-boundedness of the maximal operator

Necessary and sufficient conditions for the boundedness of the anisotropic Riesz potential

in anisotropic modified Morrey spaces

Malik S. Dzhabrailov and Sevinc Z. Khaligova

Abstract

We prove that the anisotropic fractional maximal operator Mα,σ

and the anisotropic Riesz potential operator Iα,σ, 0 < α <|σ| are bounded from the anisotropic modified Morrey space Le1,b,σ(Rⁿ) to the weak anisotropic modified Morrey space WLeq,b,σ(Rⁿ) if and only if, α/|σ| ≤1−1/q≤α/(|σ|(1−b)) and from Lep,b,σ(Rⁿ) to Leq,b,σ(Rⁿ) if and only if, α/|σ| ≤1/p−1/q≤α/((1−b)|σ|) . In the limiting case

|σ|(1−b)

α ≤p≤ ^|σ|_α we prove that the operator Mα,σ is bounded from Lep,b,σ(Rⁿ) to L∞(Rⁿ) and the modified anisotropic Riesz potential operator Ieα,σ is bounded from Lep,b,σ(Rⁿ) to BM Oσ(Rⁿ) .

1 Introduction

For x∈Rⁿ and t >0 , let B(x, t) denote the open ball centered at x of radius t and ^{B(x, t) =Rⁿ\B(x, t) . Let 0≤b≤1 , σ= (σ₁,· · ·, σ_n) with σ_i >0 for i= 1,· · · , n, |σ|=σ₁+· · ·+σ_n and t^σx≡(t^σ1x₁, . . . , t^σnx_n) for t >0 . For x∈Rⁿ and t >0, let E_σ(x, t) =Qn

i=1(x_i−t^σi, x_i+t^σi) denote the open parallelepiped centered at x of side length 2t^σi for i = 1,· · · , n.

Key Words: anisotropic Riesz potential, anisotropic fractional maximal function, anisotropic modified Morrey space, anisotropic BMO space

2010 Mathematics Subject Classification: Primary 42B20, 42B25, 42B35 Received: September 2011.

Revised: December 2011.

Accepted: June 2012.

111

By [3, 10], the function F(x, ρ) =Pn

i=1x²_iρ^−2σi , considered for any fixed x∈Rⁿ, is a decreasing one with respect to ρ >0 and the equation F(x, ρ) = 1 is uniquely solvable. This unique solution will be denoted by ρ(x) . Define ρ(x) =ρ and ρ(0) = 0 . It is a simple matter to check that ρ(x−y) defines a distance between any two points x, y∈Rⁿ . Thus Rⁿ , endowed with the metric ρ, defines a homogeneous metric space ([3, 5, 10]). Note that ρ(x) is equivalent to |x|σ= max

1≤i≤n|xi|^σi1 .

One of the most important variants of the anisotropic maximal function is the so-called anisotropic fractional maximal function defined by the formula

M_α,σf(x) = sup

t>0

|E_σ(x, t)|^−1+α/|σ|

Z

E_σ(x,t)

|f(y)|dy, 0≤α <|σ|, where |Eσ(x, t)| = 2ⁿt^|σ| is the Lebesgue measure of the parallelepiped Eσ(x, t) .

It coincides with the anisotropic maximal function Mσf ≡M0,σf and is intimately related to the anisotropic Riesz potential operator

Iα,σf(x) = Z

Rⁿ

f(y)dy

|x−y|^|σ|−ασ

, 0< α <|σ|.

If σ =1, then Mα ≡Mα,1 and Iα ≡Iα,1 is the fractional maximal operator and Riesz potential, respectively. The operators Mα, Mα,σ , Iα

and Iα,σ play important role in real and harmonic analysis (see, for example [4] and [35]).

Definition 1.1. Let 0 ≤ b ≤1, 1 ≤p < ∞ and [t]1 = min{1, t}. We denote by Lp,b,σ(Rⁿ) anisotropic Morrey space, and by Lep,b,σ(Rⁿ) the mod- ified anisotropic Morrey space, the set of locally integrable functions f(x), x∈Rⁿ, with the finite norms

kfk_L

p,b,σ = sup

x∈Rⁿ, t>0

t^−b|σ|

Z

Eσ(x,t)

|f(y)|^pdy

!1/p

,

kfk

Le_p,b,σ = sup

x∈Rⁿ, t>0

[t]^−b|σ|₁ Z

Eσ(x,t)

|f(y)|^pdy

!1/p

respectively.

Remark 1.1. Note that Lp,0,σ =Lp(Rⁿ) and Lp,1,σ =L_∞(Rⁿ). If b <0 or b >1, then Lp,b,σ= Θ, where Θ is the set of all functions equivalent to 0 on Rⁿ . In the case σ≡1= (1, . . . ,1) and b= ^λ_n we get the classical Morrey space Lp,λ(Rⁿ) =L_p,λ

n,1(Rⁿ), 0≤λ≤n.

