CC BY
10
УДК 517.5
Sharp Theorems on Multipliers and Distances in Harmonic Function Spaces in Higher Dimension
MiloS ArsenoviC*
Faculty of Mathematics, University of Belgrade, Studentski Trg., 16, 11000 Belgrade,
Serbia
Romi F. Shamoyan^
Bryansk University, av. 50 October, 7, 241035, Bryansk,
Russia
Received 22.12.2011, received in revised form 22.01.2012, accepted 05.03.2012 We present new sharp results concerning multipliers and distance estimates in various spaces of harmonic functions in the unit ball of Rn.
Keywords: multipliers, harmonic functions, Bergman spaces, mixed norm spaces, distance estimates.
1. Introduction and Preliminaries
The aim of this paper is twofold. One is to describe spaces of multipliers between certain spaces of harmonic functions on the unit ball. We note that so far there are no results in this direction in the multidimensional case, where the use of spherical harmonics is a natural substitute for power series expansion. In fact, even the case of the unit disc has not been extensively studied in this context. We refer the reader to [1], where multipliers between harmonic Bergman type classes were considered, and to [2] and [3] for the case of harmonic Hardy classes. Most of our results are present in these papers in the special case of the unit disc.
The other topic we investigate is distance estimates in spaces of harmonic functions on the unit ball. This line of investigation can be considered as a continuation of papers [4-6].
Let B be the open unit ball in Rn, S = dB is the unit sphere in Rn, for x G Rn we have
x = rx', where r = |x| = . / J2 xj and x' G S. Normalized Lebesgue measure on B is denoted by
V^1
dx — dx i ... dx^
= rn 1drdx' so that /B dx = 1. We denote the space of all harmonic functions in an open set Q by h(Q). In this paper letter C designates a positive constant which can change its value even in the same chain of inequalities. For 0 < p < to, 0 < r< 1 and f G h(B) we set
Mp(f,r)=[ jT If (rx' )| pdx'^
1 /p
with the usual modification to cover the case p = to. Weighted Hardy spaces are defined, for a > 0 and 0 <p < to, by H£(B) = Hp = {f G h(B) : \\f ||p,a = supMp(f,r)(1 - r)a < to}. For
a = 0 the space Hg is denoted simply by Hp.
r<1
*arsenovic@matf.bg.ac.rs tshamoyan.r@gmail.com © Siberian Federal University. All rights reserved
For 0 < p < to, 0 < q < to and a > 0 and we consider mixed (quasi)-norms ||f ||p defined
by i/
Ilf ||p,q;a Mq (/'r)P(1 - r2 , f G h(B), (1)
again with the usual interpretation for p = to, and the corresponding spaces BP'q(B) = BP'q = {f G h(B) : |f ||p,q;a < to}. It is not hard to show that these spaces are complete metric spaces and that for min(p, q) > 1 they are Banach spaces. These spaces include weighted Bergman spaces App(B) = App = B|+ where p > -1 and 0 < p < to. We set A^ = for p > 0.
P
Note that = Hfor a > 0 and Bra'q = Ha for 0 < q < to, a > 0. We also have, for 0 <po < pi < to, Bg»'1 c Bg1-1, see [7].
Next we need certain facts on spherical harmonics and Poisson kernel, see [1] for a detailed exposition. Let Yj(k) be the spherical harmonics of order k, j < 1 < dk, on S. Next, )(y') =
dk Yj(fc)(x')Yj(fc)(y') are zonal harmonics of order k. Note that the spherical harmonics Yj(k), j=i
(k ^ 0, 1 ^ j ^ dk) form an orthonormal basis of L2(S, dx'). Every f G h(B) has an expansion
f (x) = f (rx') = £ rkbk ■ Yk(x'), where 6fc = (bj.,..., bdkk), Yk = (Y1(k),..., Yd(fck)) and 6fc ■ Yk k=0 k
dk . (k)
is interpreted in the scalar product sense: bk ■ Yk = £ bkYj(k). We often write, to stress
j=i
dependence on a function f G h(B), bk = bk(f) and b3k = b^ (f), in fact we have linear functionals bk, k ^ 0,1 ^ j ^ dk on the space h(B).
