ISSN 0001-4346, Mathematical Notes, 2007, Vol. 81, No. 1, pp. 121–125. c© Pleiades Publishing, Ltd., 2007.Original Russian Text c© E. V. Ignat′eva, 2007, published in Matematicheskie Zametki, 2007, Vol. 81, No. 1, pp. 140–144.

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Sobolev–Poincare-Type Inequality on Metric Spacesin Terms of Sharp-Maximal Functions

E. V. Ignat′eva*

Belorussian State University, MinskReceived May 17, 2006

DOI: 10.1134/S0001434607010129

Key words: Sobolev–Poincare inequality, maximal function, Hausdorff space, Sobolev class,Borel measure, Lebesgue point, space of homogeneous type.

Suppose that X is the Hausdorff space with quasimetric d and regular Borel measure μ related to theorder doubling condition γ > 0:

μB(x, s) ≤ csγt−γμB(x, t), x ∈ X, 0 < t < s. (1)

(Here and elsewhere, by c we denote different positive constants depending, possibly, on certainparameters, but these dependences are inessential for us.) Then the triple (X, d, μ), is called [1] a spaceof homogeneous type.

The goal of this paper is to establish analogs of the classical Poincare inequality(∫

B

∣∣∣∣f(x) − 1|B|

∫B

f dx

∣∣∣∣p

dx

)1/p

≤ cr

(∫B|∇f(x)|p dx

)1/p

(B ⊂ Rn is the ball of radius r and |B| is the Lebesgue measure of B) for functions on spaces of

homogeneous type. Such generalizations were studied in [2]–[4], where the role of the gradient wasplayed by maximal functions. Namely, for f ∈ L1

loc(X), set

Sη,Af(x) = Sηf(x) = supB�x

1η(r)μB

∫−B|f − ABf | dμ, (2)

where η : R+ → R+ is an increasing positive function and sup is taken over all balls B = B(y, r)containing the point x. Here AB : L1

loc(X) → L1loc(X) is the linear operator defined for each ball B ⊂ X

and satisfying the conditions (see [3]):

1) for any function f ∈ L1loc(X) and for any ball B ⊂ X,

|ABf(y)| ≤ c|f |B , where fB =1

μB

∫B

f dμ =∫−B

f dμ; (3)

2) if B′ ⊂ B, then AB′(ABf) = ABf and, for all λ ∈ R and B ⊂ X, AB(λ) = λ.

As a simple example, we can indicate the mean values ABf = fB of a function f ∈ L1loc(X) on the

ball B. In the case X = Rn, for AB we can take projections onto the subspace of polynomials of degree

at most a given number k ∈ N.Maximal functions Sηf for η(t) = tα, α > 0, were studied in [3]. Operators of such kind first appeared

in the papers by Calderon [5] and Calderon–Scott [6]. Operators from [5], [6] were later generalized andapplied to various problems in the monograph [7], where the maximal functions Sηf , η(t) = tα, α > 0,

*E-mail: ignatieval@mail.ru

121

122 IGNAT′EVA

were studied in detail for X = Rn and AB was taken to be the projection operator onto the subspace of

polynomials of degree [α] or max{n ∈ Z : n < α}.The use of such maximal functions allows us to define analogs of Sobolev classes on an arbitrary

metric measure space and to provide equivalent descriptions (see, for example, [8] and [9], where Sobolevclasses of first and higher order were defined, respectively, in terms of inequalities of Poincare type).

The main result of the present paper is as follows.

