Table Of ContentSHARP ESTIMATES OF THE POTENTIAL KERNEL FOR THE HARMONIC
OSCILLATOR WITH APPLICATIONS
ADAMNOWAK ANDKRZYSZTOFSTEMPAK
1
1
0 Abstract. Weprovequalitativelysharpestimatesofthepotentialkernelfortheharmonicoscil-
2 lator. TheseboundsarethenusedtoshowthattheLp−Lq estimatesoftheassociated potential
v operator obtained recently byBongioanni and Torrea [2] are in fact sharp.
o
N
4
1. Introduction
2
The study of the potential theory for the d-dimensional harmonic oscillator
]
A
= ∆+ x 2,
C H − k k
. hasrecentlybeeninitiatedbyBongioanniandTorrea[2]. Themulti-dimensionalHermitefunctions
h
t hk are eigenfunctions of and we have hk = (2k +d)hk. The operator has a natural self-
a H H | | H
adjoint extension, here still denoted by , whose spectral decomposition is given by the h .
m k
H
The integral kernel G (x,y) of the Hermite semigroup exp( t ) : t > 0 is known explicitly
[ t { − H }
to be (see [7] for this symmetric variant of the formula)
1
v ∞
8 Gt(x,y) = e−(2n+d)t hk(x)hk(y)
3 n=0 k=n
7 X |X|
5 = 2πsinh(2t) −d/2exp 1 tanh(t) x+y 2+coth(t) x y 2 .
1. − 4 k k k − k
(cid:18) (cid:19)
h i
1 Given σ > 0, cons(cid:0)ider the neg(cid:1)ative power σ, which is a contraction on L2(Rd). It is easily
11 seen that −σ coincides in L2(Rd) with the pHot−ential operator
H
:
v
(1) σf(x) = σ(x,y)f(y)dy,
i
X I RdK
Z
r where the potential kernel is given by
a
1
(2) σ(x,y) = ∞G (x,y)tσ 1dt.
t −
K Γ(σ)
Z0
Note that all the spaces Lp(Rd), 1 p , are contained in the natural domain of σ consisting
≤ ≤ ∞ I
of those functions f for which the integral in (1) converges x-a.e., see [5, Section 2].
The main result of the paper, Theorem 2.4 below, provides qualitatively sharp estimates of the
potential kernel (2). As an application of this result, we prove sharpness of the Lp Lq estimates
−
for the potential operator (1) obtained recently by Bongioanni and Torrea [2, Theorem 8], see
Theorem 3.1.
Recall that an operator T defined on Lp(Rd) for some 1 p , with values in the space of
measurable functions on Rd, is said to be of weak type (p,q≤), 1 ≤q∞< , provided that
≤ ∞
q
(3) x Rd: Tf(x) > λ C f /λ ,
p
|{ ∈ | | }| ≤ k k
(cid:16) (cid:17)
2010 Mathematics Subject Classification: Primary 47G40; Secondary 31C15.
Key words and phrases: harmonic oscillator, potential operator, potential kernel,Lp−Lq estimate.
Research of both authors supported byMNiSW Grant N N201 417839.
1
2 A.NOWAKANDK.STEMPAK
with C > 0independentof f Lp(Rd) andλ > 0. Therestricted weak type(p,q)of T means that
(3)holdsforf = χ ,whereE∈isanymeasurablesubsetofRd offinitemeasure. Bydefinition,weak
E
type (p, ) coincides with strong type (p, ), i.e. the estimate Tf C f , f Lp(Rd).
p
In terms∞of Lorentz spaces, the weak type (∞p,q) is equivalent to thke boku∞nd≤edneksskfrom∈Lp(Rd) to
Lq, (Rd),andtherestrictedweaktype(p,q)ischaracterized bytheboundednessfromLp,1(Rd)to
∞
Lq, (Rd),see[1,Chapter4,Section 4]. Strongtype(p,q)meansofcoursetheLp-Lq boundedness.
∞
The notation X . Y will be used to indicate that X CY with a positive constant C
≤
independent of significant quantities; we shall write X Y when simultaneously X . Y and
≃
Y . X. We will also use the notation X Y exp( cZ) to indicate that there exist positive
≃≃ −
constants C,c and c , independent of significant quantities, such that
1 2
C 1Y exp( c Z) X CY exp( c Z).
