VARIABLE LEBESGUE ALGEBRA ON A LOCALLY COMPACT GROUP

Abstract

For a locally compact group 𝐻 with a left Haar measure, we study the variable Lebesgue algebra L^(p(.))(𝐻) with respect to convolution. We show that if L^(p(.))(𝐻) has a bounded exponent, then it contains a left approximate identity. We also prove a necessary and sufficient condition for L^(p(.))(𝐻) to have an identity. We observe that a closed linear subspace of L^(p(.))(𝐻) is a left ideal if and only if it is left translation invariant.

Keywords

• variable lebesgue space
• bounded exponent
• approximate identity
• haar measure