We denote the Poisson kernel for the unit ball by P(x,y'), it is given by
dk 1 1 _ | |2
P(x,y') = Py (x) = V rkV Y(k)(y')Y(k) (x') = -1-, x = rx' G B, y' G S,
j j |x _ y'|n
k=0 j=1 n 1 y 1
where wn is the volume of the unit ball in Rn. We are going to use also a Bergman kernel for A^ spaces, this is the following function
Qb (x,y)=2V r(P +\+k + n/2) rkpk zXk )(y'), x = rx', y = py ' G B, p > 0. (2) k= r(p +1)r(k + n/2) ^ y Hy \)
For details on this kernel we refer to [7], where the following theorem can be found. Theorem 1 ( [7]). Let p ^ 1 and p ^ 0. Then for every f G A^ and x G B we have
f (x) = ['I Qb (x, y)f (py ' )(1 _ p2)B pn-1dpdy ', y = py '.
Jo Jsn-1
This theorem is a cornerstone for our approach to distance problems in the case of the unit ball. The following lemma gives estimates for this kernel, see [7,8]. Note the Bergman kernel can be also defined for all p > _1
C
Lemma 1. 1. Let p > 0. Then, for x = rx ', y = py ' G B we have |Qb(x,y)| ^
|px _ y ' |n+B ' 2. Let p > _1. Then
r C
Js 1 |QB(rx ',y)|dx ' < (1 _ rp)'+B, |y| = P, 0 ^ r< 1.
3. Let ß>n - 1, , 0 * r < 1 and y' G Sn-1. Then
dx' C
*
JSn-i |rx' - y'^ (1 - r-y-n+1' Lemma 2 ( [7]). Let a > —1 and X > a + 1. Then
/ ' (11-r)" dr < C(1 — p)a+1-X, 0 < p< 1.
Jo (1 — rPF
Lemma 3. Let G(r), 0 ^ r < 1, be a positive increasing function. Then, for a > —1, ft > —1, Y ^ 0 and 0 < q ^ 1 we have
( i Gir)-^—^rad,A * C f G(r)q (1, 1 radr, 0 * p< 1. (3)
Vi0 (1 - pr)1 ) Jo (1 - pr)qY ' '
A special case of the above lemma appears in [9], for reader's convenience we produce a proof. Proof. We use a subdivision of I = [0,1) into subintervals Ik = [rk, rk+1), k > 0, where rk = 1 — 2-k. Since 1 — prk x 1 — prk+1, 0 < p < 1, we have
J =( [ G(r) ,(1 — r)f radA = |V I G(r) (1 — r)f radr1 <
\Jo ^(1 — PrY< J (1 — Pr)Y )
<g(LG<r)^q <gG'<->(X (I—y <
< CX2-kq/3Gq(rk+i)2-kq(1 — prk+i)-qY < CX2-kq^Gq(rk+l)2-kq(1 — prk)-qY < k^0 k^0
(1 — r)^q+q-1 ra dr f1 (1 — r)Pq+q-1
* CEGq(rk+i* Cl -
□
Lemma 4. For S > -1, y > n + £ and ß > 0 we have
i \Qß(x,y)|^(1 - |y|)Ädy * C(1 - \x\)Ä-Y+n, x G B. J B
Proof. Using Lemma 1 and Lemma 2 we obtain:
f \Qß(x,y)\^(1 -\y\)Ädy * C f (1 - lyl)*dy * Jb Jb \prx - y \
* C IV - p)s [ --f-— dy'dp * C f V - p)s (1 - rp)n-Y-1dp * C(1 - r)n+S-Y.
Jo Js \prx' - y'\Y Jo
□
Lemma 5 ( [7]). For real s,t such that s > —1 and 2t + n > 0 we have
^ 1 r(s + 1)r(n/2 + t)
2 r(s + 1 + n/2 + t) '
/ (1 - r2) Jo
2-,s 2t+n-1, _
dr = --
We set R++1 = {(x,t) : x e Rn,t > 0} C Rn+1. We usually denote the points in R++1 by z = (x, t) or w = (y, s) where x, y e Rn and s, t > 0. For 0 < p < to and a > —1 we consider spaces
»n+1) = Aa = < f e h(R++1) : f |f(x,t)|ptadxdt < TO I .