− 1 2
− ≤ ≤ −
Further, in a number of places, we will use natural and self-explanatory generalizations of the
≃≃
relation, for instance in connection with certain integrals involving exponential factors. In such
cases the exact meaning will be clear from the context. By convention, is understood as
≃≃ ≃
whenever there are no exponential factors involved.
We write log+ for the positive part of the logarithm, and , for the operations of taking
∨ ∧
maximum and minimum, respectively.
2. Estimates of the potential kernel
We begin with two technical results describing the behavior of the integrals
I (T) = ∞tAexp( t)dt, T > 0,
A
−
ZT
S
J (T,S) = tAexp( t)dt, 0 < T < S < .
A
− ∞
ZT
Notice that I (T) dominates J (T,S). The lemma below is a refinement of [5, Lemma 2.1], see
A A
also [6, Lemma 1.1].
Lemma 2.1. Let A R and γ > 0 be fixed. Then
∈
(4) I (γT) TAexp( γT), T 1,
A
≃ − ≥
and for 0< T < 1
TA+1, A< 1
−
I (γT) log(2/T), A= 1 .
A
≃ −
1, A> 1
−
Proof. We assume that γ = 1. From theproof it will be clear that the estimates are true for any
γ > 0. The case 0 < T < 1 was treated in the proof of [5, Lemma 2.1], so we consider T 1 and
≥
focus on showing (4). The lower bound in (4) is straightforward, we have
2T 2T
I (T)> tAe tdt & TA e tdt = TA e T e 2T & TAe T, T 1.
A − − − − −
− ≥
ZT ZT
(cid:0) (cid:1)
It remains to prove the upper bound,
(5) ∞tAe tdt . TAe T, T 1,
− −
≥
ZT
SHARP ESTIMATES OF THE POTENTIAL KERNEL 3
and here we assume that A> 0, since for A 0 we have tA TA, t > T 1, and the conclusion
≤ ≤ ≥
is trivial. Choosing T such that for T T one has
A A
≥
1
∞tAe tdt ∞tAe tdt,
− −
≤ 2
Z2T ZT
we can write
2T 1
∞tAe tdt tAe tdt+ ∞tAe tdt CTAe T + ∞tAe tdt, T T .
− − − − − A
≤ ≤ 2 ≥
ZT ZT Z2T ZT
This implies (5) for T T and consequently for all T 1. (cid:3)
A
≥ ≥
Lemma 2.2. Let A R and γ > 0 be fixed. Then for 0 < T < S 2T we have
∈ ≤
(6) TA(S T)exp( 2γT) . J (γT,γS) . TA(S T)exp( γT),
A
− − − −
while for S > 2T > 0 we have J (γT,γS) I (γT) when S 2, and
A A
≃ ≥
TA+1, A< 1
−
J (γT,γS) log(S/T), A= 1
A
≃ −
SA+1, A> 1
−
when 0 < S < 2.
Proof. As in the proof of Lemma 2.1, it is enough to deal with the case γ = 1. The bounds for
T < S 2T follow since then StAe tdt TA Se tdt and
≤ T − ≃ T −
R S R
(S T)e 2T e tdt (S T)e T.
− − −
− ≤ ≤ −
ZT
Assume now that S > 2T. Clearly, J (T,S) < I (T). On the other hand, if T 1 then
A A
≥
2T 2T
J (T,S) > tAe tdt & TA e tdt & TAe T & I (T),
A − − − A
ZT ZT
the last estimate being a consequence of (4). When 0 < T < 1, we distinguish two subcases. If
S 2, then again J (T,S) & 2tAdt & I (T). If 2T < S < 2, then J (T,S) StAdt, and
≥ A T A A ≃ T
evaluating the last integral we arrive at the claimed bounds for J (T,S). (cid:3)
A
R R
We note that (4) and (6) may be written slightly less precisely as
I (γT) exp( cT), T 1,
A
≃≃ − ≥
J (γT,γS) TA(S T)exp( cT), 0 <T < S 2T,
A
≃≃ − − ≤
respectively. This fact will be used in the sequel without further mention.
