We build a new class of
Banach function spaces, whose
function norm is
where
ρp(x) denotes the norm of the Lebesgue space of exponent
p(x) (assumed measurable and possibly infinite), constant with respect to the variable of
f, and
δ is measurable, too. Such class contains some known Banach
spaces of
functions, among which are the classical and the small Lebesgue
spaces, and the Orlicz space
L(logL)α,
α>0.
Furthermore we prove the following Hölder-type inequality
where
ρp[⋅]),δ[⋅](f) is the norm of fully measurable grand Lebesgue
spaces introduced by Anatriello and Fiorenza in
[2]. For suitable choices of
p(x) and
δ(x) it reduces to the classical Hölder's inequality for the
spaces EXP1/α and
L(logL)α,
α>0.