In the theory of partial differential equations, together with weighted Lp,w(Rⁿ) spaces, Morrey spaces Lp,λ(Rⁿ) play an important role. Morrey spaces were introduced by C. B. Morrey in 1938 in connection with certain problems in elliptic partial differential equations and calculus of variations (see [25]). Later, Morrey spaces found important applications to Navier-Stokes ([22], [36]) and Schr¨odinger ([26], [28], [29], [31], [32]) equations, elliptic prob- lems with discontinuous coefficients ([8], [11]), and potential theory ([1], [2]).

An exposition of the Morrey spaces can be found in the book [20].

The modified Morrey space Lep,b,σ(Rⁿ) firstly was defined and investigated by [19] (see also [4]).

Note that

Lep,0,σ(Rⁿ) =Lp,0,σ(Rⁿ) =Lp(Rⁿ),

Lep,b,σ(Rⁿ)⊂Lp,b,σ(Rⁿ)∩Lp(Rⁿ) and max{kfk_Lp,b,σ,kfk_Lp} ≤ kfk

Le_p,b,σ (1.1)

and if b <0 or b >1 , then Lp,b,σ(Rⁿ) =Lep,b,σ(Rⁿ) = Θ .

Definition 1.2. [6] Let 1≤p <∞,0≤b≤1. We denote by W L_p,b,σ(Rⁿ) the weak anisotropic Morrey space and by WLep,b,σ(Rⁿ) the weak modified anisotropic Morrey space as the set of locally integrable functions f(x),x∈ Rⁿ with finite norms

kfk_{W L}

p,b,σ = sup

r>0

r sup

x∈Rⁿ, t>0

t^−b|σ| |{y∈Eσ(x, t) : |f(y)|> r}|^1/p ,

kfk_W

Le_p,b,σ = sup

r>0

r sup

x∈Rⁿ, t>0

[t]^−b|σ|₁ |{y∈E_σ(x, t) : |f(y)|> r}|1/p

respectively.

Note that

W L_p(Rⁿ) =W L_p,0,σ(Rⁿ) =WLe_p,0,σ(Rⁿ), Lp,b,σ(Rⁿ)⊂W Lp,b,σ(Rⁿ) and kfk_{W L}

p,b,σ ≤ kfk_L

p,b,σ, Lep,b,σ(Rⁿ)⊂WLep,b,σ(Rⁿ) and kfk_W

Le_p,b,σ ≤ kfk

Le_p,b,σ.

The anisotropic result by Hardy-Littlewood-Sobolev states that if 1 <

p < q < ∞, then I_α,σ is bounded from L_p(Rⁿ) to L_q(Rⁿ) if and only if α =|σ|

1 p−¹_q

and for p= 1 < q <∞, Iα,σ is bounded from L1(Rⁿ) to W Lq(Rⁿ) if and only if α=|σ|

1−¹_q

. Spanne (see [33]) and Adams

[1] studied boundedness of the Riesz potential Iα for 0< α < n in Morrey spaces Lp,λ. Later on Chiarenza and Frasca [9] was reproved boundedness of the Riesz potential Iα in these spaces. By more general results of Guliyev [13] (see also [14, 17]) one can obtain the following generalization of the results in [1, 9, 33] to the anisotropic case.

Theorem A.Let 0< α <|σ| and 0≤b <1, 1≤p < ^(1−b)|σ|_α . 1) If 1 < p < ^(1−b)|σ|_α , then condition ¹_p − ¹_q = _(1−b)|σ|^α is necessary and sufficient for the boundedness of the operator I_α,σ from L_p,b,σ(Rⁿ) to L_q,b,σ(Rⁿ).

2) If p= 1, then condition 1−¹_q = _(1−b)|σ|^α is necessary and sufficient for the boundedness of the operator I_α,σ from L_1,b,σ(Rⁿ) to W L_q,b,σ(Rⁿ).

If α= (1−b)|σ| ¹_p−¹_q

, then b= 0 and the statement of Theorem A re- duces to the aforementioned anisotropic result by Hardy-Littlewood-Sobolev.

Recall that, for 0< α <|σ|,

Mα,σf(x)≤2^{n(|σ|α−1)}Iα,σ(|f|)(x), (1.2) hence Theorem A also implies the boundedness of the fractional maximal operator Mα,σ . It is known that the anisotropic maximal operator Mσ is also bounded from Lp,b,σ to Lp,b,σ for all 1 < p < ∞ and 0 < b < 1 , which isotropic case proved by F. Chiarenza and M. Frasca [9].