Theorem 1. Suppose that p > 0, 0 < α < γ/p, q = γp/(γ − αp), η ∈ Ω[α, γ/p), f ∈ L1loc(X). If x0 is

the Lebesgue point of the function f , then

|f(x0) − AB(x0,r)f(x0)| ≤ cη(r)(Sηf(x0))1−αp/γ

( ∫−B(x0,r)

(Sηf)p dμ

)α/γ

. (4)

In particular, for any ball B = B(x, r),(∫

−B|f − ABf |q dμ

)1/q

≤ cη(r)( ∫

−2adB

(Sηf)p dμ

)1/p

. (5)

Here Ω[α, β), 0 ≤ α < β < ∞, is the class of positive increasing functions η : R+ → R+ for whichη(t)t−α is almost increasing and η(t)t−β+δ almost decreases for some 0 < δ < β − α. (A functionη : R+ → R+ is said to be almost increasing (decreasing) if there exists a constant c ≥ 1 such that,for all 0 < t1 < t2, the following inequality holds: η(t1) ≤ c η(t2) (respectively, η(t2) ≤ cη(t1)).) Further,suppose that Ω[α,∞) =

⋃β>α Ω[α, β).

Note that, under more rigid constraints on the measure μ and η(t) = tα, α > 0, 1 < p < γ/α,inequality (5) was given in [3]. The special case ABf = fB of Theorem 1 was proved in [4]. The proof ofTheorem 1 is based on the scheme of proof in [4].

Lemma 1. Suppose that f ∈ L1loc(X), 0 < r1 ≤ r2. Then

1) if x0 ∈ X, then

‖AB(x0,r1)f − AB(x0,r2)f‖L∞(B(x0,r1)) ≤ c

(r2

r1

)γ ∫−B(x0,r2)

|f − AB(x0,r2)f | dμ;

2) if B(x1, r1) ∩ B(x2, r2) �= ∅, then, for all points z1, z2 ∈ B(x1, r1) ∩ B(x2, r2), the followinginequality holds:

|AB(x1,r1)f(z1) − AB(x2,r2)f(z1)| ≤ c

(r2

r1

)γ ∫−B(z2,R)

|f − AB(z2,R)f | dμ,

where R = 2adr2.

Proof. 1) For any point z ∈ B(x0, r1), consider the difference

|AB(x0,r1)f(z) − AB(x0,r2)f(z)| = |AB(x0,r1)f(z) − AB(x0,r1)(AB(x0,r2)f)(z)|

= |AB(x0,r1)[f − AB(x0,r2)f ](z)| ≤∫−B(x0,r1)

|f − AB(x0,r2)f | dμ

≤ c

(r2

r1

)γ ∫−B(x0,r2)

|f − AB(x0,r2)f | dμ.

2) We can easily see that B(xi, ri) ⊂ B(z2, R), i = 1, 2; then μB(z2, R)/(μB(xi, ri)) ≤ c(R/ri)γ(see Lemma 1 in [4]) and

|AB(x1,r1)f(z1) − AB(x2,r2)f(z1)|= |(AB(x1,r1)f − AB(z2,R)f)(z1)| + |(AB(x2,r2) − AB(z2,R)f)(z1)|

MATHEMATICAL NOTES Vol. 81 No. 1 2007

SOBOLEV–POINCARE-TYPE INEQUALITY ON METRIC SPACES 123

≤∫−B(x1,r1)

|f − AB(z2,R)f | dμ +∫−B(x2,r2)

|f − AB(z2,R)f | dμ

≤ c

(r2

r1

)γ ∫−B(z2,R)

|f − AB(z2,R)f | dμ.

If x ∈ X is the Lebesgue point of a function f ∈ L1loc(X), then, by definition,

limr→+0

∫−B(x,r)

|f − f(x)| dμ = 0. (6)

For each function f ∈ L1loc(X), relation (6) is satisfied μ-almost everywhere [1]. Also, the equality

limr→+0

AB(x,r)f(x) = f(x) (7)

is also satisfied μ-almost everywhere. Indeed, for x ∈ X,

|AB(x,r)f(x) − f(x)| = |AB(x,r)[f − f(x)](x)| ≤ c

∫−B(x,r)

|f − f(x)| dμ.

