On the projections of mutual Lq,t-spectrum Текст научной статьи по специальности «Физика»

94

Probl. Anal. Issues Anal. Vol. 6(24), No. 2, 2017, pp. 94-108

DOI: 10.15393/j3.art.2017.4231

UDC 517.518.1, 517.518.126, 517.518.13

B. Selmi, N. Yu. SvETQYA

ON THE PROJECTIONS OF MUTUAL Lq t-SPECTRUM

Abstract. In this paper we are interested in the mutual Lq,t-spectrum relatively to two Borel probability measures having the same compact support and also in the study of their behavior under orthogonal projections.

Key words: orthogonal 'projection, dimension spectra, m,utua,l multifractal analysis

2010 Mathematical Subject Classification: 28A20, 28A80

1. Introduction. The notion of singularity exponents or spectrum and generalized dimensions are the major components of the multifractal analysis. They were introduced to characterize the geometry of measure and are linked with the multifractal spectrum. The multifractal spectrum is the map that affects the Hausdorff or packing dimension of the isoHolder set

EY m J ^ C o C I" lQg /jBr (x) A i• log vBr (x) a

E(a, p)=\ x G Sß n Sv; lim —--=a and lim —--=ß

r^Q log r r^Q log r

for a given a,0 > 0. Here S^ is the topological support of the probability measure ^ on Rn and Br (x) is the closed ball of center x and radius r. For (q, t) E R2 the mutual Lq,t-spectrum of v) is defined as the mapping

log ( sup { E^(Br (Xi))qv(Br (Xi))

(q,t) = lim-i--- ,

r^o log r

where the supremum is taken over all centered packing of S^ if Sv by balls of radius r. It is easy to check that (q, t) is a concave function [24] of (q, t) over R2; for q, t > 1 it has an integrand expression.

©Petrozavodsk State University, 2017 [MiiI!H|

For q, t > 1 and for equal compact supports S^, Sv we have r^(q, t) = liminf logr lo^/ ^(x))q-1 v(Br(y))t-1d^(x)dv(y).

This equation unifies the mutual multifractal spectra to the mutual Lq,t-spectrum (q, t) via the Legendre transform [22, 23], i.e.,

dimH (E ( a, ft)) = inH qa + t^ — r^v(q, t)

In this paper we provide the mutual Lq,t-spectrum relatively to two compactly supported Borel probability measures on Rn. We write Pn for the set of compactly supported Borel probability measures on Rn. Let v E Pn be such that S^ = Sv = K. For (q, t) E R2 define

l^v(q, t) = lin^inf logr log Jj ^(Br (x))qv(Br (y)№(x)dv(y),

K2 nBr (D2)

and

l„,*(q, t) = limsup log j j n(Br(x))qv(Br(y))*d^(x)dv(y),

r^0 log r J J

K2 n Br (D2)

where D2 = {(x, x) E R2 : x E R} is the diagonal ray in R2 and Br(D2) = = {x E Rn : dist(x, D2) < r} is the closed r-neighborhood of D2.

If l (q, t) = l(q, t), then their common value at (q, t) is denoted by (q, t) and called the mutual Lq,t-spectrum of ^ and v. Note that these quantities are strictly related to the mutual multifractal analysis [22], [25] - [27] and the mixed multifractal analysis [18], introduced by Olsen.

In the recent decade there has been a great interest in understanding the fractal dimensions of projections of sets and measures. Recently, the projectional behavior of dimensions and multifractal spectra of sets and measures have generated a large interest in the mathematical literature [1] - [4], [7] - [13], [15, 19, 20]. The first significant work in this area was the result by Marstrand [15] who proved a well-known theorem: the Hausdorff dimension of a planar set is preserved under orthogonal projections.

S n X

Let us mention that Falconer and Mattila [8] and Falconer and Howroyd [7] have proved that the packing dimension of the projected set or measure will be the same for almost all projections. However, despite these substantial advances for fractal sets, only very little is known about the multifractal structure of projections of measures, except a paper by O'Neil [19] and some more recent papers by Barral and Bhouri [2]. The result of O'Neil was later generalized by Selmi et al. in [4]-[6], [21].