We now apply Lemmas 2.1 and 2.2 to prove qualitatively sharp estimates of the integral
1
E (T,S) = tAexp Tt 1 St dt, 0 < T,S < .
A −
− − ∞
Z0
(cid:0) (cid:1)
The following result provides, in particular, a refinement and generalization of [3, Lemma 2.4].
Lemma 2.3. Let A R be fixed. Then
∈
TA+1, A < 1
−
E (T,S) exp c T(T S) 1+log+ 1 , A = 1 ,
A ≃≃ − ∨ × T(T S) −
(cid:16) p (cid:17) (S 1)−A−1,∨ A > 1
∨ −
uniformly in T,S > 0.
4 A.NOWAKANDK.STEMPAK
Proof. We first estimate E (T,S) in terms of the integrals I and J . For 0 < S 2T we have
A A A
≤
1
E (T,S) tAexp( cTt 1)dt TA+1 ∞u A 2e udu = TA+1I (cT),
A − − − − A 2
≃≃Z0 − ≃ ZcT − −
where the second relation follows by the change of variable t = cT/u. When S > 2T we change
the variable t = u T/S and get
p T (A+1)/2 √S/T
E (T,S) = uAexp √TS(u+u 1) du + ,
A − 1 2
S − ≡ J J
(cid:16) (cid:17) Z0
(cid:0) (cid:1)
where and come from splitting the integration over the intervals (0,1) and (1, S/T),
1 2
J J
respectively. Then
p
T (A+1)/2 1
uAexp c√TSu 1 du TA+1 ∞ z A 2e zdz
1 − − − −
J ≃≃ S − ≃
(cid:16) (cid:17) Z0 Zc√TS
(cid:0) (cid:1) = TA+1I c√TS
A 2
− −
and (cid:0) (cid:1)
T (A+1)/2 √S/T cS
uAexp c√TSu du S A 1 zAe zdz
2 − − −
J ≃≃ S − ≃
(cid:16) (cid:17) Z1 Zc√TS
(cid:0) (cid:1) = S A 1J c√TS,cS .
− − A
Summing up, we have (cid:0) (cid:1)
E (T,S) TA+1I c T(T S) +χ S A 1J c√TS,cS ,
A A 2 S>2T − − A
≃≃ − − ∨ { }
uniformly in S,T > 0. In the next ste(cid:0)ppwe describe(cid:1)the behavior of the t(cid:0)wo terms h(cid:1)ere by means
of Lemmas 2.1 and 2.2.
From Lemma 2.1 it follows that
TA+1I c T(T S) TA+1exp c T(T S) , T(T S) 1,
A 2
− − ∨ ≃≃ − ∨ ∨ ≥
(here, and also in ana(cid:0)lopgous places(cid:1)below, c on the(cid:0)left-hpand side sh(cid:1)ould be understood as a given
constant) and
TA+1, A< 1
−
TA+1I c T(T S) log( 4 ), A= 1 , T(T S) 1.
−A−2(cid:0) p ∨ (cid:1) ≃ TTST(T(A∨+S1))/2, A> −−1 ∨ ≤
∨
The term S−A−1JA(c√TS,cS) comes int(cid:0)o pla(cid:1)y when S > 2T, and in this case we use Lemma 2.2
to write the bounds
S A 1J c√TS,cS χ Φ +χ Φ ,
− − A S 2 1 S<2 2
≃ { ≥ } { }
where
(cid:0) (cid:1)
(T/S)(A+1)/2, A < 1
−
Φ1 = S−A−1IA c√TS , Φ2 =log(TS), A = −1 .
1, A > 1
(cid:0) (cid:1)
−
By Lemma 2.1,
Φ S A 1exp c√TS , TS 1,
1 − −
≃≃ − ≥
(T/S)(A+1)(cid:0)/2, A<(cid:1) 1
−
Φ log( 4 ), A= 1 , TS 1.
