On a two-weighted estimation of maximal operator

Annali di Matematica (2011) 190:263–275 DOI 10.1007/s10231-010-0149-y

On a two-weighted estimation of maximal operator in the Lebesgue space with variable exponent Farman I. Mamedov · Yusuf Zeren

Received: 8 July 2009 / Accepted: 14 May 2010 / Published online: 26 May 2010 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2010

Abstract We study two-weight inequalities with general-type weights for HardyLittlewood maximal operator in the Lebesgue spaces with variable exponent. The exponent function satisfies log-Holder-type local continuity condition and decay condition in infinity. The right-hand side weight to the certain power satisfies the doubling condition. Sawyer-type two-weight criteria for fractional maximal functions are derived. Keywords Maximal functions · Weighted Lebesgue spaces · Variable exponent · Two-weight inequality Mathematics Subject Classification (2000)

42B25 · 46E30

1 Introduction In the mathematical description of a number of elasticity and fluid dynamics problems such as problems associated with the modeling of an electrorheological fluids with dissipation, the modeling of a fluid with pressure tensor depending on temperature distribution and so on, there occur energy integrals of the form  |Df(x)| p(x) dx En

F. I. Mamedov (B) Department of Mathematics, Education Faculty, Dicle University, 21280 Diyarbakir, Turkey e-mail: [email protected] F. I. Mamedov Institute Mathematics and Mechanics National Academy of Science, 1141 Baku, Azerbaijan Y. Zeren Department of Mathematics, Yildiz Technical University, Istanbul, Turkey e-mail: [email protected]

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(see [22,25]). These and other examples of physical processes stimulate interest in the theory of generalized Lebesgue spaces as well as in the investigation into problems of the boundedness of classical integral operators in these spaces (for a survey of this topic see [7,9,10,18,24]). It can be said that differential equations with a nonstandard growth condition, to which the above-mentioned problems reduce, have become an independent area of investigation of nonlinear equations, where the questions of solution existence and regularity are studied intensively (see [1,3,11,20,25]). The subject of the investigation of the present paper is the two-weighted inequality in the norms of generalized Lebesgue spaces L p(.)         1/ p(.) M f  p(.) ≤ C ω1/ p(.) f  p(.) (1.1) v L

L

for the maximal operator 1 M f (x) = sup |B| B x

 f (y)dy. B

This topic was the subject of investigation in the recent papers [14] and [15]. For example, the following necessary and sufficient condition was proved in [14] for a fractional  maximal operator with a variable exponent of order α(.), Mα(.) f (x) = sup B x |B|α(x)/n−1 B f (y)dy to be bounded: ∃C > 1, ∀B ⊂ E n    p v Mα(.) χ B σ dx ≤ C σ dx; (1.2) B

B

where the exponent p(x) ≡ p = const, the function σ = ω property D∞ : ∃C > 1, ∀B ⊂ E n

1 − p−1

(x) has the doubling

σ (2B) ≤ Cσ (B) This result generalizes to the case of a variable exponent α(.), q = p the well-known Sawyer’s result [23] on the boundedness of the fractional maximal operator Mα f (x) from L p (ω) into L q (v): ⎛ ⎞q/ p   v (Mα χ B σ )q dx ≤ C ⎝ σ dx ⎠ , q≥ p>1 (1.3) B

B

If the doubling condition (or the reverse doubling condition) on the function σ is fulfilled, then this condition can be rewritten without the operator Mα(.) in the form of the so-called Aαpq condition ⎛ 1 |B|α/n−1/ p+1/q ⎝ |B|



⎞1/q ⎛ vdx ⎠

B

⎝ 1 |B|



⎞1/ p σ dx ⎠

≤ C,

(1.4)

B

where α/n−1/ p+1/q ≥ 0. Simple proofs of these statements can be found in [4,12]. In [14], the authors also consider the case of boundedness L p → L q(.) ; q(x) ≥ q − > p > 1, p(x) ≡ const, where the necessary and sufficient condition is expressed in the norms of spaces L p(.) :      1/q(.)  χ B |B|α(.)/n−1  q(.) ω−1/ p χ B  L p ≤ C, (1.5) v L

