We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
$\boldsymbol {L}^{\boldsymbol {2}}$ ESTIMATE OF COMMUTATORS OF BOCHNER–RIESZ OPERATORS FOR HERMITE OPERATOR
Let H be the Hermite operator 
$-\Delta +|x|^2$ on 
$\mathbb {R}^n$. We prove a weighted 
$L^2$ estimate of the maximal commutator operator 
$\sup _{R>0}|[b, S_R^\lambda (H)](f)|$, where 
$ [b, S_R^\lambda (H)](f) = bS_R^\lambda (H) f - S_R^\lambda (H)(bf) $ is the commutator of a BMO function b and the Bochner–Riesz means 
$S_R^\lambda (H)$ for the Hermite operator H. As an application, we obtain the almost everywhere convergence of 
$[b, S_R^\lambda (H)](f)$ for large 
$\lambda $ and 
$f\in L^p(\mathbb {R}^n)$.
We present new estimates in the setting of weighted Lorentz spaces of operators satisfying a limited Rubio de Francia condition; namely $T$
is bounded on $L^{p}(v)$
for every $v$
in an strictly smaller class of weights than the Muckenhoupt class $A_p$
. Important examples will be the Bochner–Riesz operators $BR_\lambda$
with $0<\lambda <{(n-1)}/2$
, sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with $\Omega \in L^{r}(\Sigma )$
, $1 < r < \infty$
.