We continue of this research studying the behavior of the upper and lower mutual Lq,t-spectra under orthogonal projections onto a lower dimensional linear subspace. We employ theoretical methods first used in this context by Kaufman in [14] and later generalized in [16].

2. Preliminaries. Let m be an integer with 1 < m < n and Gn,m represent the Grassmannian manifold of all m-dimensional linear subspaces of Rn. By Yn,m denote the invariant Haar measure on Gn mm such that Yn,m (Gn,m) = 1. The projection map : Rn ^ V for V E Gn,m is the usual orthogonal projection onto V. Then {nV, V E Gn,m} is compact in the space of all linear maps from Rn to Rm, and identification of V with induces a compact topology for Gn,m. Also, for a Borel probability measure ^ with compact support supp ^ on Rn and for V E Gn,m define the projection of ^ onto V by

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

Note that ^ is compactly supported and supp = (supp for all V E Gn,m, then for any continuous function f : V —> R

J fd^v = J f (nv (x))d^(x)

provided that these integrals exist (for more details see [9]). The convolution is defined for 1 < m < n and r > 0 by

: Rn —> R,

x ——> Tn,m{V E Gn,m : |nv(x)| < rj, where Yn,m is the rotation-invariant probability measure on Gn,m. Define

: Rn —> R,

x ——> min|1 , rm|x|-m|.

This ^^(x) is equivalent to (x). We write this equivalence as 0r(x) x (x). For a probability measure ^ and V E Gn,m we have

ßr'm* (x) = ß * ^m(x) x ß * = J ßVBr(xv)dV,

and

(x) = J min |1 , rm|x -So, integrating by parts and converting into spherical coordinates (see [9])

ßrm (x)= mr'"/ u-m-1 ßB.(x)du.

r

The following straightforward estimates concern the behaviour of ßr'm* (x) as r ^ 0.

Lemma 1. [9] Let 1 < m < n and ß eP„. For all x G Rn

crm < ßr'm* (x)

for all sufficiently small r, where c > 0 is independent of r. Lemma 2. [9] Let ß G Pn.

1) For all x G Rn and r > 0

ßBr (x) < ßr'n* (x).

2) Let £ > 0. For ß-almost all x

r"eßBr (x) > ßr'n* (x),

if r is sufficiently small.

We use the properties of ßr,m* (x) to obtain a relationship between the kernels and projected measures.

Lemma 3. [9] Let 1 < m < n, ß G Pn, £ > 0, and r be sufficiently small. 1) For all V G Gn,m and for ß-almost all x G Rn

reßr'm* (x) < ßvBr (xv).

2) For jn,m-almost all V E Gn,m and all x E Rn

r"e^r,m*(x) > Br(xv).

3. Projection results. In this section we need an alternative characterization of the upper and lower mutual Lq,t-spectra in terms of convolution. We specify this to the mutual (q, t)-dimensions relatively to ^ and v using appropriate definitions in terms of kernels.

From now on 1 < m < n are two integers and the measures v E Pn are such that S^ = Sv = K. For q, t > 0 we define

l^v(q, t) = liminf Iogr log // (^r,m* (x))q (vr,m* (y))* d^(x)dv(y),

K2nBr (D2)

I^v(q,t) =limsupl-g- log if (x))q (vrm (y))4 d^(x)dv(y).

log - J J

K2 nBr (D2)

Note that for all x, y G K and - > 0

(x) > ^Br (x) and vr'm* (y) > vBr (y). It is clear that for q > 0 and t > 0 and for a sufficiently small -

I^v(q,t) < (q,t) and (q,t) < (q,t). (1)

From Lemma 1 we see that for all x, y G Rn and for any sufficiently small -

c-m < (x) and cC-m < vr'm* (y), where c, cC > 0 are independent of -. This leads to

Imv(q,t) < Im(q,t) < m(q + t). (2)

Proposition 1. Let e > 0, v G Pn.

