Abstract:
Sufficient conditions
on the kernel
and the grandizer
that ensure the boundedness of integral operators
with homogeneous kernels
in grand Lebesgue spaces on $\mathbb R^n$
as well as an upper bound for their norms
are obtained.
For
some classes of grandizers,
necessary conditions
and lower bounds
for the norm of these operators
are also obtained.
In the case of a radial kernel,
stronger estimates
are established in terms of one-dimensional
grand norms of spherical means of the function.
A sufficient condition for the boundedness of the operator
with homogeneous kernel
in classical Lebesgue spaces
with arbitrary radial weight
is obtained.
As an application,
boundedness in grand spaces of the one-dimensional operator of fractional
Riemann–Liouville integration
and of a multidimensional Hilbert-type operator
is studied.
Keywords:integral operator with homogeneous kernel, grand Lebesgue space,
two-sided estimate, spherical mean, Hilbert-type operator,
fractional integration operator.
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