In this paper we study the fractional maximal integral and the Riesz po- tential in the modified Morrey space. In the case p= 1 we prove that the operators M_α,σ and I_α,σ are bounded from Le_1,b,σ(Rⁿ) to WLe_q,b,σ(Rⁿ) if and only if, α/|σ| ≤1−1/q≤α/((1−b)|σ|) . In the case 1< p < ^(1−b)|σ|_α we prove that the operators M_α,σ and I_α,σ are bounded from Le_p,b,σ(Rⁿ) to Leq,b,σ(Rⁿ) if and only if, α/|σ| ≤1/p−1/q≤α/((1−b)|σ|) . In the limiting case ^|σ|(1−b)_α ≤p≤ ^|σ|_α we prove that the operator Mα,σ is bounded from Lep,b,σ(Rⁿ) to L_∞(Rⁿ) and the modified anisotropic Riesz potential operator Ie_α,σ is bounded from Le_p,b,σ(Rⁿ) to BM O_σ(Rⁿ) .

The structure of the paper is as follows. In section 2 the boundedness of the anisotropic maximal operator in anisotropic modified Morrey space Lep,b,σ

is proved. The main result of the paper is the Hardy-Littlewood-Sobolev inequality in anisotropic modified Morrey space for the anisotropic Riesz po- tential, established in section 3. In section 4 we prove that the operator Ie_α,σ is bounded from Le_p,b,σ(Rⁿ) to BM O_σ(Rⁿ) for ^|σ|(1−b)_α ≤p≤^|σ|_α .

By A . B we mean that A ≤ CB with some positive constant C independent of appropriate quantities. If A . B and B . A, we write A≈B and say that A and B are equivalent.

2 L e

-boundedness of the maximal operator

Define ft^σ(x) =:f(t^σx) and [t]1,+= max{1, t}. Then

kft^σk_L

p=t^−|σ|p kfkL_p, kft^σk_L

p,b,σ =t^−|σ|p sup

x∈Rⁿ, r>0

r^−b|σ|

Z

Eσ(t^σx,tr)

|f(y)|^pdy

!^1/p

=t^(b−1)|σ|p kfkL_p,b,σ, and

kft^σk

Le_p,b,σ = sup

x∈Rⁿ, r>0

[r]^−b|σ|₁ Z

Eσ(x,r)

|ft^σ(y)|^pdy

!^1/p

=t^−|σ|p sup

x∈Rⁿ, r>0

[r]^−b|σ|₁ Z

E_σ(t^σx,tr)

|f(y)|^pdy

!^1/p

=t^−|σ|p sup

r>0

[tr]1

[r]1

b|σ|/p

sup

x∈Rⁿ, r>0

[tr]^−b|σ|₁ Z

E_σ(t^σx,tr)

|f(y)|^pdy

!^1/p

=t^−|σ|p [t]

b|σ|

p

1,+ kfk

Lep,b,σ. (2.1)

In this section we study the Le_p,b,σ -boundedness of the maximal operator Mσ.

Lemma 2.1. Let 1≤p <∞, 0≤b≤1. Then Lep,b,σ(Rⁿ) =Lp,b,σ(Rⁿ)∩Lp(Rⁿ) and

kfk

Le_p,b,σ = maxn kfk_L

p,b,σ,kfk_L

p

o .

Proof. Let f ∈Lep,b,σ(Rⁿ) . Then from (1.1) we have that f ∈Lp,b,σ(Rⁿ)∩ Lp(Rⁿ) and maxn

kfk_L

p,b,σ,kfk_L

p

o≤ kfk

Lep,b,σ .

Let now f ∈Lp,b,σ(Rⁿ)∩Lp(Rⁿ) . Then

kfk

Lep,b,σ = sup

x∈Rⁿ,t>0

[t]^−b|σ|₁ Z

E_σ(x,t)

|f(y)|^pdy

!^1/p

= max





 sup

x∈Rⁿ,0<t≤1

t^−b|σ|

Z

Eσ(x,t)

|f(y)|^pdy

!^1/p ,sup

x∈Rⁿ,t>1

Z

Eσ(x,t)

|f(y)|^pdy

!^1/p





≤maxn kfk_L

p,b,σ,kfk_L

p

o .

Therefore, f ∈Lep,b,σ(Rⁿ) and the embedding

Lp,b,σ(Rⁿ)∩Lp(Rⁿ)⊂ Lep,b,σ(Rⁿ) is valid.

Thus

Lep,b,σ(Rⁿ) =Lp,b,σ(Rⁿ)∩Lp(Rⁿ) and kfk

Le_p,b,σ = maxn kfk_L

p,b,σ,kfk_L

p

o .

Analogously proved the following statement.

Lemma 2.2. Let 1≤p <∞, 0≤b≤1. Then

WLep,b,σ(Rⁿ) =W Lp,b,σ(Rⁿ)∩W Lp(Rⁿ) and

kfk_W

Lep,b,σ = maxn kfk_{W L}

p,b,σ,kfk_{W L}

p

o .

To prove our main result in this section we need the following statement.

Theorem 2.1. [23] 1. If f ∈ L1,b,σ(Rⁿ), 0 ≤ b < 1, then Mσf ∈ W L1,b,σ(Rⁿ) and

kMσfkW L_1,b,σ≤Cb,σkfkL_1,b,σ, where Cb,σ depends only on n, b and σ.