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and it is not required to impose the logarithmic condition on the function α(.). In [15], for a maximal operator M to be bounded in L p(.) (ω, ), the sufficient condition is given in the form ⎞⎛ ⎞ p− −1 ⎛   1 1 1 − ⎟⎜ ⎟ ⎜ sup ⎝ ωdx ⎠ ⎝ ω p− −1 dx ⎠ ≤C (1.6) |B(x, |B(x, r )| r )| x∈,r >0 B(x,r )

B(x,r )

where p − = inf { p(x) : x ∈ }, the exponential function p(.) satisfies the local logarithmic condition of regularity | p(x) − p(y)| ≤

1 C , x, y ∈ , |x − y| < . |x − ln − y| 2

(1.7)

In the previous work of the same authors [16], the power case ω(x) = |x − x0 |γ p(x) , x0 ∈ , γ = const, was considered, where the exponent p(x) satisfies condition (1.7) in . It  is proved there that for a maximal operator M to be bounded in L p(.) |x − x0 |γ p(.) ,  , n it is necessary and sufficient that − p(xn 0 ) < γ < p (x , which means that in this case the 0) exponent γ = const depends only on the values at the point x0 . However, as the authors remark, this range of values becomes narrower if condition (1.6) is considered in the class of power weights. Note that the boundedness in L p(.) () of a maximal operator was proved in [8] for bounded domains, and in [6] for the whole space E n . Along with condition (1.6), the authors of [6] also use the requirement that ∃C > 1, p(∞) > 1 | p(x) − p(∞)| ≤

C . ln (e + |x|)

An analogous result in [21] was proved using the weak condition  c1/| p(x)− p(∞)| dx < ∞, ∃0 < c < 1, p(∞) > 1

(1.8)

(1.9)

En

In our results, it is required that the exponent p(.) would satisfy condition (1.8) and the following modified version of condition (1.7): 1 − p(x)−1

Let σ = ω(x)

, then for x, y ∈ Br , σ (Br ) < | p(x) − p(y)| ≤

1 2

we have

C − ln σ (Br )

(1.10)

Condition (1.10) was also considered in [13] in proving the boundedness in L p(.) (dμ) of the maximal function  1 Mμ f (x) = sup f dμ B x μ (B) B

with measureμ, satisfying the doubling condition. The methods of our investigation are based on the systematic application of Lemma 2.1 below, where the special Calderon-Zygmund partitioning has been applied. That is somewhat similar to the one (used in [8]): ⎛ ⎞ p−   1 1 − | f (y)| dy ≤ C ⎝ | f (y)| p(y)/ p dy ⎠ + S(x), ∃S(x) ∈ L 1 and C > 0 |B| |B| B

B

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is adapted to the weighted case. In this way, we present the test function f as a sum of two summands f 1 and f 2 that correspond to the “biggest” and “smallest” parts. The estimation of smallest part presented by the function f 2 remembers us the production of S(x). At the same time, the estimation procedure for f 1 is reduced to the application of the constant exponents weight results. Also, our approaches are similar to the methods of the recent paper [5], where they are used in proving the boundedness of the maximal operator Mα , 0 < α < n from L p(.) intoL q(.) , 1/ p(.) − 1/q(.) = α/n, when conditions (1.7) and (1.8) are fulfilled for the function p(.).

2 Definitions, notation and auxiliary statements Throughout the paper, all the notations are either standard or defined whenever necessary. E n is an n-dimensional Euclidean space of points. x = (x1 , x2 , . . . , xn ). χ E is the characteristic function of a measurable set E. We denote by B an arbitrary ball in E n . The symbol B(x, ρ) denotes a ball with center at the point x and of radius ρ > 0. For the given measurable set  E and measurable function ω, |E| denotes the Lebesgue measure E, ω(E) = E ωdx. For given 1 < p < ∞, p = p/( p − 1) denotes the conjugate number to p. Finally, C denotes the positive constant which may change its values at each appearance. For the open set , we define L p(.) () as the class of measurable functions f :  → R, for which the modular  I p(.) ( f ) = | f (x)| p(x) dx 