[ J R++1 J
A£(R+ ') = Aa = < f e h(R+ -) : / If (x,
/R++1
Also, for p = to and a > 0, we set
A~(R++1) = ={ f e h(R++1): sup If(x,t)|ta < to}> .
flc»n+1
These spaces have natural (quasi)-norms, for 1 < p < to they are Banach spaces and for 0 < p < 1 they are complete metric spaces.
for R+
We denote the Poisson kernel for R++1 by P(x, t), i.e.
P(x,t) = cn-t-^, x e Rn,t > 0.
|x|2 + t2 Y 2
For an integer m ^ 0 we introduce a Bergman kernel Qm(z,w), where z = (x,t) e R++1 and w = (y,s) e R++1, by
(_2)m+1 dm+1
Qm (z, w) = -;-—-—rP(x _ y, t + s).
The terminology is justified by the following result from [7].
a ++ n ++ 1
Theorem 2. Let 0 < p < to and a> _1. If 0 <p ^ 1 and m ^--(n +1) or
p
a +1
1 ^ p < to and m >--1, then
p
f (z) = / f (w)Qm(z,w)smdyds, f e la, z e R++1. (4)
7r++1
The following elementary estimate of this kernel is contained in [7]:
m+ 1
|Qm(z,w)| < C [|x _ y|2 + (s +t)2] ~ , z =(x,t),w =(y,s) e R++1. (5)
2. Multipliers on Spaces of Harmonic Functions
In this section we present our results on multipliers between spaces of harmonic functions on the unit ball. The following definitions are needed to formulate these theorems.
Definition 1. For a double indexed sequence of complex numbers c = {j : k ^ 0,1 ^
k (f )j
dfc • (k)
j ^ dk} and a harmonic function f (rx') = ^k ^k rkj(f)Yj )(x') we define (c * f)(rx')
fc=0j=1 dk • • (k)
5k 5k rkcjbk(f )Yj( )(x'), rx' e B, if the series converges in B. Similarly we define convolution
k=0j=1
dk ' • (k)
of f,g e h(B) by (f * g)(rx') = 5k dk rkb^(f )bk(g)Y( )(x'), rx' e B, it is easily seen that f * g
is defined and harmonic in B.
k V-/ / k Vi// j
k=Qj=1
Definition 2. For t > 0 and a harmonic function f (x) = J2 6k(f )Yk(x ') on the unit ball we
k=0
define a fractional derivative of order t of f by the following formula:
(Af )(x) = Vrk r(k + n/2 +t) 6fc(f) • yk(x'), x = rx' e B. V ; k= r(k + n/2)r(t) feu; V
Clearly, for f e h(B) and t > 0 the function Ath is also harmonic in B.
Definition 3. Let X and Y be subspaces of h(B). We say that a double indexed sequence c is a multiplier from X to Y if c * f e Y for every f e X. The vector space of all multipliers from X to Y is denoted by MH (X, Y).
Clearly every multiplier c e MH(X, Y) induces a linear map Mc : X ^ Y. If, in addition, X and Y are (quasi)-normed spaces such that all functionals are continuous on both spaces X and Y, then the map Mc : X ^ Y is continuous, as is easily seen using the Closed Graph Theorem. We note that this holds for all spaces we consider in this paper : Aa, B^'9 and Ha.
Lemma 6. Let f, g e h(B) have expansions
to dk to dk
f (rx') = £ rfc£ &lYj(V), g(rx ') = £ rfc£ cjY/V).
k=0 j=1 i=0 i=1
Then we have
„ to dk
/(g * Py )(rx ')f (px')dx' = £ rkpk^ &£c£Yj(k)(y'), y' e S, 0 < r, p < 1.
k=0 j=1
Moreover, for every m > -1, y ' e S and 0 ^ r, p < 1 we have
f (g * Py')(rx')f (px ')dx ' = 2 [ i Am+1(g * Py')(rRx ')f (pRx')(1 - R2)mRn-1dx 'dR. ./s ./0 ./s
Proof. The first assertion of this lemma easily follows from the orthogonality relations for
(k)
spherical harmonics Yj . Using Lemma 5 and orthogonality relations we have
n-1.
