A note on the effect of projections on both measures and the generalization of q-dimension capacity Текст научной статьи по специальности «Математика»

38

Probl. Anal. Issues Anal. Vol. 5(23), No. 2, 2016, pp. 38-51

DOI: 10.15393/j3.art.2016.3290

UDC 517.518.1, 517.518.126, 517.518.13

Bilel Selmi

A NOTE ON THE EFFECT OF PROJECTIONS ON BOTH MEASURES AND THE GENERALIZATION OF q-DIMENSION CAPACITY

Abstract. In this paper, we are concerned both with the properties of the generalization of the Lq-spectrum relatively to two Borel probability measures and with the generalized q-dimension Riesz capacity. We are also interested in the study of their behaviors under orthogonal projections.

Key words: orthogonal 'projection, Hausdorff measure and dimension, capacity, dimension spectra

2010 Mathematical Subject Classification: 28A20, 28A75, 28A80, 31C15

1. Introduction. The notions of singularity exponents of spectra and generalized dimensions are the major components of the multifractal analysis. Recently, the projectional behavior of dimensions and multifractal spectra of measures have generated a large interest in the mathematical literature [1]-[9]. The first of these results was obtained by Marstrand in [7], where he proved that a Borel set E of the plane satisfies

dimH (E) = min (dimH E, 1), for a.e. line V,

where dimH denotes the Hausdorff dimension (see [10]). This statement was later generalized by Kaufman [6]. Further more, in [8], Mattila proved that for a Borel measure ^ of Rn

dimH = min (dimH m),

for a.e. vector subspace with dimension m.

Hunt and Kaloshin [11] introduced a new potential-theoretic definition of the dimension spectrum Dq of a probability measure for q > 1 and

©Petrozavodsk State University, 2016 IMlIiHI

explained its relation with prior definitions. This definition was applied to prove that if 1 < q < 2 and ^ is a Borel probability measure with compact support in Rn, then under almost every linear transformation from Rn to Rm, the q-dimension of the image of ^ is min(m, Dq(^)). In particular, the q-dimension of ^ is preserved providing m > Dq (^), for 1 < q < 2. This results was later generalized by Bahroun and Bhouri in [12].

Readers familiar with potential theory will have encountered the definition of the s-capacity of a set E, i.e.

r 1 -1

Cs(E) = inf |/s(^): ^ eM(E)}

where M(E) is the set of Radon measures ^ with compact support on E C C Rn such that 0 < ^(Rn) < to and Is is the s-energy of ^ (see section 2). This makes us able to recall the defintion of capacitary dimension of a set E as follows

C(E) = sup {s : Cs(E) > 0} = inf {s : CS(E) = 0}.

We point out, in contrast to the Hausdorff measures (denoted by Hs), that any bounded set of Rn has finite s-capacity, for all s > 0. Especially, for E C Rn with HS(E) < to, we have CS(E) = 0 and C(E) < dim#(E). Matilla has compared, in [13, 14], the capacitary dimension of a Borel set E and its orthogonal projection.

In this paper, we study the behavior of the generalized Lq-spectrum relatively to two measures on Rn and compare it (the spectrum) to its correspondant under an orthogonal projection. Moreover, we focus on the generalized (s, q)-Riesz capacity of a subset of Rn. We define the generalized q-dimension Riesz capacity and show that the q-dimension is preserved under almost every orthogonal projection.

2. Preliminaries. Let m be an integer with 0 < m < n and Gn,m the Grassmannian manifold of all m-dimensional linear subspaces of Rn. Denote by Yn,m the invariant Haar measure on Gn,m such that jn,m (Gn,m) = = 1. For V e Gn,m, we define the projection map : Rn —> V as the usual orthogonal projection onto V. Then, the set , V E Gn,m} is compact in the space of all linear maps from Rn to Rm and the identification of V with induces a compact topology for Gn,m. Also, for a Borel probability measure ^ with compact support supp^ C Rn and for V E Gn,m, we denote by ^v, the projection of ^ onto V, i.e.

W(A)= Mn-1 (A)) VA c V.

Since / is compactly supported and supply = ny (supp/) for all V E E Gn,m, then, for any continuous function f: V —> R, we have,

J fd/y = J f (ny (x))d/(x),

whenever these integrals exist.

From now on, we consider a compactly supported Borel probability measure / on Rn with topological support S^ and a Borel probability measure v on S^. First, we give a generalization of the Lq-spectrum, for q > 0, relatively to two compactly supported Borel probability measures / and v, by

1

"r^o" log r~° Jsu

(q) = liminf log / /(B(x,r))qdv(x).

D^(q) = lim —]-log I /(B(x,r))qd/(x).