is finite. If p + = sup { p(x) : x ∈ E n } < ∞, then the expression 

f p(.) := inf λ > 0 : I p(.) ( f /λ) < 1 defines the norm in L p(.) (), thereby making L p(.) () a Banach space. Moreover, the convergence in norm is equivalent to the convergence in modular, and f p(.) ≤ 1 if and only if I p(.) ( f ) ≤ 1. If p − = inf { p(x) : x ∈ E n } > 1, then the space L p(.) () becomes reflexive. For the definition and other properties of the space L p(.) (), we refer to [19]. The main result of this paper is proved by means of the following lemma on the coverings for homogeneous spaces. We give this lemma as it is stated in [2]. Let (X, μ, d) be a homogeneous space with quasi-metric d : X, X → R, where d(x, y) ≤ K (d(x, z) + d(z, y)) for any x, y, z ∈ X , and d(x, y) = 0 ↔ x = y. It is assumed that the countable additive measure μ satisfies the doubling condition. For d of the metric ball B = B(x, r ), we denote by B˜ the ball B(x, 5K 3 r ), while Bˆ denotes the ball B(x, 15K 5 r ). ˆ ≤ Aμ(B). Then the following statement is true. Let μ( B) Lemma 2.1 Let μ(X ) = ∞. For any nonnegative integrable function f with bounded  support, b ≥ 2 A3 > 1 and any k ∈ Z such that k = y ∈ X : bk+1 ≥ M f (y) > bk  = ∅, there exists a sequence of balls {Bi }i∈N satisfying the following conditions: k ∈

∞ 

B˜ ik ;

(2.1)

i=1

Bik ∩ B kj = ∅ i f i  = j;

123

(2.2)

On a two-weighted estimation of maximal operator in the Lebesgue space

267

for any Bik there exists xik ∈ Bik such that if rik is the radius of Bik , r ≥ 5K 2 rik and xik ∈ B(y, r ) = B, then  1 bk+1 ≥ M f (xik ) ≥  k  f dy; (2.3) B  i

If x ∈ /

∞ ∞  

Bik

j B˜ i and M f (x) < ∞, then M f (x) < bk ;

(2.4)

j=k i=1

   Let I kj = (l, m) ∈ Z × N : l > k + 2, B˜ ml ∩ B˜ kj  = ∅ and Akj = (l,m)∈I k Bml , then j

2μ(Akj ) ≤ μ(B kj );

⎛



Let E kj = B˜ kj − Akj , then 2μ(E kj ) ≥ μ(B kj ) and μ ⎝ X \

(2.5)

⎞ Elk ⎠ = 0;

(2.6)

k, j ∞ ∞  

Let F jk = B kj − Akj , then μ(F jk ) ≥ μ( B˜ kj )/2 A and

k=−∞ j=1

χ F k (x) ≤ 3. j

(2.7)

We will make use of this lemma in the case of Euclidean space E n , i.e. for X = E n , dμ = dx, K = 1, A = 15n , b ≥ 2 · 15n .

3 The main result The main result of the paper is Theorem 3.1 Let p : E n → [1, ∞) be a measurable function, p− > 1, satisfying conditions (1.8) and (1.10). Let furthermore v, ω : E n → [0, ∞) be the measurable functions v, σ ∈ L 1,loc ; σ = ω(x)−1/( p(x)−1) , and the following conditions be fulfilled for a sufficiently large m > 0: 1) σ ∈ A∞ ;

−m

2) (1 + |x|)

(3.1) σ ∈ A∞ ; 

3) ∃C > 0 ∀B ⊂ E n

⎛ v (1+|x|)−m ⎝

B

 4) ∃C > 0 ∀B ⊂ E n B

⎛ v(x) ⎝

1 |B|



1 |B|



(3.2)

⎞ p(x) σ dy ⎠

 dx ≤ C

B

B

(3.3)

⎞ p(x) σ dy ⎠

(1+|x|)−m σ dx;

 dx ≤ C

B

σ dx

(3.4)

B

Then inequality (1.1) holds for any function f ≥ 0. Proof Let f be an arbitrary function for which    1/ p(.)   p(.) ≤ 1, fω L

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F. I. Mamedov, Y. Zeren

then

  I p(.) ω1/ p(.) f ≤ 1.