1 ≃ TS − ≤
S A 1, A> 1
− −
−
SHARP ESTIMATES OF THE POTENTIAL KERNEL 5
Toproceed,itisconvenient toconsidereach of thecases A< 1, A = 1,andA> 1separately.
− − −
If A< 1, then
−
T (A+1)/2 TA+1exp c T(T S) , T(T S) 1
E (T,S) χ + − ∨ ∨ ≥
A ≃≃ {2>S>2T} S (TA+1, (cid:0) p (cid:1) T(T S) < 1
(cid:16) (cid:17) ∨
TA+1exp c√TS , TS 1
+χ χ − ≥ .
{S>2T} {S≥2}( TS (A+1)/(cid:0)2, (cid:1) TS < 1
Here the first and third terms are insi(cid:0)gni(cid:1)ficant in comparison to the second one. In case of the
third summand, this is because A < 1 and T (A+1)/2 < TA+1 for TS < 1. A similar argument
− S
is used for the first one. The required estimates of E (T,S) follow.
A
(cid:0) (cid:1)
If A= 1, then
−
S exp c T(T S) , T(T S) 1
E (T,S) χ log + − ∨ ∨ ≥
−1 ≃≃ {2>S>2T} T (log (cid:0)T(T4pS) , (cid:1) T(T ∨S) < 1
∨
exp (cid:0) c√TS(cid:1) , TS 1
+χ χ − ≥ .
{S>2T} {S≥2}(log (cid:0)T4S , (cid:1) TS < 1
SimilarlyasinthecaseofA< 1,herealsothefir(cid:0)sta(cid:1)ndthirdtermsareinsignificantincomparison
−
to the second one. This is clear for the third summand, and for the first one this is because
log S < log( 4 ) when S < 2. Thus the desired bounds of E (T,S) also follow.
T TS 1
−
Finally, we consider the case A> 1, which is less direct than the previous two. We have
−
TA+1exp c T(T S) , T(T S) 1
E (T,S) χ + − ∨ ∨ ≥
A ≃≃ {2>S>2T} ( TTS (A+(cid:0)1)/2, p (cid:1) T(T ∨S) < 1
∨
(cid:0) T(cid:1)A+1exp c√TS , TS 1
+χ χ − ≥ .
{S>2T} {S≥2}(S−A−1, (cid:0) (cid:1) TS < 1
Observe that here the relation remains valid if the sum of the first and the third terms is
≃≃
replaced by the comparable (in the sense of ) expression
≃
TA+1exp c√TS , TS 1
χ − ≥ .
{S>2T}((S 1)−A(cid:0)−1, (cid:1) TS < 1
∨
Taking into account that TA+1exp( c√TS) S A 1exp( c√TS) for TS 1, we conclude
− −
− ≃≃ − ≥
that
(T S) A 1exp c T(T S) , T(T S) 1
− −
E (T,S) ∨ − ∨ ∨ ≥
A ≃≃ ( TTS (A+1)/2, (cid:0) p (cid:1) T(T ∨S)< 1
∨
(cid:0) (cid:1) S A 1exp c√TS , TS 1
− −
+χ − ≥ .
{S>2T}((S 1)−A−(cid:0)1, (cid:1) TS < 1
∨
Now, if T S and T(T S) = T2 < 1, then ( T 1/2 = 1 1/(S 1), while for T < S and
≥ ∨ T S ≃ ∨
T(T S)= TS < 1, we have ( T 1/2 = (T 1/2 <∨1/(S 1). Therefore,
∨ T S S (cid:1) ∨
∨
(T S(cid:1)) A 1exp(cid:1) c T(T S) , T(T S) 1
− −
E (T,S) ∨ − ∨ ∨ ≥ .
A ≃≃((S 1)−A−1, (cid:0) p (cid:1) T(T S) < 1
∨ ∨
We claim that this implies
E (T,S) (S 1) A 1exp c T(T S) ,
A − −
≃≃ ∨ − ∨
(cid:0) p (cid:1)
6 A.NOWAKANDK.STEMPAK
which are precisely the required estimates.