2. If f ∈Lp,b,σ(Rⁿ), 1 < p <∞,0 ≤b <1, then Mσf ∈Lp,b,σ(Rⁿ) and

kMσfkL_p,b,σ ≤Cp,b,σkfkL_p,b,σ, where Cp,b,σ depends only on n, p, b and σ.

Our main theorem in this section is the following statement:

Theorem 2.2. 1. If f ∈Le1,b,σ(Rⁿ), 0≤b <1, then Mσf ∈WLe1,b,σ(Rⁿ) and

kMσfk_W

Le_1,b,σ ≤C1,b,σkfk

Le_1,b,σ, where C1,b,σ depends only on b and σ.

2. If f ∈Lep,b,σ(Rⁿ), 1 < p <∞,0≤b <1, then Mσf ∈Lep,b,σ(Rⁿ) and

kMσfk

Lep,b,σ ≤Cp,b,σkfk

Lep,b,σ, where Cp,b,σ depends only on p, b and σ. Proof. It is obvious that (see Lemmas 2.1 and 2.2)

kMσfk

Le_p,b,σ = maxn

kMσfk_L

p,b,σ,kMσfk_L

p

o

for 1< p <∞ and kM_σfk_W

Le_1,b,σ= maxn

kM_σfk_{W L}

1,b,σ,kM_σfk_{W L}

1

o

for p= 1 .

Let 1 < p < ∞. By the boundedness of Mσ on Lp(Rⁿ) and from Theorem 2.1 we have

kM_σfk

Le_p,b,σ ≤max{C_p,σ, C_p,b,σ} kfk

Le_p,b,σ.

Let p= 1 . By the boundedness of Mσ from L1(Rⁿ) to W L1(Rⁿ) and from Theorem 2.1 we have

kMσfk_W

Le1,b,σ≤max{C1,σ, C1,b,σ} kfk

Le1,b,σ.

3 Hardy-Littlewood-Sobolev inequality in modified Mor- rey spaces

The following Hardy-Littlewood-Sobolev inequality in modified Morrey spaces is valid.

Theorem 3.3. Let 0< α <|σ|, 0≤b <1−_|σ|^α and 1≤p < ^(1−b)|σ|_α . 1) If 1< p < ^(1−b)|σ|_α , then condition _|σ|^α ≤ ¹_p−¹_q ≤_(1−b)|σ|^α is necessary and sufficient for the boundedness of the operator Iα,σ from Lep,b,σ(Rⁿ) to Le_q,b,σ(Rⁿ).

2) If p= 1<^(1−b)|σ|_α , then condition _|σ|^α ≤1−¹_q ≤ _(1−b)|σ|^α is necessary and sufficient for the boundedness of the operator Iα,σ from Le1,b,σ(Rⁿ) to WLe_q,b,σ(Rⁿ).

Proof. 1) Sufficiency. Let 0< α < |σ|, 0 < b < 1− _|σ|^α , f ∈ Le_p,b,σ(Rⁿ) and 1< p < ^(1−b)|σ|_α . Then

Iα,σf(x) = Z

Eσ(x,t)

+ Z

{Eσ(x,t)

!

f(y)|x−y|^α−|σ|_σ dy≡A(x, t) +C(x, t).

For A(x, t) we have

|A(x, t)| ≤ Z

E_σ(x,t)

|x−y|^α−|σ|_σ |f(y)|dy

≤

∞

X

j=1

2^−jt^α−|σ|

Z

Eσ(x,2^−j+1t)\Eσ(x,2^−jt)

|f(y)|dy.

Hence

|A(x, t)|.t^αM f(x). (3.1) In the second integral by the H¨older’s inequality we have

|C(x, t)| ≤ Z

{Eσ(x,t)

|x−y|^−β_σ |f(y)|^pdy

!1/p

× Z

{Eσ(x,t)

|x−y|(^βp+α−|σ|)^p0

σ dy

!^1/p0

=J1·J2.

For J2 we obtain

J₂. Z ∞

t

r^|σ|−1+(^βp+α−|σ|)^p0dr _p¹0

≈t^{βp+α−|σ|p} , (3.2) where β <|σ| −αp.

Let b|σ|< β <|σ| −αp. For J1 we get J1=X^∞

j=0

Z

E_σ(x,2^j+1t)\Eσ(x,2^jt)

|x−y|^−β_σ |f(y)|^pdy^1/p

≤t^−βpkfk

Lep,b,σ

X^∞

j=0

2^−βj[2^j+1t]^b|σ|₁ ^1/p

=t^−βpkfk

Le_p,b,σ











2^b|σ|t^b|σ|

[log_22t¹]

P

j=0

2^{(b|σ|−β)j}+

∞

P

j=[log_22t¹]+1

2^−βj1/p

, 0< t < ¹₂, ^∞

P

j=0

2^−βj1/p

, t≥ ¹₂

≈t^−βpkfk

Le_p,b,σ

(

t^b|σ|+t^β1/p

, 0< t < ¹₂,

1, t≥ ¹₂

≈kfk

Le_p,b,σ

(

t^b|σ|−βp , 0< t < ¹₂, t^−βp, t≥¹₂

= [2t]

b|σ|

p

1 t^−βp kfk

Lep,b,σ. (3.3)