(3.5)

To prove inequality (1.1), we must show that the inequality   I p(.) v 1/ p(.) f ≤ C is valid. For the function f , we assume f = f1 + f2 ;

f 1 = f χ f ≥σ ,

f 1 = f χ f 0 is some number later will be specified. For the fixed ball Bx , we have ⎛ ⎜ 1 ⎝ |Bx |



⎞ p(x) ⎟ f 2 dy ⎠

⎛ ⎜ 1 =⎝ |Bx |

Bx

⎛ ⎜ 1 ≤⎝ |Bx |



⎞ p(x) ⎛ ⎟ σ dy ⎠

Bx



⎜ ⎝

1 μ(Bx )

⎞ p(x) 

⎟ σ dy ⎠

 Bx

1 + W (x)λ

⎞ p(x) f2 ⎟ dμ⎠ σ − p(x)

Bx

by virtue of σ ∈ A∞ , ⎛ ⎜ 1 ≤⎝ |Bx |



⎞ p(x) 

⎟ σ dy ⎠

1 + |x|λα

− p(x)

Bx

by virtue of (1.8), ⎛ − p(∞) ⎜ 1 ≤ C 1 + |x|λα ⎝ |Bx | 

123

 Bx

⎞ p(x) ⎟ σ dy ⎠

.

(3.9)

On a two-weighted estimation of maximal operator in the Lebesgue space

If

1 |Bx |

⎛ ⎜ 1 ⎝ |Bx |

 Bx



271

 −1 and p(x) < p(∞), then ( f 2 /σ )dμ > 1 + W (x)λ ⎞p(x) ⎛ ⎟ f 2 dy⎠

⎜ 1 =⎝ |Bx |

Bx



⎞p(x)⎛ ⎟ σ dy⎠

⎜ 1 ⎝ μ(Bx )

Bx

⎛

⎞p(∞)⎛

 Bx

 p(∞)− p(x) ⎜ 1  ≤ 1 + W (x)λ ⎝ |Bx |

⎜ 1 ⎝ μ(Bx )

f2 ⎟ dμ⎠ σ 

⎞ p(x) ⎛

 Bx

⎜ 1 ⎝ μ(Bx )

⎟ σ dy⎠

Bx

⎞p(x)− p(∞) f2 ⎟ dμ⎠ σ  Bx

⎞p(∞) f2 ⎟ dμ⎠ σ (3.10)

by virtue of the condition σ ∈ A∞ , we have the estimate C −1 |x|α ≤ W (x) ≤ C |x|β for the function W (x), then ⎛   p(∞)− p(x) ⎜ 1 ≤ C 1 + |x|λβ ⎝ |Bx |

⎞ p(x) ⎛



⎟ σ dy ⎠

⎜ ⎝

Bx

1 μ(Bx )

 Bx

⎞ p(∞) f2 ⎟ dμ⎠ σ

by virtue of (1.8), ⎛ ⎜ 1 ≤C⎝ |Bx |

⎞ p(x) ⎛



⎟ σ dy ⎠

Bx

If p(x) ≥ p(∞), then by virtue of ⎛ ⎜ 1 ⎝ |Bx |



⎞ p(x) ⎟ f 2 dy ⎠

Bx

⎜ ⎝

1 |Bx |



⎛ ⎜ 1 ≤⎝ |Bx |

Bx

1 μ(Bx )

⎞ p(∞)



f2 ⎟ dμ⎠ σ

Bx

.

(3.11)

( f 2 /σ ) dμ ≤ 1, from (3.10) we obtain



⎞ p(x) ⎛ ⎟ σ dy ⎠

Bx

⎜ ⎝

1 μ(Bx )

 Bx

⎞ p(∞) f2 ⎟ dμ⎠ σ

.