To justify the claim, it is enough to recall that A > 1 and observe that if T S and
− ≥
T(T S)= T2 1, then
∨ ≥
(T S) A 1exp c T(T S) = T A 1exp( cT) (T 1) A 1exp( cT)
− − − − − −
∨ − ∨ − ≃ ∨ −
(cid:0) p (cid:1) (S 1)−A−1exp( cT),
≃≃ ∨ −
while if T < S and T(T S)= TS 1 (this forces S >1), then
∨ ≥
(T S) A 1exp c T(T S) = S A 1exp c√TS (S 1) A 1exp c√TS .
− − − − − −
∨ − ∨ − ≃ ∨ −
The proof is finish(cid:0)ed. p (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:3)
We are now in a position to prove qualitatively sharp estimates of the potential kernel.
Theorem 2.4. For σ > 0 we have
x y 2σ d, σ < d/2
−
k − k
σ(x,y) exp c x y ( x + y ) 1+log+ 1 , σ = d/2 ,
K ≃≃ − k − k k k k k × x y ( x + y )
(1+ x+ky−)kdk2σk,k k σ > d/2
(cid:0) (cid:1) −
k k
uniformly in x,y Rd.
∈
Proof. We decompose
1
Γ(σ) σ(x,y) = G (x,y)tσ 1dt+ ∞G (x,y)tσ 1dt σ(x,y)+ σ(x,y).
K Z0 t − Z1 t − ≡ J0 J∞
For 0 < t < 1 we have tanht t, cotht t 1, sinh2t t, and therefore
−
≃ ≃ ≃
σ(x,y) E c x y 2,c x+y 2 .
J0 ≃≃ σ−d/2−1 k − k k k
This combined with Lemma 2.3 shows that the e(cid:0)stimates from the sta(cid:1)tement hold with σ(x,y)
K
replaced by σ(x,y). Further, taking into account that tanht 1 cotht for t > 1, we see that
J0 ≃ ≃
σ(x,y) exp c( x 2 + y 2) .
J∞ ≃≃ − k k k k
Thus J0σ(x,y) dominates J∞σ(x,y) in the above(cid:0)decomposition, in(cid:1)the sense that
σ(x,y) . E c x y 2,c x+y 2
σ d/2 1
J∞ − − k − k k k
for a sufficiently small constant c > 0. The conc(cid:0)lusion follows. (cid:1) (cid:3)
3. Sharpness of the Lp-Lq boundedness of the potential operator
Given 0 < σ < d/2, define the region
1 1 1 1 2σ 1 1 2σ
R = , : 0 1 and 0 1 +
p q ≤ p ≤ ∨ p − d ≤ q ≤ ∧ p d
(cid:26) (cid:27)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
1 1 1 2σ 1 1 2σ 2σ 2σ
, : 0 1 and = + ,0 , 1,1
p q ≤ p ≤ − d q p d ∪ d − d
(cid:18)(cid:26) (cid:27) (cid:26) (cid:27)(cid:19)
/ (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
contained in the unit (1,1)-square [0,1]2, see Figure 1.
p q
The following result enhances [2, Theorem 8], see also [5, Theorem 2.3].
Theorem 3.1. Let d 1, 0< σ < d/2 and 1 p,q . Then σ: Lp(Rd) Lq(Rd) boundedly
≥ ≤ ≤ ∞ I →
if and only if (1, 1) lies in the region R.
p q
On the other hand, σ is not even of restricted weak type (p,q) when (1,1) / R, except for the
I p q ∈
two cases (1, 1) = (0, 2σ) and (1, 1) = (1,1 2σ) in which weak type (p,q) inequalities hold, and
p q d p q − d
the singular case (1, 1) =(2σ,0), in which the restricted weak type is true, but weak type fails.
p q d
SHARP ESTIMATES OF THE POTENTIAL KERNEL 7
Figure 1. Mapping properties of σ for 0 < σ < d/2.
I
Before giving the proof we take the opportunity to present a short argument showing [2, (21)
and (41)], the result we will apply in a moment.
Lemma 3.2. Given σ > 0,
σ(x, ) (1 x ) 2σ, x Rd.