Besides, note that, for any function η ∈ Ω[α, β), α > 0, the following inequalities hold:∫ t

0η(s)

ds

s≤ cη(t),

∫ ∞

tη(s)s−β−1 ds ≤ cη(t)t−β , t > 0. (8)

Proof of Theorem 1. Suppose that

u(x, s) =1

η(s)

∫−B(x,s)

|f − AB(x,s)f | dμ.

Then, by the first assertion of Lemma 1, the following inequality holds:

u(x, t) ≤ cu(x, s) for t ≤ s ≤ 2t. (9)

Suppose that Bk = B(x0, 2−kr), k ≥ 0; then, by (7) and Lemma 1, we have

|f(x0) − AB(x0,r)f(x0)| =∣∣∣∣∞∑

k=0

(ABk+1f(x0) − ABk

f(x0))∣∣∣∣ ≤ c

∞∑k=0

η(2−kr)u(x0, 2−kr). (10)

Hence, from the condition η ∈ Ω[α, γ/p) and (9), we obtain

|f(x0) − AB(x0,r)f(x0)| ≤ cη(r)u(x0, r) + c

∞∑k=1

η(2−kr)u(x0, 2−kr)

≤ cη(r)u(x0, r) + c

∞∑k=1

∫ 2−(k−1)r

2−krη(s)u(x0, s)

ds

s

≤ cη(r)u(x0, r) + c

∫ r

0η(s)u(x0, s)

ds

s. (11)

To estimate the last integral, denote, for brevity,

I =(∫

−B(x0,r)

(Sηf)p dμ

)1/p

and consider two cases.1. Suppose that Sηf(x0) ≤ I; then, in view of the condition η ∈ Ω[α, γ/p) and (8),∫ r

0η(s)u(x0, s)

ds

s≤ Sηf(x0)

∫ r

0η(s)

ds

s≤ cη(r)Sηf(x0)

= cη(r)(Sηf(x0))1−αp/γ(Sηf(x0))αp/γ ≤ cη(r)(Sηf(x0))1−αp/γIαp/γ . (12)

MATHEMATICAL NOTES Vol. 81 No. 1 2007

124 IGNAT′EVA

2. Suppose that Sηf(x0) > I. Set

t = r

(I

Sηf(x0)

)p/γ

< r

and split the last integral in (11) into two parts∫ r

0η(s)u(x0, s)

ds

s=

(∫ t

0+

∫ r

t

)η(s)u(x0, s)

ds

s≡ I1 + I2,

which will be estimated differently. The integral I1 is estimated as in (12):

I1 ≤ cη(t)Sηf(x0) ≤ cη

(r

(I

Sηf(x0)

)p/γ)Sηf(x0). (13)

To estimate I2, note that

u(x0, s) ≤ Sηf(x), x ∈ B(x0, s).

Averaging this inequality over the ball B(x0, s) and using the doubling condition (1), we obtain

u(x0, s) ≤(∫

−B(x0,s)

(Sηf)p dμ

)1/p

≤(

μB(x0, r)μB(x0, s)

)1/p

I ≤ c

(r

s

)γ/p

I. (14)

Therefore, by (8), we have

I2 ≤ cIrγ/p

∫ r

tη(s)s−γ/p−1 ds ≤ cI

(r

t

)γ/p

η(t)

≤ cIrγ/pη

(r

(I

Sηf(x0)

)p/γ)(r

(I

Sηf(x0)

)p/γ)−γ/p

≤ cη

(r

(I

Sηf(x0)

)p/γ)Sηf(x0).

Combining this with (13), we see that, in case 2 being studied, the following estimate holds:∫ r

0η(s)u(x0, s)

ds

s≤ cη

(r

(I

Sηf(x0)

)p/γ)Sηf(x0).

Therefore, in view of the fact that η(t)t−α is almost increasing, the following relation holds:∫ r

0η(s)u(x0, s)

ds

s≤ cη(r)

(I

Sηf(x0)

)αp/γ

Sηf(x0) = cη(r)(Sηf(x0))1−αp/γIαp/γ .

