Let T be an operator defined on the space of locally integrable functions on
the real line with the following properties: the values of T are continuous functions
on the real line, T is sublinear and commutes with the translations, and T is
semilocal. Assume that X is a measure space which is totally -finite and
t U is a
one-parameter group of measure-preserving transformations of X . Let us also
assume that for every measurable function f on X the function
( ) t f U x
is
measurable in the product of X with the real line. Given a function f on X let
F(t, x) f (Ut x)
. If f is the sum of two functions which are bounded and
integrable, respectively, then
F(t, x)
is locally integrable function of t for almost
all x and thus G(t, x) T(F(t, x)) is a well-defined continuous function of t for
almost all x . Define
# T f G(0, x)
. Let
( ) n S
and
( ) n K
be the sequences of
operators with the above properties and define
sup | | n n Sf S f
and
sup | | n n Kf K f
. Let
# # sup | | n n S f S f
and
# # sup | | n n K f K f
. We
prove that if there exists a constant C 0 such that p p
Sf C Kf
for all
( ) p f L (ℝ),
1 p
, then
# #
p p
S f C K f
for all
( ) p f L X
,
1 p
Mathematics Subject Classification: 47A35, 28D05, 47A64.

Keywords :Translation Invariant Operator, Calderón Transfer Principle, Operator
Inequality.