1 −
kK · k ≃ ∨k k ∈
Proof. Using the identity (see [7, Proposition 3.3])
1
exp( t )1(x) = G (x,y)dy = (cosh2t) d/2exp tanh(2t) x 2 , x Rd,
t −
− H Rd − 2 k k ∈
Z (cid:16) (cid:17)
we may write
1
σ(x,y)dy = ∞ G (x,y)dytσ 1dt
t −
ZRdK Γ(σ) Z0 ZRd
1 1
= ∞(cosh2t) d/2exp tanh(2t) x 2 tσ 1dt.
− −
Γ(σ) − 2 k k
Z0 (cid:16) (cid:17)
Here we split the integration to the intervals (0,1) and (1, ) and denote the resulting integrals
by and , respectively. Then, uniformly in x Rd, ∞
0
J J∞ ∈
1 x 2
exp ct x 2 tσ 1dt = x 2σ k k e ctsσ 1dt x 2σ x 2σ 1
0 − − − − −
J ≃≃ − k k k k ≃ k k k k ∧
Z0 Z0
and (cid:0) (cid:1) (cid:0) (cid:1)
∞e tdexp c x 2 tσ 1dt = C exp c x 2 .
− − d,σ
J∞ ≃≃Z1 − k k − k k
The conclusion follows. (cid:0) (cid:1) (cid:0) (cid:1) (cid:3)
Proof of Theorem 3.1. We first focus on strong type inequalities. Then, in view of [2, Theorem
8], what remains to prove are the following two items.
8 A.NOWAKANDK.STEMPAK
(a) σ is not Lp Lq bounded for 2σ < 1 < 1 and 0< 1 < 1 2σ.
I − d p q p − d
(b) σ is not Lp Lq bounded for 0 < 1 < 1 2σ and 1 + 2σ 1 < 1.
I − p − d p d ≤ q
To justify (a), we fix p and q satisfying the assumed conditions and define
f(y)= χ y 2σ d/q.
y <1 − −
{k k }k k
ThisfunctionisinLp(Rd)since (2σ+d/q)p+d > 0. However, σf / Lq(Rd). Indeed,considering
− I ∈
x such that x < 1 and using the lower bound from Theorem 2.4 we get
k k
σf(x)& x y 2σ d y 2σ d/qdy & x 2σ d y 2σ d/qdy = C x d/q,
− − − − − − −
I k − k k k k k k k k k
Zkyk<kxk/2 Zkyk<kxk/2
and the function x χ x d/q does not belong to Lq(Rd).
x <1 −
Proving (b) we m7→ay a{sksukme}kthakt (1,1) lies on the critical segment 1 = 1 + 2σ, 0< 1 < 1 2σ.
p q q p d p − d
The case when 1 > 1 + 2σ is implicitly contained in what follows, and can also be covered by a
q p d
simplified counterexample not involving the logarithmic factor. Define
f(y)= χ y >e y −d/p log y −1/p−2σ/d.
{k k }k k k k
We have (cid:0) (cid:1)
f(y)pdy = C ∞r 1(logr) 1 2σp/ddr < ,
d − − −
ZRd| | Ze ∞
so f Lp(Rd). We claim that σf / Lq(Rd). Assuming that x > 2e and using the lower bound
∈ I ∈ k k
from Theorem 2.4 we write
σf(x) & x y 2σ−dexp c x y ( x + y ) y −d/p log y −1/p−2σ/ddy
I k − k − k − k k k k k k k k k
Zkxk/2<kyk<kxk
(cid:0) (cid:1) (cid:0) (cid:1)
& x −d/p log x −1/p−2σ/d x y 2σ−dexp 2c x y y dy.
k k k k k − k − k − kk k
Zkxk/2<kyk<kxk
(cid:0) (cid:1) (cid:0) (cid:1)
As we shall see in a moment, the last integral is comparable with x 2σ. Thus
−
k k
σf(x)& x −d/p−2σ log x −1/p−2σ/d = x −d/q log x −1/q, x > 2e,
I k k k k k k k k k k
and the claim follows. (cid:0) (cid:1) (cid:0) (cid:1)
It remains to analyze the last integral, which we denote by . Changing the variable y =
J
x z/ x we get
− k k
= x 2σ z 2σ de 2c z dz,
− − − k k
J k k k k
ZDx
where the set of integration is D = z Rd : x 2/2 < x x z < x 2 . We now observe
x
that D contains the ball B = x R{d :∈ x x k/4k z <k xk2/k4−. Ikndeekd,kif}z B then
x x x
{ ∈ k k k − k k k } ∈
x 2 x x 3 x x 3
k k < k k z x x x x z k k z + x x < x 2.