I = 2 [ [ Am+1(g * Py )(rRx ')f (pRx ')(1 - R2)mRn-1dx 'dR =
./o J S
= 2 f1 V rkpkR2k+n-1(i - r2)m r(k + n/2 + m + 1) * j jdR io k=0 p ( ) r(k + n/2)r(m +1) k k j
oo dk
^ bkck j
fc=0 j=1
Erfc /E bkck Y(fcV),
which proves the second assertion. □
We note that (g * Py)(rx') = (g* Px')(ry') and At(g* Py)(x) = (Atg* Py)(x), these easy to prove formulae are often used in our proofs.
In this section fm,y stands for the harmonic function fm,y(x) = Qm(x, y), y e B. We often write fy instead of fm,y. Let us collect some norm estimates of fy.
o
Lemma 7. For 0 < p ^ to and m > 0 we
Mœ(fm,y ,r) < C (1 — |y|r)— n— m
Mfy ,r) < C (1 — |y|r)— 1— m,
\\fm,y y < C (1 — |y|)a 1— m, m > a — 1, a > 0,
\\fm,y\\b£'~ < C (1 — |y l)a n— m, m > a — n, a > 0,
\\fm,y Wa^ < C (1 — |y |)a m m > a > —1,
\\fm,y Wh^ < C (1 — |y |)a 1— m, m > a — 1, a > 0.
(6)
(7)
(8) (9)
(10) (11)
Proof. Using Lemma 1 we obtain
Mœ(fm,y,r) = max IQm(y,rx')| < max
C
x'es
es |prx ' — y'|
n+m
= C (1 — r|y|)
—n—m
which gives (6). The estimate (7) follows from Lemma 1. The estimates (8), for finite p, and (10) follow from Lemma 2 and (7). Similarly, for finite p (9) follows from (6) and Lemma 2. Next, using (7),
fm,y Uffi < C SUp (1 — r)a(1 — rp)
—m — 1
p = |y|-
0^r<1
The function ^(r) = (1 — r)a(1 — rp)—m—1 attains its maximum on [0,1] at
a
r0
1 — (1 — P)
p(1 + m — a) '
as is readily seen by a simple calculus, and this suffices to establish (11) and therefore (8) for p = to. Finally, (9) directly follows from Lemma 1. □
In this section we are looking for sufficient and/or necessary condition for a double indexed sequence c to be in MH(X, Y), for certain spaces X and Y of harmonic functions. We associate to such a sequence c a harmonic function
dk
gc(x) = g(x) = rkY, 4Yjk)(x ' ), x = rx ' G B, k^0 j=1
(12)
and express our conditions in terms of gc. Our main results give conditions in terms of fractional derivatives of gc, however it is possible to obtain some results on the basis of the following formula, contained in Lemma 6:
(c . f)(rV)=/(g . P„)(rx')f(ry')dy'.
J S
(13)
Using continuous form of Minkowski's inequality, or more generally Young's inequality, this formula immediately gives the following proposition.
Proposition 1. Let c = {ck : k ^ 0,1 ^ j ^ dk} be a double indexed sequence and let g(x) =
S rk dk ckYjk\x') be the corresponding harmonic function. If
k'^0 j = 1
I |(g . Py')(rx ')lpdx' < C, y' G S, 0 < r < 1,
S
then c G Mh(H 1,Hp).
More generally, if 1/q + 1/p = 1 + 1/r, where 1 ^ p,q,r ^ to, a + 7 = ft, a, ft, 7 ^ 0 and g G Hp, then c G Mh (Hq, Hrp).
The first part of the following lemma, which gives necessary conditions for c to be a multiplier, is based on [9].
Lemma 8. Let 0 < p, q ^ to, 1 ^ s ^ to and m > a — 1. Assume a double indexed sequence c = {j : k ^ 0,1 ^ j ^ dk} is a multiplier from Bp1 to Bq's and g = gc is defined in (12). Then the following condition is satisfied:
Ns(g)= sup sup(1 — p)m+1-a+pf f |Am+1(g * Px')(py ')|sdx 7 < to, (14)
0^p<1 y'es \Js J
where the case s = to requires usual modification.