1) Let q, t > 0. For all V G Gn,m we have

J J (Br (x v ))q vv (Br (yv ))4d^v (xv )dvv (yv ) >

> (x))q(vr'm*(y))'d^(x)dv(y)

for all sufficiently small -.

2) Let 0 < q, t < 1. For Y;;,m-almost all V x W E G;,m we have JJ (Br(xv))qVW(Br(yw))4d^v(xv)dvw(yw) <

< cJJ (^r,r(x))q(vr,m*(y))4d^(x)dv(y)

for all sufficiently small r and C > 0 independent of r. Proof. 1) For all V E Gn,m and x, y E K

(x) < ^T*(xv) and Vrr (y) < vJT*(yv).

Take e > 0 and r > 0. From Lemma 3 we see that for all V E Gn,m and -almost all xv E V

^v Br (xv) > re (xv), and for vv-almost all yv E V

Vv Br (yv ) > reV£m* (yv ).

This means that

Ij(^r,r (x))q(Vr,m* (y))4d^(x)dv(y) <

<JJ^ (xv))q^ (yv))4d^v(xv)dvv(yv) <

< r-e(q+i^y ^v(Br (xv))qVv (Br (yv)№v(xv)dvv(yv). 2) For 0 < q, t < 1, using Lemma 3.11 in [17], we obtain

I = U (^r,m* (x))q(Vr,r (y))4d^(x)dv(y) =

K2nBr (D2)

min {1, rm|x - u|-m} dju(u)^ x

K 2 nBr (D2)

t

x ^y min {1, rm|y - v| m} dv(v)^ d^(x)dv(y) >

> C J J (^J 7n,m {V e : |nv (x) - nv (u)| < r} d|(u)^ x

K2nBr (D2)

x ^y Yn,m {W e : |nw(y) - nw(v) | < r} dv(v)^ d|(x)dv(y) >

> C JJ (^j | {u e Rn : |nv (x) - nv (u)| < r} d7n,m (V)) x K 2nBr (D2)

x v {v e Rn : |nw(y) - nw(v)|< r} d7„,m(W^ d|(x) dv(y). The Jensen inequality and the Fubini Theorem imply

I > ci ^(Br(xv))qvw(Br(yw))*d|v(xv)dvw(yw)

for some c and Ci independent of r. □

Corollary 1. For all q, t > 0 and V e Gn m we have

liminf -- ( log

r—log r

// (|r'm* (x))q (vrm (y))id^(x)dv (y) // |v (Br (xv ))q vv (Br (yv ))* d|v (xv )dvv (yv )_

> 0.

For 0 < q, t < 1, y;; m-almost all V x W G and sufficiently small

r > 0

lim -- ( log

r—^o log r

ff (|r'm* (x))q (vr'm* (y))*d|(x)dv (y)

// |v (Br (xv ))q vw (Br (yw ))*d|v (xv )dvw (yw )_ Proof. Follows directly from Proposition 1. □ Theorem 1. Let v G

1) For all q, t > 0 and V G Gn,m

(i) (q,t) < I^v(q,t) < min (m(q + t),/^(q,t)),

(ii) ,vv(q,t) <1 i^v(q,t) < min (m(q +t),z(q,t^.

2) For all 0 < q, t < 1 and y^ m-almost all V x W G

(i) ,vw (q,t) = I^v (q, t), '

(ii) z^v ,vw (q,t) = ^v (q,t).

= 0.

Proof. This follows from (1), (2) and Corollary 1. □

Take an r > 0 and denote the set of r-mesh cubes in Rn by Cr. These cubes have the form

n

H [kjr, (kj + 1)r[, j=i

where kj G Z.