From (3.2) and (3.3) we have

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

b|σ|

p

1 t^α−|σ|p kfk

Le_p,b,σ. (3.4)

Thus for all t >0 we get

|Iα,σf(x)|.t^αMσf(x) + [t]

b|σ|

p

1 t^α−|σ|p kfk

Le_p,b,σ

≤minn

t^αMσf(x) +t^α−|σ|p kfk

Le_p,b,σ, t^αMσf(x) +t^{α−(1−b)|σ|p} kfk

Le_p,b,σ

o .

Minimizing with respect to t, at t=h

(Mσf(x))⁻¹kfk

Le_p,b,σ

ip/((1−b)|σ|)

and

t=h

(M_σf(x))⁻¹kfk

Lep,b,σ

i^p/|σ|

we get

|Iα,σf(x)|.min







Mσf(x) kfk

Le_p,b,σ

!1−_(1−b)|σ|^pα

, Mσf(x) kfk

Le_p,b,σ

!1−_|σ|^pα



 kfk

Le_p,b,σ.

Then

|Iα,σf(x)|.(Mσf(x))^p/qkfk^1−p/q

Le_p,b,σ . Hence, by Theorem 2.2, we have

Z

E_σ(x,t)

|I_α,σf(y)|^qdy.kfk^q−p

Le_p,b,σ

Z

E_σ(x,t)

(M_σf(y))^pdy .[t]^b|σ|₁ kfk^q

Le_p,b,σ,

which implies that Iα,σ is bounded from Lep,b,σ(Rⁿ) to Leq,b,σ(Rⁿ) . Necessity. Let 1< p < ^(1−b)|σ|_α , f ∈Lep,b,σ(Rⁿ) and Iα,σ bounded from Lep,b,σ(Rⁿ) to Leq,b,σ(Rⁿ) . Then from (2.1) we have

kf_tσk

Le_p,b,σ =t^−|σ|p [t]

b|σ|

p

1,+ kfk

Le_p,b,σ, and

Iα,σft^σ(x) =t^−αIα,σf(t^σx), (3.5)

kIα,σf_tσk

Leq,b,σ =t^−α sup

x∈Rⁿ, r>0

[r]^−b|σ|₁ Z

E_σ(x,r)

|Iα,σf(t^σy)|^qdy

!^1/q

=t^{−α−|σ|q} sup

r>0

[tr]₁ [r]1

b|σ|/q

sup

x∈Rⁿ, r>0

[tr]^−b|σ|₁ Z

E_σ(t^σx,tr)

|Iα,σf(y)|^qdy

!^1/q

=t^{−α−|σ|q} [t]

b|σ|

q

1,+ kIα,σfk

Le_q,b,σ.

By the boundedness of Iα,σ from Lep,b,σ(Rⁿ) to Leq,b,σ(Rⁿ) kIα,σfk

Leq,b,σ =t^α+

|σ|

q [t]⁻

b|σ|

q

1,+ kIα,σft^σk

Leq,b,σ

≤t^α+|σ|q [t]⁻

b|σ|

q

1,+ kft^σk

Le_p,b,σ

=t^{α+|σ|q −|σ|p} [t]

b|σ|

p −^b|σ|_q

1,+ kfk

Le_p,b,σ, where Cp,q,b,σ depends only on p, q, b and σ.

If ¹_p < ¹_q +_|σ|^α , then in the case t→0 we have kIα,σfk

Leq,b,σ = 0 for all f ∈Lep,b,σ(Rⁿ) .

As well as if ¹_p > ¹_q+_(1−b)|σ|^α , then at t→ ∞ we obtain kIα,σfk

Leq,b,σ = 0 for all f ∈Lep,b,σ(Rⁿ) .

Therefore _|σ|^α ≤ ¹_p−¹_q ≤_(1−b)|σ|^α .

2)Sufficiency. Let f ∈Le1,b,σ(Rⁿ) . We have

|{y∈E_σ(x, t) : |I_α,σf(y)|>2β}|

≤ |{y∈Eσ(x, t) : |A(y, t)|> β}|

+|{y∈E_σ(x, t) : |C(y, t)|> β}|. Then

C(y, t) =

∞

X

j=0

Z

Eσ(y,2^j+1t)\Eσ(y,2^jt)

|f(z)||y−z|^α−|σ|_σ dz

≤t^α−|σ|kfk

Le1,b,σ

∞

X

j=0

2^{−(|σ|−α)j}[2^j+1t]^b|σ|₁ =t^α−|σ|kfk

Le1,b,σ

×











2^b|σ|t^b|σ|

[log_22t¹]

P

j=0

2(b|σ|−|σ|+α)j+

∞

P

j=[log_22t¹]+1

2^{−(|σ|−α)j}, 0< t < ¹₂,

∞

P

j=0

2^{−(|σ|−α)j}, t≥¹₂

≈t^α−|σ|kfk

Le1,b,σ

t^b|σ|+t^|σ|−α, 0< t < ¹₂,

1, t≥¹₂

≈kfk

Le1,b,σ

t^{b|σ|+α−|σ|}, 0< t <¹₂, t^α−|σ|, t≥ ¹₂

= [2t]^b|σ|₁ t^α−|σ| kfk

Le_1,b,σ.