Also, let 0 < p ^ to, 1 ^ s ^ to and m > a — 1. If a double indexed sequence c = {j : k ^ 0,1 ^ j ^ dk} is a multiplier from Bp1 to Hp, then the above function g satisfies condition (14).
Proof. Let c e MH(BP'1, Bqs), and assume both p and q are finite, the infinite cases require only small modifications. We have ||Mcf < C||f ||Bp,i for f in Bg'1. Set hy = Mcfy, then we have
k
hy(x) = Erkpk E r(k + :/22)+(: + 1)ckYj(k)(y')Yj(k)(x'), x = rx ' e B, (15)
moreover
|hybr < C||fy||bs.i. (16)
This estimate and Lemma 8 give
llhy|Ur < C(1 — |y|)a-m-1, y e B. (17)
Note that hy(x) = Am+1(g * Py')(px), using monotonicity of Ms(hy, r) we obtain:
1/s , r 1 \ -1/9
V (p2) = ( Js |Am+1(g * Px' )(p2 y' )|s dx = QP (1 — r)Pq-1rn-1dr)
x (^V — |Am+1(g * Py')(p2x')|sdx 'y/ drj <
< C(1 — p)-^ (y (1 — r)P9-1rn-1Ms9(hy,r)dr^ <
p
p
< C(1 — p)-p||hy. (18)
Combining (18) and (17) we obtain
QT |Am+1(g * Px')(p2y ')|sdx 7 < C(1 — p)a-p-m-1,
which is equivalent to (14). The case s = to is treated similarly.
Next we consider c e MH(Bgg'1 , Hp), assuming 0 < p < to. Set hy = Mchy = g * fy. We have, by Lemma 7,
llfyIbs-1 < C(1 — |y|)a-m-1, y e B, and, by continuity of Mc, ||hy ||Hs < C ||fy ||Bp,i. Therefore
11hy||*. < C(1 — |y|)a-m-1, y e B.
Setting y = py' we have
Iy' (p2)= |Am+1(g . Px' )(p2y ' )|sdx'^ 7 = ( ji |Am+1(g . Py )(px ' Wdx 7 =
= Ms(hy ,p) < (1 — |y|)— P\\hy Wh- . The last two estimates yield
QS |Am+1(g . Px')(p2y'Wdx7 < C(1 — |y|)a—m—1, |y| = p
which is equivalent to (14). □
Theorem 3. Let 1 < p ^ q ^ to, and m > a — 1. Then for a double indexed sequence c = {j : k ^ 0,1 ^ j ^ dk} the following conditions are equivalent:
1. c G Mh(BP!1,Bq'1).
2. The function g(x) = rk 2 C~ÎYjk\x') is harmonic in B and satisfies the following
condition
Ni(g) < to. (19)
Proof. Since necessity of (19) is contained in Lemma 8 we prove sufficiency of condition (19). We assume p and q are finite, the remaining cases can be treated in a similar manner. Take f € Bp1 and set h = Mcf. Applying the operator Am+1 to both sides of equation (13) we obtain
Am+ih(rx) = Am+i(g * Py')(x)f (ry')dy'. (20)
■J S
Now we estimate the L1 norm of the above function on |x| = r:
Mi(Am+ih,r2) ^ I Mi(Am+i(g * Py'),r)|f (ry')|dy ' <
■J S
< Mi(f,r)sup |Am+i(g * Py')(rx')|dx' <
y'esJs
< Mi(f,r)Ni(g)(1 - r)a-p-m-1. (21)
Since,
f Mp(h, r2)(1 - r)pp-1rn-1dr < of (1 - r)p(m+1)Mp(Am+1h, r2)(1 - r)pp-1rn-1dr,
J 0 J 0
see [7], we have
\\Hlv,i < O i V - r)p(m+i )Mf(Am+1 h, r2)(1 - r)pp-1 rn-1 dr < J 0
< ONp(g) f Mf(f, r)(1 - r)ap-1 rn-1 dr = ONP (g)\\f f A ,
Jo Ba
and therefore \\hyBp,i < \\f \\Bp,i. Since \\h\\B,,i < O\\h\\Bp,i the proof is complete. □
Next we consider multipliers from BOO;1 to Hp, in the case 0 < p < 1 we obtain a characterization of the corresponding space.