Proposition 2. If q > —1 and t > — 1, then

iminf --

r^Q log r

4,*(q,t)=liminfl0g- log £ v(C)t+\

Cec

r

1

(q,t)=limsup--log V ^(C)q+1 v(C)t+1.

r^Q log r CeCr

Proposition 2 is a consequence from the following lemma. Lemma 4. Let v G Pn. If q > 0 and t > 0, then

Jf ^(B^nr(x))qV(B^r(y))'d^(x)dv(y) > £ MC)q+1 v(C)i+1 >

K2nBr (D2) CeCr

> C (n,q,t) y[ ^(Br (x))q v (Br (y))' d^(x)dv (y).

K2 nBr (D2)

Proof. Let q > 0 and t > 0, then

MC )q+1v (C )t+1 = £ MC )q v (C y J J dM*)dv (y) =

r CeCr C2

= £ U MC)qv(C)'d^(x)dv(y) <

CeCr C2

< £ II ^(B^nr(x))q(vB^nr(y))'d^(x)dv(y) =

CeCr C2nBr(D2)

= // (x))q v (BVnr (y))td^(x)dv (y).

K 2 nBr (D2 )

For the other inequality we have

MBr(x))qv(Br(y))'dMx)dv(y) <

K2nBr (D2)

< £ J J ^(Br (x))q v (Br (y))' d^(x)dv (y) <

CeCr C2nBr(D2)

< £ U MC)qv(C)'d^(x)dv(y) =

CeCr C2nBr(D2)

II U^(X)Uv(y) <

c ecr

t+1

£ v(C)' /f Mx)dv(y) < £ M^1 v(C)

C eCr C2nBr (D2) C eCr

v (C)^1 <

< £ £ mcO

cecr \i=1

q+1 /

£ v(Cj )

<

v=

< 3n(q+t) £ £ ^(Ci)q+1v(Cj)t+1 < c(n,q,t) £ MC)q+1v(C)t+1, Cecr i,j=1 cecr

where C is the cube of side 3r, concentric with C. □

Proposition 3. Let v G Pn. For all q, t > 0 and V G Gn,m we have

1) (q,t) < (q,t),

2) I^v ,vv (q,t) < I(q, t).

Proof. Assume that S^ = Sv C BR(0) with R > 0. Let V E Gn,m be given, 0 < r < 1 and (C^ be a set of r-mesh sub-cubes of V that cover the unit cube in V with center at the origin. For each i let (Cjj be a column of cubes of side r above Cj, so that (Ci,^ij. are a set of r-mesh cubes which cover the unit cube with center at the origin. Let q > 0 and t > 0; then

£]>>(Ci,j )q+1v (Ci,j )i+1 <

i j

( \ (

£ MCi,j )q+1 j

£v (Ci,j ) j

t+1

<

(

9+1/ \ t+1

<E Ei(Ci,j)) Ev(Ci,j)

v j

V j

So,

EEl(Ci,j)9+1 v(Ci,j)t+1 < E|v(Ci)9+1vv(C)

t+i

Taking the lower and upper limits gives the desired result. □

Definition. For a measure | on Rn and for p > 1 we say that | e Lp (Rn) if there is a function f e Lp (Rn) such that f is the Radon-Nikodym derivative of | with respect to Ln for i-a.e. x.

Now let us obtain conditions that projections of a measure belong to Lp for some p > 1. Consider a compactly supported Borel probability measure | on Rn. For s, m < s < n define the s-energy of | by

/s (i) = II|x - y|-s di(x)di(y).

Mattila [17] proved that if Is (|) is finite for m < s < n, then for almost all V e Gn,m the measure |v is absolutely continuous with respect to the m-dimensional Lebesgue measure Ly on V (denoted by Rm), where Ly(E) = Lm(E n V) for E C Rn, and |v e L2(V).

Proposition 4. [9] Let | be a compactly supported Radon measure on Rn. Let m < s < n. Suppose that Is(|) < to. Then |v is absolutely continuous with respect to Cy, with |v in L2(V) for jn,m-almost all V e Gn,m. Moreover, for jn,m-almost all V e Gn,m

2m

1) if m < s < 2m then |v e Lp (V) for all p satisfying 1 < p <

2m - s

2) if 2m < s < n then the Radon-Nikodym derivative of I V with respect to Ly is bounded and essentially continuous.