This quantity appears as a generalization of the q-spectral dimension defined, for q > 0, by

1

r^o q log r

It is clear that, if / = v, then T^,^(q) = qD^(q).

The q-spectral dimension D^(q) allows us to measure in certain cases the degree of singularity and in other ones the degree of regularity of measures (see, for example, [11], [15]-[21]).

The generalised Lq-dimension (q) is strictly related to the relative multifractal analysis, the multifractal variation measure, the relative Renyi dimention and multifractal variation for projections of measures developed by Olsen, Cole, Svetova and Selmi et al. [22]-[24], [2]. Other works were carried in this sense in probability and symbolic spaces [25]-[29]. We note that some researchers such as El Naschie [30]-[34], Ord et al. [35] have achieved many valuable results on the same subject and application.

For example if v is a Gibbs measure for the measure i.e. there exists a measure v on S^, a constant K > 1 and tq E R such that for every x E S^ and every 0 < r < 0

K-1/(B(x,r))q(2r)tq < v(B(x,r)) < K/(B(x,r))q(2r)tq

(q) represents the C^ function of Olsen's multifractal formalism [17]. In this case Bahroun and Bhouri compared the multifractal spectrum of a measure / and its projections /y (see [12]).

In [12], Bahroun and Bhouri investigated the behaviour of the generalized Lq-spectrum relatively to ß and v under orthogonal projection and proved that, for q > 0 and Yn,m-almost every V E Gn,m, we have the following

1) If 0 < q < 1 and (q) < mq, then ,Uv (q) = (q);

2) If q > 1 and T^ (q) < m, then T^v ,Uv (q) = T^,v (q).

The (s, q)-energy of ß relatively of v, denoted by Is,q (ß, v), is given by

I.q (ß,V )= i (i )q dV (X).

Js^J | x - y |q y

This definition allows the application of some techniques developed by Bahroun and Bhouri in [12].

Remark. It's clear that, if q = 1 and ß = v, then the (s, q)-energy of ß relatively of v reduces to the standard notion of the s-energy of ß, given by

Is(ß) = J j I x - y |-s dß(y)dß(x)-

Frostman [36] showed that the Hausdorff dimension of a Borel subset E of Rn is the supremum of the positive reals s for which there exists a Borel probability measure ß charging E and for which the s-energy of ß is finite. This characterization is used by Kaufmann [6] and Mattila [8] to prove their results on the preservation of the Hausdorff dimension.

Proposition 1 generalizes this notion to the q-dimension spectrum T^,v (q), for q > 0, and thus allows the methods of potential theory to be applied to this part of the spectrum.

Proposition 1. [12] For q > 0, we have

1) T^v(q) = inf {s > 0 : Is,q(ß, v) = to},

2) T^v(q) = sup {s > 0 : Is,q(ß, v) < to}.

Minkowski dimensions:

For a non-empty bounded subset E of Rn we define the upper Minkowski dimension as

A(E) = inf {s : lim sup N (E) rs = 0}

r^Q

where 0 < r < to and Nr (E) is the least number of balls with radius r needed to cover E. In a similar manner, we define the lower Minkowski dimension as

A(E) = inf {s : liminf Nr(E) rs = 0}.

It is clear that dim#(E) < A(E) < A(E). Whenever these two limits are equal, we call the common value the Minkowski dimension of E.

3. Projection results. In the following theorem, we investigate the relationship between the generalization of the Lq-spectrum relatively to two Borel probability measures ^ and v and study their behaviors under orthogonal projections.

Theorem 1. For q > 0 and jn,m-a.e. V G Gn,m, the following holds

1) If 0 < q < 1, then

min ((1 + A(S^)) V^v(q),mq) < (q) < T^>u(q).

2) If q > 1, then

min (^(1 + A(S^))-1 (q),m) < ,vv(q) < Tu,v(q).

Remark.

1) The techniques used in the proof of the first assertion are similar to those of the proof of Theorem 3.1 in [11]. The proof of the second one is almost identical to that of assertion 2 of Theorem 2.1 in [12]. For more details, the reader can see the appendix.

2) In the case where A(S^) = 0, we obtain the main theorem of Bahroun and Bhouri in [12].

Generalization of the q-dimension capacity. Let ^ be a locally finite Borel measure Rn and v is a Borel probability measure on S^. For E C Rn,

we set

M(E) = : supp^ C E, supp^ is compact, ^(E) = 1}. We define the (s, q)-Riesz capacity of E for s > 0 and q > 0, by

Csq (E) = [ inf ( inf {lsq v) s ' qV 7 LveM(£)l ^eM(E)1 s'qVP' 7

Note that, in this definition, the potentials and the energies may be infinite: we adopt the convention ^ =0. The generalized capacity Cs,q is an outer measure on Rn, this means that a Borel set E has a positive (s, q)-capacity if and only if there are two measures / and v in M(E) such that Is,q(/, v) < to.