The first summand on the right-hand side of (11) can be estimated by inequality (14) in which we musttake s = r:

u(x0, r) = [u(x0, r)]1−αp/γ [u(x0, r)]αp/γ ≤ [Sηf(x0)]1−αp/γIαp/γ ,

and hence inequality (4) is proved.To prove (5), we write∫

−B|f − ABf |q dμ =

∫−B|[f(y) − AB(y,r)f(y)] + [AB(y,r)f(y) − ABf(y)]|q dμ(y)

≤∫−B|f(y) − AB(y,r)f(y)|q dμ(y) +

∫−B|AB(y,r)f(y) − ABf(y)|q dμ(y) ≡ I1 + I2.

To estimate the integral I1, note that B(y, r) ⊂ B(x, 2adr) for y ∈ B(x, r). Therefore, by inequality (4)proved above, we obtain

I1 ≤ cηq(r)∫−B

(Sηf(y))q(1−αp/γ)

(∫−B(y,r)

(Sηf)p dμ

)qα/γ

dμ(y)

MATHEMATICAL NOTES Vol. 81 No. 1 2007

SOBOLEV–POINCARE-TYPE INEQUALITY ON METRIC SPACES 125

≤ cηq(r)( ∫

−B(x,2adr)

(Sηf)p dμ

)qα/γ ∫−B

(Sηf(y))p dμ(y)

≤ cηq(r)( ∫

−B(x,2adr)

(Sηf)p dμ

)q/p

.

The expression under the sign of the integral I2 can be estimated using assertion 2) of Lemma 1:

|AB(y,r)f(y) − ABf(y)| ≤ c

∫−B(x,2adr)

|f − AB(x,2adr)f | dμ ≤ cη(r)Sηf(z)

for all y ∈ B and z ∈ B(x, 2adr). Therefore, averaging over z ∈ B(x, 2adr) leads to the inequalities

|AB(y,r)f(y) − ABf(y)| ≤ cη(r)( ∫

−B(x,2adr)

(Sηf)p dμ

)1/p

, y ∈ B, (15)

and hence

I2 ≤ cηq(r)(∫

−B(x,2adr)

(Sηf)p dμ

)q/p

.

The theorem is proved.

Note that, for αp = γ, η ∈ Ω[α,∞), the following analogs of estimates (4), (5) hold:

|f(x0) − AB(x0,r)f(x0)| ≤ cη(r)( ∫

−B(x0,r)

(Sηf)p dμ

)1/p

× max(

1, ln[Sηf(x0)

( ∫−B(x0,r)

(Sηf)p dμ

)−1/p]),

∫−B

exp(

c|f − ABf |

η(r)

(∫−2adB

(Sηf)p dμ

)−1/p)dμ ≤ c

for any ball B = B(x, r).In the case αp > γ, η ∈ Ω[α,∞), for any Lebesgue point x0 of the function f , we have

|f(x0) − AB(x0,r)f(x0)| ≤ cη(r)( ∫

−B(x0,r)

(Sηf)p dμ

)1/p

.

(see [4] in the case ABf = fB).

REFERENCES1. R. R. Coifman and G. Weiss, Bull. Amer. Math. Soc. 83 (4), 569 (1977).2. J. Dorronsoro, Trans. Amer. Math. Soc. 297, 669 (1986).3. P. Cifuentes, J. Dorronsoro, and J. Sueiro, Trans. Amer. Math. Soc. 332 (1), 331 (1992).4. I. A. Ivanishko and V. G. Krotov, Trudy Inst. Mat. NAN Belarus 14 (1) (2005).5. A. P. Calderon, Studia Math. 44, 167 (1972).6. A. P. Calderon and R. Scott, Studia Math. 62, 75 (1978).7. R. DeVore and R. Sharpley, Mem. Amer. Math. Soc. 47, 1 (1984).8. P. Hajłasz and P. Koskela, Mem. Amer. Math. Soc. 145, 1 (2000).9. Y. Liu, G. Lu and R. L. Wheeden, Ann. Math. 323, 157 (2002).

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