,λ≤C∥∇Lu∥_BeL^s

pθ,λ

, for every u∈C₀^∞(Hⁿ) where 1/Q≤1/p−1/q≤1/(Q−λ).

The Dirichlet problem for the Kohn-Laplacian onHn belongs to Jerison [17, 18]. In particular, our results lead to the following apriori estimate for the sub-Laplacian equationLf =g.

Theorem 3.6. Let 0< s <1,1≤θ≤ ∞,g∈BLe^s_pθ,λ(Hn)andLf =g 1) If0≤λ < Q−2,1< p < ^Q−₂^λ and _Q² ≤ ¹_p−¹_q ≤_Q²_−λ, then there exists a constant C >0such that

∥f∥_BeLs

qθ,λ≤C∥g∥_BeLs pθ,λ

.

, 2) If0≤λ < Q−1,1< p < Q−λand _Q¹ ≤¹_p−¹_q ≤_Q¹_−λ, then there exists a constant C >0 such that

∥Xif∥_BeL^s

qθ,λ ≤C∥g∥_BeL^s

pθ,λ

, i= 1,2,· · ·, n,

∥Yif∥_BLe^s_qθ,λ ≤C∥g∥BeL^s_pθ,λ, i= 1,2,· · · , n.

The proof of Theorems 3.5 and 3.6 are similar to Theorem 3.3.

4 Preliminaries

Defineft(u) =:f(δtu) and [t]1,+= max{1, t}. Then

∥ft∥L_p,λ =t^−Qp sup

u∈Hn, r>0

( r^−λ

∫

B(δtu,tr)

|f(v)|^pdV(v) )1/p

=t^λ−Qp ∥f∥L_p,λ,

∥ft∥_Lep,λ = sup

u∈Hn, r>0

( [r]⁻₁^λ

∫

B(u,r)

|ft(v)|^pdV(v) )1/p

=t^−Qp sup

u∈Hn, r>0

( [r]⁻₁^λ

∫

B(δ_tu,tr)

|f(v)|^pdV(v) )1/p

=t^−Qp sup

r>0

([tr]1

[r]1

)λ/p

sup

u∈Hn, r>0

( [tr]⁻₁^λ

∫

B(δ_tu,tr)

|f(v)|^pdV(v) )1/p

=t^−Qp[t]

λ p

1,+∥f∥_Lep,λ.

In this section we study theLep,λ-boundedness of the maximal operatorM. Lemma 4.1. Let 1≤p <∞,0≤λ≤Q. Then

Lep,λ(Hn) =Lp,λ(Hn)∩Lp(Hn) and ∥f∥_Lep,λ = max

{∥f∥_Lp,λ,∥f∥_Lp} .

Proof. Letf ∈Lep,λ(Hn). Then from (2) we have thatf ∈Lp,λ(Hn)∩Lp(Hn) and max

{∥f∥L_p,λ,∥f∥L_p

}≤ ∥f∥Lep,λ.

Let nowf ∈Lp,λ(Hn)∩Lp(Hn). Then

∥f∥_Lep,λ = sup

u∈Hn,t>0

( [t]⁻₁^λ

∫

B(u,t)

|f(v)|^pdV(v) )1/p

= max



 sup

u∈Hn,0<t≤1

( t^−λ

∫

B(u,t)

|f(v)|^pdV(v) )1/p

,

sup

u∈Hn,t>1

(∫

B(u,t)

|f(v)|^pdV(v) )1/p





≤max

{∥f∥L_p,λ,∥f∥L_p

} .

Therefore,f ∈Le_p,λ(Hn) and the embeddingL_p,λ(Hn)∩L_p(Hn)⊂≻Le_p,λ(Hn) is valid.

ThusLe_p,λ(Hn) =L_p,λ(Hn)∩L_p(Hn) and∥f∥Lep,λ = max

{∥f∥L_p,λ,∥f∥L_p

} .

Analogously proved the following statement.