The generalised capacity Cs,q(E) plays a role in the study of potential theory, for example one can compare the generalised capacity to the variational q-capacity, the relative q-capacity and the Riesz capacity in metric spaces (see [37]-[39]).

Now, we define the generalized q-dimension Riesz capacity by

Cq(E) = sup {s : Cs,q(E) > 0} = inf {s : Cs,q(E) = 0}. (1)

In the following theorem, we show that Cq (E) is preserved under almost every orthogonal projection.

Theorem 2. Let E C Rn. For q > 0 and jn,m-a.e. V E Gn,m, one has the following:

1) If 0 < q < 1 and Cq(E) < mq, then Cq(ny (E)) = Cq(E).

2) If q > 1 and Cq(E) < m, then Cq(ny (E)) = Cq(E).

Remark. We can define an other type of (s, q)-Riesz capacity of E by setting, for s > 0 and q > 0,

Cs,q (E )=[ inf {/a,q

L ue.M(E) I. J

<! Is q (w, wH

-v-eM(E)

This allows us to define the generalized q-dimension Riesz capacity by Cq(E) = sup {s : Cs,q(E) > 0} = inf {s : Cs,q(E) = 0}.

1) Taking q =1, Cs,q reduces to the standard notion of the s-Riesz capacity. Particularly, we obtain Ci (E) = C(E).

2) The generalized q-dimension Riesz capacity is preserved under almost every orthogonal projection. This means that Cq (E) satisfies the assertions of Theorem 2.

To prove Theorem 2, we need some preliminary lemmas.

Lemma 1. Let 0 < q < 1 and 0 < s < mq.

1) There is a constant c, depending only on m,n and s, such that for

E c Rn,

/ (nv (E))-1d7n,m(V) < c (E)-1. Jv

2) If Csq(E) > 0, then Cs,g(nv(E)) > 0, for 7n,m-a.e. V G Gn,m.

Proof. We will prove Assertion 1). The second is its immediate consequence.

Let ^ and v be two compactly supported Radon measures on Rn, such that

Sp, Sv C E and ^(E) = v(E) = 1.

Then, and vv are two compactly supported Radon measures on Rm, such that

Spv, Svv C nv (E) and ^v (nv (E)) = vv (nv (E)) = 1. Consequently, Cs,g(nv(E))-1 < Is,q(^v, vv). By Fubini-Tonelli's theorem and the fact that 0 < q < 1, we have

/ Cs,q(nv(E))-1d7n,m(V) < Is,q(^v, vv)d7n,m(V) = vv

= /7(7, ■ *)q(V)dv(x) <

J Jv \ J | nv (x - y) | q /

< i ( i /, (V)) qdv(x) =

7 | nv(x - y) |q /

f f ^m(V)),s ,(y)Vdv(x).

Jv | nv (x - y) | q /

Since s < mq,

i dYn,m(V) s < _(2)

Jv | nv(x - y) | s | x - y | s

where c is a constant depending only on m, n and s (see corollary 3.12 in

[13]).

Hence

/ Cs,q(nv(E))-1 d7n,m(V) < c Is,qv).

v

By taking the infimum over all such / and v, we are done. □ Lemma 2. Let q > 1 and 0 < s < m.

1) There is a constant c depending only on m, n and s, such that for

E C Rn,

/ Cs,q (ny (E))-1d7n,m(V) < C Cs,q (E)-1. y

2) If Cs,q(E) > 0, then Cs,q(ny (E)) > 0, for 7n,m-a.e. V E Gn,m.

Proof. By Fubini-Tonelli's theorem, Minkowski's inequality, inequality (2) and the fact that q > 1, we get

/ Cs,q(ny (E))-1d7n,m(V) < / Is,q(/y, vy)d7n,m(V) = yy

y V./ | ny (x - y) | q

d/(y) Vj ( \ ^

dYn,m(V)dv(x) <

<

□

Lemma 3.

f dYn,m (V) ly 1 ny(x - y) |s

1 N q

q

d/(y H dv(x) < c /s,q(/, v).

1) Let 0 < q < 1 and 0 < s < mq. There is a constant c depending only on m, n and s such that, for E C Rn, we have

C-1 Cs,q (E) < / Cs,q (ny (E))d7n,m (V) < Cs,q (E) . y

2) Let q > 1 and 0 < s < m. There is a constant c1 depending only on m, n and s such that for E C Rn, we have

C-1 Cs,q (E) < / Cs,q (ny (E))d7n,m (V) < Cs,q (E) . y

Proof.