Theorem 2. Let m < s < n, Is(|) < to and Is(v) < to. Then for Yn,m-almost all V e Gn,m 1) if 2m < s < n then

I^v(q,t) = ^W,vv (q,t) = m(q +t) for 0 <q,t< to;

2) if m < s < 2m then

(i) for 0 < q, t <

2m 2m — s

- 1

,vv (q,t) = 7 ?v ,vv (q,t) = m(q +1),

(ii) for q, t > ^^ - 1

/ 2m—s

s(q +1 + 2)

< ,vv (q,t) < 7,vv (q,t) < m(q + ^

(m) for q > 2m-s - 1 >t

m (q + t + 1 - (2m -2m < ,vv(q, t) < V,vv(q, t) < m(q+t),

(iv)

for t> 2mms -1 > q,

m (q + t + 1 - (2m 2m + < ,vv(q,t) < V,VV(q, t) < m(q+t).

Proof. It is a simple consequence of Theorem 1, Proposition 4, and the following Lemma. □

Lemma 5. Fix a p > 1. Suppose that ß, v G Lp(Rn), q, t > 0. Then n(q + t + 2) (1 - ^ , if q +1 > p and t + 1 > p;

n I q

(q + t +1 - ^p1 ) , if q +1 > p and t + 1 < p;

(q,t) > <

_ t+i\

n(q + t +1 - ^ J , if q +1 <p and t + 1 > p;

n(q +1),

if p > q + 1 and p > t + 1 .

Proof. Let f = E Lp (Rn) and g = E Lp (Rn). The Holder

dLn v 7 y dLn v 7

inequality gives

2

9+1

t+1

E l(C)9+1v (C)t+1 = En fdLn ) ( i

gdLn

<

< rn(9+1)(1-P)rn(t+1)(1-P) E

q+i

P

t+i p

fp dLn

Cecr \C

gpdLn | <

C

< rn(9+t+2)(1-P)

Ce£r

q+i

P

t+i p

fp dLn

C

E

C e£r

<

C

fr«(9+t+2)(1-p )x

x

I \CeC

E /fPdLn

q+i . P /

C

E /gpdcn

\CGCr C

t+i p

if q +1 > p, t + 1 > p;

C1 r /

x

n(q+t+2)(1-P )r-n(1-^ )

p r p

q+i P

I VC^r C

f pdLn)

E /gpdcn \ce£r C

t+i p

<<

C2r /

x

n(q+t+2)(1-P) r-n(1-q+i)

E /f'dC

i C

p y r

q+i P

p ^ X

E /gpdcn

\cecr C

t+i p

if q +1 > p, t + 1 < p;

if q +1 < p, t + 1 > p;

C3 r /

x

n(9+t+2)(1-P)r-n(1-q+i) r-n(1-m)

E fPdC

i C

p r p r

q+i P (

px

t+i

\ce£r C

if p > q + 1 , p > t + 1

r

p

<

' C1rn(q+t+2)(l-p), if q +1 > p, t + 1 > p; C2rn(q+t+1-^), if q +1 > p, t +1 < p;

C3rn(q+4+1-^), if q +1 <p, t +1 > p;

C4rn(q+4), ifp > q +1,p>t + 1,

where C1, C2, C3, and C4 are independent of r. Since r is sufficiently small, we obtain the required inequality taking the lower limit. □

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Received August 29, 2017. In revised form, November 27, 2017. Accepted November 29, 2017. Published online December 27, 2017.

Faculty of sciences of Monastir Department of mathematics 5000 Monastir, Tunisia E-mail: bilel.selmi@fsm.rnu.tn

Petrozavodsk State University

33, Lenina st., Petrozavodsk 185910, Russia

E-mail: nsvetova@petrsu.ru