Theorem 4. Let f ^ 0, 0 < p ^ 1, s ^ 1 and m > a — 1. Then, for a double indexed sequence c = {cj : k ^ 0,1 ^ j ^ dk} the following two conditions are equivalent:
1. c G Mh).
2. The function g(x) = rk 2 CtY^^ ') is harmonic in B and satisfies the following
fc^0 j=1
condition:
N(g) < to. (22)
Proof. The necessity of condition (22) is contained in Lemma 8. Now we turn to the sufficiency of (22). We chose f G Bp1 and set h = c * f. Then, by Lemma 6:
h(r2x ') = 2 i [ Am+1(g * Pç)(rRx ')f (rR£)(1 — R2)mRn-1d£dR (23)
./o ./s
and this allows us to obtain the following estimate:
Ms(h,r2) < 2 I (1 — R2)mRn-1 f Am+1(g * Pç )(rRx')f (rR£)d£ ■Jo J S
< 2 / (1 — R2)mR"-1M1(f,rR)sup ||Am+1(g * P5)(rRx')||L.dR <
Jo çes
< CNs(g J (1 — R)mM1(f,rR)(1 — rR)a"^"m-1dR <
Jo
< CNs(g)(1 — W M1 (f, rR)(1 — R)m(1 — rR)a-m-1dR.
Jo
dR <
Note that M1(f, rR) is increasing in 0 ^ R < 1, therefore we can combine Lemma 3 and the above estimate to obtain:
Mp(h,r2) <cNP(g)(i-r)-ßpjo Mf(/,rR)(j-rR);:_i;+pdR< < CNp(g)(1 - r)-pß f Mf(/,R)(1 - R)ap-1dR <
Jo
< CNp(g)(1 - r)
o
II /llp
Therefore Ms(h, r2) ^ CNs(g)(1 — r) ß||/||bp,i, which completes the proof of the Theorem. □
3. Estimates for Distances in Harmonic Function Spaces in the Unit Ball and Related Problems in R++1
In this section we investigate distance problems both in the case of the unit ball and in the case of the upper half space.
Lemma 9. Let 0 < p < to and a > —1. Then there is a C — such that for every
/ G Aa(B) we have
|/(x)| < C(1 — |x|)-||/|U* , x G B.
B
Proof. We use subharmonic behavior of |f |p to obtain
If(x)lp < TTTrnl 1 ,, If(y)lpdy <
(1 - lxl)n JB(x,)
^ o ^J^T jB{x ^ If (y)lp (1 - lyl)ady < O (1 - |x|)-a-n\f \\Aa.
□
This lemma shows that Ava is continuously embedded in A^+n and motivates the distance problem
p
that is investigated in Theorem 5.
Lemma 10. Let 0 <p < to and a > -1. Then there is O = Op,a,n such that for every f € Aa
lf(x,t)| < Cy-\\f \\a*. (24)
and every (x,t) € R++1 we have
The above lemma states that A^, is continuously embedded in A^+n+1, its proof is analogous
p
to that of Lemma 9.
For e > 0, t> 0 and f € h(B) we set
Ue,t(f) = Ue,t = {x € B : lf(x)l(1 - |x|)4 > e}.
a ++ n (a ++ n a \ Theorem 5. Let p > 1, a > -1, t = - and ft > mad--1, — . Set, for f €
p V p p J
A^+n (B):
P
ti(f) = distA~+n (f,Apa),
P
t2(f )=infje > 0: lQp(x,y)|(1 - |y|)p-tdyj (1 - |x|)adx < to| .
Then ti(f) x t2(f).
Proof. We begin with inequality t1(f) > t2(f). Assume t1(f) < t2(f). Then there are 0 < e1 < e and f1 € Ava such that \\f - f1\A~ ^ e1 and
f(f lQp (x, y) l (1 -|y|)P-td^ (1 - |x|)adx = +to. J®yue,t(f) J
Since (1 - |x|)t|fi(x)| > (1 - |x|)t|f(x)| - (1 - |x|)t|f(x) - fi(x)| for every x € B we conclude that (1 - |x|)t|f1(x)| > (1 - |x|)t|f(x)| > (1 - |x|)t|f (x)| - e1 and therefore
(e - ei)xu€,t(f)(x)(1 - |x|)-t < |fi(x)|, x € B.
