ON THE BOUNDEDNESS AND THE NORM OF A CLASS OF INTEGRAL OPERATORS∗
2015-02-10周立
(周立)
Department of Mathematics,Huzhou University,Huzhou 313000,China
E-mail:lfzhou@hutc.zj.cn
ON THE BOUNDEDNESS AND THE NORM OF A CLASS OF INTEGRAL OPERATORS∗
Lifang ZHOU(周立芳)
Department of Mathematics,Huzhou University,Huzhou 313000,China
E-mail:lfzhou@hutc.zj.cn
integral operators;sufcient condition;necessary condition;operator norm; hypergeometric functions
2010 MR Subject Classifcation47B38;47G10
1 Introduction
It is well known that the weighted harmonic Bergman projection
was discussed while studying the Toeplitz operator on harmonic Bergman space,see[8].And see[9]for the Berezin-type transform with λ=0.Therefore,to give the boundedness and the norm of the integral operators,such as Sλand Berezin-type operator Bλdefned above,we need to consider such class integral operators
where a,b,c∈R.
(3)Ta,b,cis bounded on L∞,if and only if
Corollary 1.2Suppose a,b,c and λ∈R,and c=n+a+b.
(i)Let 1<p<∞,a,b,λ be such that
Then we have
(ii)Suppose p=1.If a,b,λ are such that
then we have
If a,b,λ are such that
then we have
(iii)Suppose p=∞,a>0,b>-1.If b≤a-n or b≥a-2,then
If a-n<b<a-2,then
2 Preliminaries
A number of hypergeometric functions will appear throughout.We use the classical notation2F1(α,β;γ;z)to denote
with γ/=0,-1,-2,···,where
And the hypergeometric series in(2.1)converges absolutely for all the value of|z|<1.
We list a few formulas for easy reference(see[26,Chapter II]):
The following integral formulas concerning the hypergeometric functions are signifcant for our main results.
Lemma 2.1([6,Lemma 2.2])For γ∈R and α>-1,we have
Corollary 2.2(Forelli-Rudin estimates,see[3,Lemma 4.4])Let α>-1 and β∈R. Then for any x∈Bn,
where a(x)≈b(x)means that the ratio a(x)/b(x)has a positive fnite limit as|x|→1-.
Lemma 2.3([6,Lemma 2.1])Suppose Reλ>0,Reδ>0 and Re(λ+δ-α-β)>0. Then
Lemma 2.4Let α>0,β>0,γ∈R,and n+α+β-2γ-1>0,we have
ProofUsing Lemma 2.1 in the inner integral of(2.8),we have
Then(2.7)shows the result.
Lemma 2.5([27,Theorem 3.6])Suppose that(X,µ)is a σ-fnite measure space and K(x,y)is a nonnegative measurable function on X×X and T the associated integral operator
Let 1<p<∞and 1/p+1/q=1.If there exist positive constants C and a positive measurable function u on X such that
for almost every x in X,and
for almost every y in X,then T is bounded on Lp(X,µ)with‖T‖≤C.
3 Proof of Theorem 1.1
The proof of Theorem 1.1 will be divided into two steps.
Step IProve(2)and(3)in Theorem 1.1.Let T∗
a,b,cdenote the adjoint operator of Ta,b,c, and
Since
is a fnite number by Lemma 2.1.From(2.6),we see that
which gives case(2).And case(3)can be obtained in the same way as case(2).
Step IIProve(1)in Theorem 1.1.Suppose 1<p<∞,and q is the conjugate number of p such that 1/p+1/q=1 in this step.
we will take
where
With α=b+1+(σ-(λ+1))/p,β=a+σ/q+(λ+1)/p,γ=c/2,applying Lemma 2.4 to the left-hand of last inequality,we frstly get
Then the arbitrariness of σ>0 implies
When c=n+a+b,by(2.8),the inequality(3.2)is to be that
Letting σ→0+in the last inequality,we can see that the limit
is a fnite non-negative real number.Thus,we can conclude that-pa<λ+1<p(b+1)since the limit of the denominator,
is a fnite non-negative real number under the condition of(3.3).
Using Lemma 2.1 and(2.3),we frst calculate the integral
where
Thus,applying(2.4),we have
Similar argument gives
4 Proof of Corollary 1.2
whose monotonicity implies the result of(ii)in Corollary 1.2.And the same line as(ii)gives (iii)in Corollary 1.2.?
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∗Received May 20,2014;revised November 27,2014.Supported by the National Natural Science Foundation of China(11426104,11271124,11201141,11301136,and 61473332),Natural Science Foundation of Zhejiang province(LQ13A010005,LY15A010014)and Teachers Project of Huzhou University(RP21028).
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