Lemma 4.2. Let 1≤p <∞,0≤λ≤Q. Then

WLe_p,λ(Hn) =W L_p,λ(Hn)∩W L_p(Hn) and

∥f∥_WLep,λ = max

{∥f∥W L_p,λ,∥f∥W L_p

} .

Theorem 4.7. [21] 1. If f ∈L_1,λ(Hn), 0≤λ < Q, then M f ∈W L_1,λ(Hn) and

∥M f∥W L_1,λ≤Cλ∥f∥L_1,λ, where Cλ depends only on nandλ.

2. If f ∈L_p,λ(Hn),1< p <∞,0≤λ < Q, thenM f ∈L_p,λ(Hn)and

∥M f∥Lp,λ ≤C_p,λ∥f∥Lp,λ, where C_p,λ depends only on n,pandλ.

The following statement is valid:

Theorem 4.8. 1. If f ∈Le1,λ(Hn),0≤λ < Q, thenM f ∈WLe1,λ(Hn)and

∥M f∥_WLe1,λ ≤C1,λ∥f∥Le1,λ, whereC1,λ depends only onλ.

2. If f ∈Le_p,λ(Hn),1< p <∞,0≤λ < Q, thenM f ∈Le_p,λ(Hn) and

∥M f∥_Lep,λ ≤C_p,λ∥f∥_Lep,λ, whereCp,λ depends only onpandλ.

Proof. It is obvious that (see Lemmas 4.1 and 4.2)

∥M f∥_Lep,λ = max

{∥M f∥L_p,λ,∥M f∥L_p

}

for 1< p <∞and

∥M f∥_WLe1,λ= max

{∥M f∥W L_1,λ,∥M f∥W L₁

} forp= 1.

Let 1< p <∞. By the boundedness ofM onLp(Hn) and from Theorem 4.7 we have

∥M f∥Lep,λ ≤max{C_p, C_p,λ} ∥f∥Lep,λ.

Letp= 1. By the boundedness ofM fromL1(Hn) toW L1(Hn) and from Theorem 4.7 we have

∥M f∥_WLe1,λ≤max{C1, C1,λ} ∥f∥_Le1,λ.

Lemma 4.3. Let 0 < α < Q. Then for 2|u| ≤ |v|,u, v ∈Hn, the following inequality is valid

|v⁻¹u|^α−Q− |v|^α−Q≤2^Q−α+1|v|^α−Q−1|u|. (8) Proof. From the mean value theorem we get

|v⁻¹u|^α−Q− |v|^α−Q≤|v⁻¹u| − |v|·ξ^α−Q−1, where min{

|v⁻¹u|,|v|}

≤ξ≤max{

|v⁻¹u|,|v|} .

We note that ¹₂|v| ≤ |v⁻¹u| ≤ ³₂|v|,and|v⁻¹u| − |v|≤ |u|. Thus the proof of the lemma is completed.

5 Proof of the Theorems

Proof of Theorem 2.1.

1) Sufficiency. Let 0 < α < Q, 0 < λ < Q−α, f ∈ Le_p,λ(Hn) and 1< p < ^Q−_α^λ. Then

I_αf(u) = (∫

B(u,t)

+

∫

{B(u,t)

)

f(v)|uv⁻¹|^α−QdV(v)≡A(u, t) +C(u, t).

ForA(u, t) we have

|A(u, t)| ≤

∫

B(u,t)

|f(v)||uv⁻¹|^α−QdV(v)

≤∑^∞

j=1

(2^−jt)α−Q∫

B(u,2^−j+1t)\B(u,2^−jt)

|f(v)|dV(v).

Hence

|A(u, t)|.t^αM f(u). (9)

In the second integral by the H¨older’s inequality we have

|C(u, t)| ≤(∫

{B(u,t)

|uv⁻¹|^−β|f(v)|^pdV(v) )1/p

×(∫

{B(u,t)

|uv⁻¹|(^β_p^+α−Q)^p′dV(v) )1/p^′

=J1·J2.