Hence
+to =f(f lQp (x, y) l (1 -|y|)p-td^ (1 - |x|)adx = JB\JUe,t(f) J
= Kl Xf lQp(x,y)l(1 - |y|)pdy^j (1 - |x|)adx <
< Oe,ei [ ( I |fi(y)||Qp (x, y) | (1 -|y|)P dyY (1 - |x|)adx = M, JB \JB /
and we are going to prove that M is finite, arriving at a contradiction. Let q be the exponent conjugate to p. We have, using Lemma 4,
I(x)= (L '/i(y)'(1 - |y|)ßIQß(x,y)|d^P =
= QB I/i(y)I(1 - |y|)ßIQß(x,y)I^(n+ß-e)IQß(x,y)|^(n +e)d^P <
< / |/i(y)|p(1 -|y|)pß|Qß(x,y)|dy(f IQß(x,y)|wdy)P/q < jb \./b /
< C(1 - |x|)-p^ |/i(y)|P(1 - |y|)pßIQß(x,y)|dy
Jb
for every e > 0. Choosing e > 0 such that a - pe > -1 we have, by Fubini's theorem and Lemma 4:
M < C f |f1(y)|p(1 — |y|W(1 — |x|)a-pe|Qp(x,y)|dxdy < J b jb
< C / |f1(y)|p(1 — |y|)ady< to. jb
In order to prove the remaining estimate t1(f) < Ct2(f) we fix e > 0 such that the integral appearing in the definition of t2(f) is finite and use Theorem 1, with ft > max(t — 1,0):
f (x) = / Qp(x, y)f (y)(1 — |y|2)pdy + / Qp(x, y)f (y)(1 — |y|2)pdy =
MUe,t(f) JUe,t (f)
= f1(x) + f2(x).
Since, by Lemma 4, |f1 (x)| < 2p fB |Qp(x,y)|(1 — |w|)p-tdy < C(1 — |x|)-t we have ||f1|U~ < Ce. Thus it remains to show that f2 e Aa and this follows from
l№ < If IIA- i ( i |Qp(x, y)|(1 — |y|2)p-id^ (1 — |x|)adx< to. □
a 4 JB\Ju€,t(f) J
The above theorem has a counterpart in the R++1 setting. As a preparation for this result we need the following analogue of Lemma 4.
Lemma 11. For 5 > —1, y > : + 1 + 5 and m e N0 we have JRn+i |Qm(z,w)| n+m+1 s5dyds ^
Ct^-7+n+1, t > 0.
Proof. Using Fubini's theorem and estimate (5) we obtain
I(t)=/ |Qm(z,w)| ^^ s5 dyds < CfVf/ f| + dy + 2 ) ds = ./R++1 ./0 \JRn [|y|2 + (s + t)2JY7
f TO
= C s5 (s + t)n-Yds = Ct5-Y+n+1. □
0
For e > 0, A > 0 and f e h(R++1) we set: Ve,A(f) = {(x,t) e R++1 : |f(x,t)|tA > e}.
a + : + 1 a + : + 1 a
Theorem 6. Let p > 1, a > —1, A = -, m e N0 and m > max--1, — .
p V p PJ
Set, for f e ATO+„+i (R++1): s1(f) = dist^^r (f, A*),
p a+n+1
P
s2(f )=infje > 0: J ^J Qm(z,w)sm-Adydsj tadxdt < to| .
Then si(f) x S2(f).
The proof of this theorem closely parallels the proof of the previous one, in fact, the role of Lemma 4 is taken by Lemma 11 and the role of Theorem 1 is taken by Theorem 2. We leave details to the reader.
The first author was supported by Ministry of Science, Serbia, project МЩ010.
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Точные теоремы о мульпликаторах и расстояние в пространствах гармонических функций высшей размерности
Милош Арсенович Роми Ф. Шамоян
Представляются новые точные результаты, связанные с мульпликаторами и оценками 'расстояния в различных пространствах гармонических функций в единичном шаре из Rn.
Ключевые слова: мультипликаторы, гармонические функции, пространства Бергмана, пространства со смешанной нормой, оценки дистанции.