2 (cid:12) 4 − − 4 k k (cid:12) ≤ k k− ≤ 4 − 4 k k k k
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:12) (cid:12)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:12) (cid:12)
Thus we have (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:12) (cid:12)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:12) (cid:12)
x 2σ z 2σ de 2c z dz x 2σ z 2σ de 2c z dz.
− − − k k − − − k k
k k ZBxk k ≤ J ≤ k k ZRdk k
Clearly, the integral over Rd here is finite. The integral over B depends on x only through x .
x
k k
Since the balls B are increasing in the sense of when x is moved away from the origin along a
x
⊂
fixed line passing through the origin, we see that the integral over B is an increasing function of
x
x , which is positive and finite. We conclude that x 2σ, x > 2e, as desired.
−
k k J ≃ k k k k
We pass to weak type and restricted weak type inequalities. Consider first the four ‘corners’ of
the boundary of R for which the associated strong type inequalities fail. If (1,1) = (1,1 2σ),
p q − d
SHARP ESTIMATES OF THE POTENTIAL KERNEL 9
thentheweak type(1, d )holdsby[5,Theorem2.3]. Noticethatthispropertycanbeexpressed
d 2σ
in terms of Lorentz spa−ces by saying that σ is bounded from L1(Rd) to Ld/(d 2σ), (Rd). Then
− ∞
I
( σ) (the adjoint operator in the Banach space sense) maps boundedly (Ld/(d 2σ), (Rd)) into
∗ − ∞ ∗
I
(L1(Rd)) = L (Rd). Further, the associate space of Ld/(d 2σ), (Rd) in the sense of [1, Chapter
∗ ∞ − ∞
1, Definition 2.3] is Ld/(2σ),1(Rd) (cf. [1, Chapter 4, Theorem4.7]), andby [1, Chapter 1, Theorem
2.9]itcanberegardedasasubspaceofthedualofLd/(d 2σ), (Rd). Since( σ) = σ bysymmetry
− ∞ ∗
I I
of the kernel, we infer that σ is of restricted weak type ( d , ). On the other hand, weak type
I 2σ ∞
( d , ) coincides, by definition, with the strong type, so σ is not of weak type ( d , ) in view
2σ ∞ I 2σ ∞
of the strong type results we already know. This clarifies the situations related to the ‘corners’
(1,1 2σ) and (2σ,0).
− d d
Taking into account (1,1) = (0, 2σ), we will show that σ is of weak type ( , d ). To do that,
p q d I ∞ 2σ
it is enough to verify the estimate
d/(2σ)
f
(7) x Rd : σf(x) > λ . k k∞ , λ > 0, f L∞(Rd).
∈ |I | λ ∈
(cid:18) (cid:19)
(cid:12)(cid:8) (cid:9)(cid:12)
But this is im(cid:12)mediate in view of the b(cid:12)ound, see Lemma 3.2,
σ(x, ) C x 2σ, x Rd,
1 −
kK · k ≤ k k ∈
since then it follows that σf(x) C x 2σ f and consequently
−
|I | ≤ k k k k∞
1/2σ
f
x Rd : σf(x) > λ x Rd : x < Ck k∞ .
∈ |I | ⊂ ∈ k k λ
(cid:26) (cid:18) (cid:19) (cid:27)
(cid:8) (cid:9)
This inclusion leads directly to (7). Finally, in case of the remaining ‘corner’ (1,1) = (1 2σ,1),
p q − d
we use an au contraire argument. If σ were of restricted weak type ( d ,1), then by the
I d 2σ
Marcinkiewicz interpolation theorem for Lorentz spaces (cf. [1, Chapter 4−, Theorem 4.13]) we
would have strong type (p,q) for all points (1, 1) lying inside the segment S with endpoints
p q
(0, 2σ) and (1 2σ,1), a contradiction with (b) above.
d − d
For all the points (1,1) lying inside the segment S, as well as those lying outside the closure
p q
of R, we apply again the interpolation argument with suitably chosen endpoints (see Figure 1).