(3.12)

From (3.9), (3.11), (3.12), we find ⎛ ⎜ 1 ⎝ |Bx |

 Bx

⎞ p(x) ⎟ f 2 dy ⎠

⎞ p(x) ⎛   1 − p(∞) ⎜ ⎟ ≤ C 1 + |x|λα σ dy ⎠ ⎝ |Bx | 

⎛ ⎜ 1 +C ⎝ |Bx |

Bx

 Bx

⎞ p(x) ⎛ ⎟ σ dy ⎠

⎜ ⎝

1 μ(Bx )

 Bx

⎞ p(∞) f2 ⎟ dμ⎠ σ

.

(3.13)

Now, repeating our previous reasoning, by virtue of Lemma 2.1, we obtain (having replaced the balls Bx by the balls B kj , where k ∈ Z, j ∈ N)

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F. I. Mamedov, Y. Zeren

 (M f 2 ) p(x) v(x)dx ≤ Cb2 p

+

k, j

En

⎡

⎛



⎛

E kj



1 + |x|λα

⎤⎛

⎞ p(x)

  ⎜ ⎟ + ⎢ ⎢ ⎜  1  σ dy ⎟ + Cb2 p ⎣ ⎝ k ⎠ B j  k k, j k Ej

− p(∞) ⎜ 1 ⎜  ⎝ k B j 

⎥⎜ ⎜ v(x)dx ⎥ ⎦⎝

Bj

1 μ(B kj )

⎞ p(x)



⎟ σ dy ⎟ ⎠

B kj



v(x)dx

⎞ p(∞) f2 ⎟ dμ⎟ ⎠ σ

B kj

:= I1 + I2 . Choosing λ sufficiently large λαp(∞)/β := m, by virtue of (3.2), (3.3) and (2.7), we have ⎞ p(x) ⎛     ⎟  − p(∞) ⎜ 1 σ (y)dy ⎜   σ dy ⎟ v 1 + |x|λα dx ≤ C I1 ≤ C   ⎠ ⎝ k αλ p(∞) B j  k k, j k k, j k 1 + |x| Bj

≤C

 k, j

≤C

F jk

Bj



σ (y)dy  p(∞) ≤ C 1 + |x|αλ

 k, j

F jk

k, j

Bj



F jk

σ (y)dy 1 + W (x)αλ/β



σ (y)dy (1 + W (x))



p(∞)λα/β

 p(∞)

σ (y)dy

≤C

≤ C1 .

(1 + W (x)) p(∞)λα/β

En

The above reasoning can be actually applied for the estimation of I2 : ⎛ ⎛ ⎞ p(x) ⎞ ⎛ ⎞ p(∞)     ⎜ ⎜ ⎟ ⎟⎜ 1 ⎟ + ⎜ v(x) ⎜  1  σ ⎟ ⎜ ⎟ I2 = Cb2 p dx ⎟ ⎝ ⎝ k ⎠ ⎠ ⎝ μ(B k ) ( f 2 /σ ) dμ⎠ B  j k, j j k k k Ej

⎛

Bj

⎞⎛

⎛

⎞ p(x)−1

   ⎜ ⎟⎜ ⎜ 1 ⎟ ⎜ σ dy ⎟ ⎜ 1  ⎜  σ⎟ ≤C v(x) ⎝ ⎠ ⎝ k  ⎝ k ⎠ B j  k B j  k k, j Bk E B j

j

Bj

⎞⎛

⎟⎜ 1 ⎜ dx⎟ ⎠ ⎝μ(B k ) j

j

 B kj

By virtue of conditions (3.4), (3.1) and (2.7), ⎞⎛ ⎞ p(∞) ⎛   ⎜ ⎟⎜ ⎟ ⎟ ⎜ σ dy ⎟ ⎜ 1 ≤C ⎠ ⎝ μ(B k ) ( f 2 /σ ) dμ⎠ ⎝ j k, j ≤C

 k, j



F jk

B kj

σ (x) (Mσ ( f 2 /σ )) p(∞) dx,

F jk

(Mσ ( f 2 /σ )) p(∞) σ dx

≤C En



≤C En

123

p(∞) 1− p(∞)

f2

σ

  dx = C En

f2 σ

 p(∞) σ dx

⎞ p(∞) ⎟ ( f 2 /σ ) dμ⎟ ⎠

.

