Commutators of parabolic fractional integrals with variable kernels in vanishing generalized variable Morrey spaces
Tóm tắt
We obtain the boundedness of parabolic fractional integral operators
$$T_{\Omega ,\alpha }$$
with variable kernels
$$\Omega (\cdot ,\cdot )$$
belonging to
$$L^{\infty }({\mathbb {R}^n}) \times L^{s}({\mathbb {S}}^{n-1}), s>n/(n-\alpha )$$
, and their commutators
$$[b,T_{\Omega ,\alpha }]$$
with BMO functions in variable exponent generalized Morrey spaces
$$M^{p(\cdot ),\varphi }$$
and variable exponent vanishing generalized Morrey spaces
$$\textrm{VM}^{p(\cdot ),\varphi }$$
. We find the sufficient conditions on the pair
$$(\varphi ,\psi )$$
which ensures the boundedness of the operators
$$T_{\Omega ,\alpha }$$
and
$$[b,T_{\Omega ,\alpha }]$$
from
$$M^{p(\cdot ),\varphi }$$
to
$$M^{q(\cdot ),\psi }$$
and from
$$\textrm{VM}^{p(\cdot ),\varphi }$$
to
$$\textrm{VM}^{q(\cdot ),\psi }$$
.
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