Letλ < β < Q−αp. ForJ1 we get

J1= (∑^∞

j=0

∫

B(u,2^j+1t)\B(u,2^jt)

|f(v)|^p|uv⁻¹|^−βdV(v) )1/p

≤t^−βp∥f∥Lep,λ

(∑^∞

j=0

2^−βj[2^j+1t]^λ₁ )1/p

=t^−βp∥f∥_Lep,λ









 (

2^λt^λ

[log∑_22t¹] j=0

2^(λ−β)j+ ∑^∞

j=[log_22t¹]+1

2^−βj )1/p

, 0< t < ¹₂, (∑∞

j=0

2^−βj )1/p

, t≥¹₂

≈t^−βp∥f∥_Lep,λ { (

t^λ+t^β )1/p

, 0< t < ¹₂,

1, t≥¹₂

≈∥f∥_Lep,λ {

t^λ−pβ, 0< t < ¹₂, t^−βp, t≥¹₂

= [2t]

λ p

1 t^−βp ∥f∥_Lep,λ. (10)

ForJ2we obtain J2=

(∫

S

dξ

∫ _∞

t

r^Q−1+(^βp+α−Q)^p′dr )_p′¹

≈t^βp+α−Qp. (11) From (10) and (11) we have

|C(u, t)|.[t]

λ p

1 t^α−Qp ∥f∥_Lep,λ. (12) Thus for allt >0 we get

|Iαf(u)|. (

t^αM f(u) + [t]

λ p

1 t^α−Qp ∥f∥_Lep,λ ) .min

{

t^αM f(u) +t^α−Qp ∥f∥_Lep,λ, t^αM f(u) +t^α−Q−λp ∥f∥_Lep,λ} .

Minimizing with respect tot,at t=

[

(M f(u))⁻¹∥f∥_Lep,λ]p/(Q−λ)

and

t= [

(M f(u))⁻¹∥f∥Lep,λ

]p/Q

we have

|I_αf(u)| ≤C₁₁min



 (

M f(u)

∥f∥Lep,λ

)1−_Q−λ^pα

, (

M f(u)

∥f∥Lep,λ

)1−^pα_Q



 ∥f∥Lep,λ. Then

|Iαf(u)|.(M f(u))^p/q∥f∥¹_Le^−p/q

p,λ

.

Hence, by Theorem 4.8, we have

∫

B(u,t)

|I_αf(v)|^qdV(v).∥f∥^q_Le^−p

p,λ

∫

B(u,t)

(M f(v))^pdV(v) .[t]^λ₁∥f∥^q_Le

p,λ

,

which implies thatIαis bounded from Lep,λ(Hn) toLeq,λ(Hn).

Necessity. Let 1 < p < ^Q_α^−λ, f ∈ Lep,λ(Hn) and the operators Iα and Iα

are bounded from Le_p,λ(Hn) toLe_q,λ(Hn).

Definef_t(u) =:f(δ_tu), [t]_1,+= max{1, t}. Then

∥ft∥_Lep,λ = sup

r>0, u∈Hn

( [r]⁻₁^λ

∫

B(u,r)

|ft(v)|^pdV(v) )1/p

=t^−Qp sup

u∈Hn, r>0

( [r]⁻₁^λ

∫

B(u,tr)

|f(v)|^pdV(v) )1/p

=t^−Qp sup

r>0

([tr]₁ [r]1

)λ/p

sup

r>0, u∈Hn

( [tr]⁻₁^λ

∫

B(u,tr)

|f(v)|^pdV(v) )1/p

=t^−Qp[t]

λ p

1,+∥f∥Lep,λ, and

Iαft(u) =t^−αIαf(δtu), Iαft(u) =t^−αIαf(δtu),

∥Iαft∥_Leq,λ =t^−α sup

u∈Hn, r>0

( [r]⁻₁^λ

∫

B(u,r)

|Iαf(δtv)|^qdV(v) )1/q

=t^−α−Qq sup

r>0

([tr]1

[r]₁ )λ/q

sup

r>0, u∈Hn

( [tr]⁻₁^λ

∫

B(δtu,tr)

|Iαf(v)|^qdV(v) )1/q

=t^−α−Qq[t]

λ q

1,+∥Iαf∥_Leq,λ.