On a two-weighted estimation of maximal operator in the Lebesgue space

273

Furthermore,   I2 ≤ C En

f2 σ

 p(∞)



 σ dx = C −1 1+W (x)λ

f 2 /σ ≤(



)

+ −1 1+W (x)λ

f 2 /σ >(

)



f2 σ f2 σ

 p(∞) σ dx  p(∞) σ dx

:= J1 + J2 Then it is obvious that  σ dx n J1 = C .   p(∞) ≤ C1 for λ > λ p(∞) 1 + W (x) En   p(x)   p(∞)− p(x)  f2 f2 σ dx J2 = C σ σ   −1 x: p(x)< p(∞), f 2 /σ >(1+W (x)λ )   p(x)   p(∞)− p(x)  f2 f2 σ dx + σ σ   −1 x: p(x)≥ p(∞), f 2 /σ >(1+W (x)λ )   p(x)     f2 p(x) 1− p(x) λ | p(x)− p(∞)| f2 σ dx + C σ dx 1 + W (x) ≤ C1 σ En



≤ C1 + C2

En



1 + |x|

En

 βλ | p(x)− p(∞)|



f2 σ

 p(x)

 σ dx ≤ C1 + C2

p

f 2 σ 1− p dx ≤ C En

by virtue of (1.8) and (3.5). Remark 3.1 When instead of E n we use the bounded domain , Theorem 3.1 remains valid under weaker assumptions, namely conditions (1.8), (3.2) and (3.3) are not any longer required. Remark 3.2 In the case v = ω = |x|γ p(x) ,  is a bounded domain, p(x) satisfies the logarithmic condition (1.7), then conditions (1.10), (3.1) and (3.4) of Theorem 3.1 will be fulfilled n in the bounded domain if − p(0) < γ < p n(0) . In other words, our conditions are exact in the class of power weights since, according to [16], this range is unimprovable. Remark 3.3 The conditions in Theorem 3.1 do not require the maximal operator that is used in Sawyer’s condition (1.3). This can be explained by the presence of conditions (3.1), (3.2), (3.3). In this context, we do not know whether the conditions (3.2) and (3.3) are essential. And it is obvious that for the case of a bounded domain, these two conditions are not required. Remark 3.4 In the case of a whole space E n and the weight function v = ω = |x|γ p(x) and the fulfillment of conditions (1.8), (1.10), we add to the condition of Remark 3.2 for γ , the n n condition − p(∞) < γ < p (∞) for p(x).

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Remark 3.5 Condition (3.3) in Theorem 3.1 can be replaced by ⎛ ⎞ p(x)    v(x) σ dy ⎜1 ⎟ σ dy ⎠ dx ≤ C ⎝ n t (1 + |x|)m (1 + |y|)m B(0,t)

B(0,t)

(3.14)

B(0,t)

Indeed, let Bx = B(x, r x ). It is obvious that for r x ≤ |x| 2 , condition (3.3) follows from condition (3.4). If r x > |x| 2 , then Bx ⊂ B(0, 3r x ). Therefore, by virtue of (3.14)  Bx

⎛ ⎜ 1 ⎝ |Bx |



⎞ p(y) ⎟ σ dy ⎠

Bx

v(y) dy ≤ (1+|x|)m

⎛

 B(0,3r x )

1 ⎜ ⎝ |B(0, 3r x )|



≤C B(0,3r x )

By virtue of (3.2), we have



⎞ p(y) ⎟ σ dy⎠

B(0,3r x )

v(y) dy. (1+|x|)m

σ dy . (1 + |y|)m

σ (1+|y|)m

∈ D∞ , B(0, 3r x ) ⊂ B(x, 5r x ), then  σ dy ≤C . (1 + |y|)m B(x,5r x )

Since

σ (1+|y|)m

∈ D∞ , we have  ≤C B(x,r x )

σ dy =C (1 + |y|)m

 Bx

σ dy . (1 + |y|)m

Thus, condition (3.3) is proved. Acknowledgments suggestions.

We thank the referee for reading the paper manuscript carefully and making some useful

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