1) Since 7n,m is an invariant Radon probability measure on Gn,m, using Holder's inequality and Lemma 1, we obtain

1=(X dYn,m(V)) =

'V

(Ca,q (nv (E))) 2 (Ca,q (nv (E))) 2 d7n,m (V) <

< / (nv(E))d7n,m(VW (Ca,q(nv(E))) 1 d7n,m(V) <

VV

< c(Cs,q (E)) W (nv (E ))d7n,m(V). Jv

Hence

C-1 Cs,g (E) < / Cs,g (nv (E))d7n,m (V). v

Now, Fix V G Gn,m. For all s > 0 and q > 0, we have

(^v, vv ) = J J | ^ dv (v) =

^ Vdv(x) > i ( i ^ s dv(x) =

1 nv(x - y) 1 " J \J 1 x - y 1 q

= Is,' (ju,v) > (E)-1. By taking the infimum over all the measures and vv, we get

Cs,q (nv (E)) < Cs,q (E).

2) The proof is similar to that of assertion 1). □ Proof of Theorem 2.

1) Take 0 < q < 1 and s < Cg(E) < mq. From we have Cs,g(E) > 0. Lemma 1 yields Cs,q(nv(E)) > 0 for 7n,m-a.e. V G Gn,m. The definition of Cq implies that s < Cq(nv(E)) for 7n,m-a.e. V. The second inequality is a consequence of assertion 1) of Lemma 3. Assertion 2) is a consequence of Lemma 2 and Assertion 2) of Lemma 3.^

4. Appendix. Proof of Theorem 1. We first need the following lemma.

Lemma 4. Suppose that s < m and there exists a constant C, depending only on n, m and s, such that for all x, y G Rn \ {0} and p > 0,

f d7n,m (V) C

'v 1 nv(x - y) |s min{ | x - y |, 1}

2

Proof of lemma 4. It is a consequence of corollary 3.12 in [13]. □

1) Fix 0 < q < 1 and choose p > A(Sp).

We will prove that for 0 < s < (1 + p) (q) < mq,

Is(1+p),q v) <

Is,q (^v , vv) < TO,

for Yn,m-a.e. V G Gn,m. The result follows from the fact that p can be arbitrarily chosen close to A(Sp).

The case where mq < (1 + A(Sp)) 1 (q) is similar.

Computing the (s, q)-energy of relatively of vv, we have

Is,q (^v, vv) =

d^v (u)

| u - v | s

dvv (v) =

dMy)

| nv(x - y) | s

dv (x).

We integrate the energy over V G Gn,m. Thanks to the fact that 0 < q < 1 and by the Fubini-Tonelli's theorem, we have

v

Is,q(^v, vv)d7n,m(V) =

v

dMy)

| nv(x - y) | q d^(y) x q

<

v V./ | nv(x - y) | f f d^(y)

dv (x)d7n,m (V) = d7n,m(V)dv(x) <

v J | nv (x - y) |q f d7n,m (V)

xdYn,m (V ) dv(x) =

s d^(y n dv(x).

/v | nv (x - y) | q J /

Now, applying Lemma 4 to the preceding inequality, we get

Is,q(^v, vv)dYn (V) <

q

I

I

q

q

q

q

q

<

C

min { | x — y |, 1} q

d^(y) ) dv(x) < +to.

Thus,

Is(1+p),q v) < TO ^ Is,q (^v ,vv) < TO, for Yn,m-a.e. V G Gn,m.

2) Fix q > 1 and choose p > ). We will show that, under the

assumption 0 < s < (1 + p) (q), we have

Is(1+p),q v) < TO ^ Is,q (^v ,vv) < TO,

for Yn,m-a.e. V G Gn,m.

By Fubini-Tonelli's theorem and Minkowski's inequality as well as the fact that q > 1, we have

v

Is,q (^v ,vv )d7n,m(V) =

v

d^(y)

<

1 nv(x - y) 1 q f d7n,m(V)

dYn,m (V)dv(x) <

lv 1 nv(x - y) |s

n dv (x).

(3)

Applying lemma 4 to the inequality (3), we get

v

Is,q(^v,vv)d7n,m(V) <

<

C

. min { | x - y |, 11 q

d^(y) ) dv(x) < +to,

which shows that /s,q (^v, vv) is finite for Yn,m-a.e. V G Gn,m. This proves the result. □

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Received July 20, 2016.

In revised form, November 18, 2016.

Accepted December 02, 2016.

Faculty of sciences of Monastir Department of mathematics

Avenue of the environment 5019 Monastir, Tunisia E-mail: bilel.selmi@fsm.rnu.tn