This shows that for those points (1,1) the operator σ is not of restricted weak type (p,q).
p q I
The proof is finished. (cid:3)
For completeness, we remark that in the case σ > d/2 the operator σ is bounded from Lp(Rd)
to Lq(Rd) for every 1 p,q , see [5, Theorem 2.3]. The behaviorIof σ in the limiting case
≤ ≤ ∞ I
σ = d/2 is described by the theorem below. This result enhances [5, Theorem 2.3] when σ = d/2.
Theorem 3.3. Let d 1 and 1 p,q . Then d/2 is bounded from Lp(Rd) to Lq(Rd) except
≥ ≤ ≤ ∞ I
for (p,q) = ( ,1) and (p,q) = (1, ). Considering the two singular cases, we have:
∞ ∞
(i) d/2 is of weak type ( ,1), but not of strong type ( ,1);
I ∞ ∞
(ii) d/2 is not of restricted weak type (1, ).
I ∞
Proof. The Lp-Lq boundedness is contained in [5, Theorem 2.3]. To show (i), we observe that the
weak type ( ,1) holds true since the proof of (7) covers also the case σ = d/2. The strong type
∞
( ,1) fails because d/21 / L1(Rd), as easily seen by means of Lemma 3.2.
∞ I ∈
It remains to verify (ii). For 0< ε < 1/e, let f (x)= χ . By the lower bound of Theorem
ε x <ε
{k k }
2.4 it follows that
1
d/2f (x)& log dy, x < 1/e,
ε
I x y ( x + y ) k k
Zkyk<ε k − k k k k k
10 A.NOWAKANDK.STEMPAK
uniformly in ε < 1/e. Therefore,
ε 1
d/2f & log y dy = C rd 1logrdr & εdlog , 0 < ε< 1/e,
ε d −
I − k k − ε
(cid:13) (cid:13)∞ Zkyk<ε Z0
and we(cid:13)conclu(cid:13)de that
d/2f 1
ε
kI k∞ & log , 0< ε < 1/e.
f ε
ε 1
k k
Letting ε 0+, we see that d/2 is not of restricted weak type (1, ). (cid:3)
→ I ∞
References
[1] C. Bennett and M. Sharpley,Interpolation of operators, Academic Press Professional, Inc., San Diego, 1987.
[2] B. Bongioanni and J. L.Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. IndianAcad.Sci.
Math. Sci. 116 (2006), 337–360.
[3] A.NowakandK.Stempak,Riesztransforms andconjugacyforLaguerrefunctionexpansions ofHermitetype,
J. Funct.Anal. 244 (2007), 399–443.
[4] A.NowakandK.Stempak,Riesztransformsformulti-dimensionalLaguerre functionexpansions,Adv.Math.
215 (2007), 642–678.
[5] A. Nowak and K.Stempak, Negative powers of Laguerre operators, Canad. J. Math. (in press).
http://dx.doi.org/10.4153/CJM-2011-040-7
[6] K. Stempak and J. L. Torrea, Poisson integrals and Riesz transforms for Hermite function expansions with
weights, J. Funct.Anal. 202 (2003), 443–472.
[7] K. Stempak and J. L. Torrea, BMO results for operators associated to Hermite expansions, Illinois J. Math.
49 (2005), 1111–1131.
Adam Nowak,
Instytut Matematyczny, Polska Akademia Nauk,
S´niadeckich 8, 00–956 Warszawa, Poland
E-mail address: [email protected]
Krzysztof Stempak,
Instytut Matematyki i Informatyki, Politechnika Wrocl awska,
Wyb. Wyspian´skiego 27, 50–370 Wrocl aw, Poland,
and
Katedra Matematyki i Zastosowan´ Informatyki, Politechnika Opolska,
Mikol ajczyka 5, 45–271 Opole, Poland
E-mail address: [email protected]
