Endpoint estimates for certain commutators of fractional and singular integrals. (English) Zbl 1018.42010
In this paper, the authors obtain the endpoint estimates for the following non-standard commutator defined by
\[
T_\alpha^Af(x)=\int_{\mathbb R^n}\frac {\Omega(x-y)}{|x-y|^{n-\alpha+m-1}} R_m(A;x,y)f(y) dy
\]
and its variant
\[
\bar T_\alpha^Af(x)=\int_{\mathbb R^n}\frac {\Omega(x-y)}{|x-y|^{n-\alpha+m-1}} Q_m(A;x,y)f(y) dy,
\]
where \(0\leq\alpha <n\), \(\Omega\in \text{ Lip}_1(S^{n-1})\) is homogeneous of degree zero, \(m\in{\mathbb N}\), \(A\) has derivatives of order \(m-1\) in \(\text{ BMO}(\mathbb R^n)\),
\[
R_m(A;x,y)=A(x)-\sum_{|\gamma|<m}\frac 1{\gamma!}D^\gamma A(y)(x-y)^\gamma,
\]
and
\[
Q_m(A;x,y)=R_{m-1}(A;x,y)-\sum_{|\gamma|=m-1}\frac 1{\gamma!}D^\gamma A(x) (x-y)^\gamma.
\]
If \(m=1\), \(T^A_\alpha\) degenerates into the classical commutator of the fractional or singular integral with the BMO function \(A\). The results of the authors indicate that \(T^A_\alpha\) and \(\bar T^A_\alpha\) have better properties when \(m\geq 2\) than the classical commutator. Moreover, the authors also show that these operators are not bounded in certain cases.
Reviewer: Dachun Yang (Beijing)
MSC:
42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
47B47 | Commutators, derivations, elementary operators, etc. |
47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |
42B30 | \(H^p\)-spaces |
42B35 | Function spaces arising in harmonic analysis |
47B38 | Linear operators on function spaces (general) |
Keywords:
commutator; Hardy space; BMO; atom; fractional integral; endpoint estimates; singular integralReferences:
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