Taking into account inequality (3.1) and Theorem 2.2, we have

|{y∈E_σ(x, t) : |A(y, t)|> β}|

≤

y∈E_σ(x, t) : M f(y)> β C1t^α

≤ C2t^α

β ·[t]^b|σ|₁ kfk

Le_1,b,σ,

where C2 = C1 ·C1,b,σ and thus if C2[2t]^b|σ|₁ t^α−|σ| kfk

Le_1,b,σ = β , then

|C(y, t)| ≤β and consequently, | {y∈Eσ(x, t) : |C(y, t)|> β} |= 0 . Then

|{y∈Eσ(x, t) : |Iα,σf(y)|>2β}|. 1

β [t]^b|σ|₁ t^αkfk

Le1,b,σ

.[t]^b|σ|₁ kfk

Le_1,b,σ

β

!_{(1−b)|σ|−α}^(1−b)|σ|

, if 2t <1

and

|{y∈E_σ(x, t) : |I_α,σf(y)|>2β}|. 1

β[t]^b|σ|₁ t^α kfk

Le_1,b,σ

.[t]^b|σ|₁ kfk

Le_1,b,σ

β

!_|σ|−α^|σ|

, if 2t≥1.

Finally we have

|{y∈E_σ(x, t) : |Iα,σf(y)|>2β}|

.[t]^b|σ|₁ min





 kfk

Le_1,b,σ

β

!_{(1−b)|σ|−α}^(1−b)|σ|

, kfk

Le_1,b,σ

β

!_|σ|−α^|σ| 





≤[t]^b|σ|₁ 1

βkfk

Le_1,b,σ

q

.

Necessity. Let I_α,σ is bounded from Le_1,b,σ(Rⁿ) to WLe_q,b,σ(Rⁿ) . From (3.5) we have

kIα,σft^σk_W

Le_q,b,σ = sup

r>0

r sup

x∈Rⁿ, τ >0

[τ]^−b|σ|₁ Z

{y∈Eσ(x,τ) :|Iα,σftσ(y)|>r}

dy

!1/q

= sup

r>0

r sup

x∈Rⁿ, τ >0

[τ]^−b|σ|₁ Z

{y∈Eσ(x,τ) :|Iα,σf(t^σy)|>rt^α}

dy

!^1/q

=t^{−α−|σ|q} sup

τ >0

[tτ]₁ [τ]1

b|σ|/q

sup

r>0

rt^α

× sup

x∈Rⁿ, τ >0

[tτ]^−b|σ|₁ Z

{y∈Eσ(t^σx,tτ) :|Iα,σf(y)|>rt^α}

dy

!1/q

=t^{−α−|σ|q} [t]

b|σ|

q

1,+ kIα,σfk_W

Le_q,b,σ.

By the boundedness of Iα,σ from Le1,b,σ(Rⁿ) to WLeq,b,σ(Rⁿ) and from (2.1) we get

kIα,σfk_W

Leq,b,σ=t^α+|σ|q [t]⁻

b|σ|

q

1,+ kIα,σf_tσk_W

Leq,b,σ

.t^α+

|σ|

q [t]⁻

b|σ|

q

1,+ kft^σk

Le1,b,σ

.t^{α+|σ|q −|σ|}[t]^b|σ|−

b|σ|

q

1,+ kfk

Le_1,b,σ.

If 1< ¹_q +_|σ|^α, then in the case t→0 we have kIα,σfk_W

Le_q,b,σ = 0 for all f ∈Le1,b,σ(Rⁿ) .

Similarly, if 1> ¹_q+_(1−b)|σ|^α , then for t→ ∞ we obtain kIα,σfk_W

Le_q,b,σ = 0 for all f ∈Le_1,b,σ(Rⁿ) .

Therefore ¹_p−¹_q =_(1−b)|σ|^α .

Corollary 3.1. Let 0< α <|σ|, 0≤b <1−_|σ|^α and 1≤p≤^|σ|_α . 1) If 1< p < ^(1−b)|σ|_α , then condition _|σ|^α ≤ ¹_p−¹_q ≤ _(1−b)|σ|^α is necessary and sufficient for the boundedness of the operator M_α,σ from Le_p,b,σ(Rⁿ) to Le_q,b,σ(Rⁿ).

2) If p= 1< ^(1−b)|σ|_α , then condition _|σ|^α ≤1−¹_q ≤_(1−b)|σ|^α is necessary and sufficient for the boundedness of the operator M_α,σ from Le_1,b,σ(Rⁿ) to WLeq,b,σ(Rⁿ).

3) If ^(1−b)|σ|_α ≤ p ≤ ^|σ|_α , then the operator Mα,σ is bounded from Lep,b,σ(Rⁿ) to L_∞(Rⁿ).

Proof. Sufficiency of Corollary 3.1 follows from Theorem 3.3 and inequality (1.2).

Necessity. (1) Let M_α,σ be bounded from Le_p,b,σ(Rⁿ) to Le_q,b,σ(Rⁿ) for 1< p < ^(1−b)|σ|_α . Then we have

M_α,σf_tσ(x) =t^−αM_α,σf(t^σx), and

kM_α,σf_tσk

Le_q,b,σ =t^{−α−|σ|q} [t]

b|σ|

q

1,+ kM_α,σfk

Le_q,b,σ.

By the same argument in Theorem 3.3 we obtain _|σ|^α ≤¹_p−¹_q ≤_(1−b)|σ|^α . (2) Let Mα,σ be bounded from Le1,b,σ(Rⁿ) to WLeq,b,σ(Rⁿ) . Then

kMα,σft^σk_W

Le_q,b,σ =t^{−α−|σ|q} [t]

b|σ|

q

1,+ kMα,σfk_W

Le_q,b,σ.

Hence we obtain _|σ|^α ≤1−¹_q ≤ _(1−b)|σ|^α .

(3) Let ^(1−b)|σ|_α ≤p≤ ^|σ|_α . Then by the H¨older’s inequality we have kM_α,σfk_L

∞ = 2⁻ⁿ sup

x∈Rⁿ, t>0

t^α−|σ|

Z

E_σ(x,t)

|f(y)|dy

≤2^−np sup

x∈Rⁿ, t>0

t^α−|σ|p [t]

b|σ|

p

1 [t]^−b|σ|₁ Z

E_σ(x,t)

|f(y)|^pdy

!^1/p

≤2^−npkfk

Lep,b,σsup

t>0

t^α−|σ|p [t]

b|σ|

p

1

= 2^−npkfk

Lep,b,σmaxn sup

0<t≤1

t^{α−|σ|(1−b)p} ,sup

t>1

t^α−|σ|p o

= 2^−npkfk

Le_p,b,σ.

4 The modified anisotropic Riesz potential in the spaces L e

( R

)

The examples show that the anisotropic Riesz potential Iα,σ are not defined for all functions f ∈ L_p,b,σ(Rⁿ) , 0 ≤ b < 1−_|σ|^α , if p ≥ ^|σ|(1−b)_α , and I_α,σ are not defined for all functions f ∈ Le_p,b,σ(Rⁿ) , 0 ≤b < 1−_|σ|^α , if p≥^|σ|(1−b)_α .

We consider the modified Riesz potential Ieα,σf(x) =

Z

Rⁿ

|x−y|^α−|σ|_σ − |y|^α−|σ|_σ χ{E_σ(0,1)(y)

f(y)dy.

Note that in the limiting case ^|σ|(1−b)_α ≤p≤ ^|σ|_α statement 1) in Theorem A does not hold. Moreover, there exists f ∈Lep,b,σ(Rⁿ) such that Iα,σf(x) =

∞ for all x∈ Rⁿ. However, as will be proved, statement 1) holds for the modified anisotropic Riesz potential Ie_α,σ if the space L_∞(Rⁿ) is replaced by a wider space BM O_σ(Rⁿ) .

The following theorem is our main result in which we obtain conditions ensuring that the modified anisotropic Riesz potential Ie_α,σ is bounded from the space Lep,b,σ(Rⁿ) to BM Oσ(Rⁿ) .

Theorem 4.4. Let 0 < α < |σ|, 0 ≤ b < 1−_|σ|^α , and ^|σ|(1−b)_α ≤ p ≤

|σ|

α , then the operator Ieα,σ is bounded from Lep,b,σ(Rⁿ) to BM Oσ(Rⁿ).

Moreover, if the integral Iα,σf exists almost everywhere for f ∈Lep,b,σ(Rⁿ),

|σ|(1−b)

α ≤p≤^|σ|_α , then Iα,σf ∈BM Oσ(Rⁿ) and the following inequality is valid

kIα,σfkBM Oσ ≤Ckfk

Lep,b,σ, where C >0 is independent of f .

Proof. For given t >0 we denote

f1(x) =f(x)χ_Eσ(0,2t)(y), f2(x) =f(x)−f1(x), (4.1) where χ_Eσ(0,2t) is the characteristic function of the set Eσ(0,2t) . Then

Ieα,σf(x) =Ieα,σf1(x) +Ieα,σf2(x) =F1(x) +F2(x), (4.2) where

F1(x) = Z

E_σ(0,2t)

|x−y|^α−|σ|_σ − |y|^α−|σ|_σ χ{E_σ(0,1)(y)

f(y)dy,

F2(x) = Z

{Eσ(0,2t)

|x−y|^α−|σ|_σ − |y|^α−|σ|_σ χ{E_σ(0,1)(y)

f(y)dy.

Note that the function f1 has compact (bounded) support and thus a1=−

Z

E_σ(0,2t)\Eσ(0,min{1,2t})

|y|^α−|σ|_σ f(y)dy is finite.

Note also that

F₁(x)−a₁= Z

E_σ(0,2t)

|x−y|^α−|σ|_σ f(y)dy

− Z

E_σ(0,2t)\Eσ(0,min{1,2t})

|y|^α−|σ|_σ f(y)dy

+ Z

Eσ(0,2t)\Eσ(0,min{1,2t})

|y|^α−|σ|_σ f(y)dy

= Z

Rⁿ

|x−y|^α−|σ|_σ f₁(y)dy=I_α,σf₁(x).

Therefore

|F1(x)−a1| ≤ Z

Rⁿ

|y|^α−|σ|_σ |f1(x−y)|dy= Z

Eσ(0,2t)

|y|^α−|σ|_σ |f(x−y)|dy.

Then

|E_σ(x, t)|⁻¹ Z

Eσ(x,t)

|F₁(y)−a₁|dy

=|Eσ(0, t)|⁻¹ Z

E_σ(0,t)

|F1(x−y)−a1|dy

≤ |Eσ(0, t)|⁻¹ Z

Eσ(0,t)

Z

Eσ(0,2t)

|z|^α−|σ|_σ |f(x−y−z)|dz

! dy

=|Eσ(0, t)|⁻¹ Z

Eσ(0,2t)

Z

Eσ(0,t)

|f(x−y−z)|dy

!

|z|^α−|σ|_σ dz

.t^−|σ|t^|σ|−αkfkL_1,1−α

|σ|,σ

Z

Eσ(0,2t)

|z|^α−|σ|_σ dz

≈kfkL_1,1−α

|σ|,σ. (4.3)

Denote

a2= Z

E_σ(0,max{1,2t})\Eσ(0,2t)

|y|^α−|σ|_σ f(y)dy.

If 2|x|σ≤ |y|σ , then

||x−y|^α−|σ|_σ − |y|^α−|σ|_σ | ≤C|x|_σ|y|^{α−|σ|−1}_σ . By the H¨older’s inequality we have

|F2(x)−a2| ≤C|x|σ

Z

{Eσ(0,2t)

|y|^{α−|σ|−1}_σ |f(y)|dy

.|x|σ

∞

X

j=0

Z

2^j+1t≤|y|σ≤2^j+2t

|y|^{α−|σ|−1}_σ |f(y)|dy

.|x|σ

∞

X

j=0

(2^j+1t)^{α−|σ|−1} Z

|y|σ≤2^j+2t

|f(y)|dy

.|x|σkfkL_1,1−α

|σ|,σ

∞

X

j=0

(2^j+2t)^{α−|σ|−1}(2^j+2t)^|σ|−α

≈|x|σt⁻¹kfkL_1,1−α

|σ|,σ. (4.4)

Therefore, from (4.3) and (4.4) we have sup

x,t

1

|E_σ(0, t)|

Z

Eσ(0,t)

Ieα,σf(x−y)−af

dy.kfkL_1,1− α

|σ|,σ. (4.5)

By the H¨older’s inequality we have kfk_L

1,1−α

|σ|,σ = sup

x∈Rⁿ, t>0

t^α−|σ|

Z

E_σ(x,t)

|f(y)|dy

≤2^pn0 sup

x∈Rⁿ, t>0

t^α−|σ|p [t]

b|σ|

p

1 [t]^−b|σ|₁ Z

Eσ(x,t)

|f(y)|^pdy

!1/p

≤2^pn0 kfk

Le_p,b,σsup

t>0

t^α−|σ|p [t]

b|σ|

p

1 (4.6)

= 2^pn0 kfk

Le_p,b,σmaxn sup

0<t≤1

t^{α−|σ|(1−b)p} ,sup

t>1

t^α−|σ|p o

= 2^pn0 kfk

Lep,b,σ.

Finally let ^|σ|(1−b)_α ≤p≤ ^|σ|_α and f ∈Lep,b,σ(Rⁿ) , then from (4.5) and (4.6) we get

Ie_α,σf

_{BM O}

σ

≤2 sup

x,t

1

|Eσ(0, t)|

Z

E_σ(0,t)

Ie_α,σf(x−y)−a_f

dy.kfk

Lep,b,σ.

The Theorem 4.4 is proved.

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Malik S. Dzhabrailov Department of Mathematics

Azerbaijan State Pedagogical University, Baku, Azerbaijan E-mail: [email protected]

Sevinc Z. Khaligova Department of Mathematics

Azerbaijan State Pedagogical University, Baku, Azerbaijan Email: [email protected]