Opposed Exponent Profiles and Real Interpolation of Measurable Subspaces

Abstract

We study real interpolation of the $G$-measurable subspaces $Y_j$ of variable Lebesgue spaces $L^{p_j(\cdot)}$. For a family of continuous exponent pairs that vary in opposite directions on the atoms of $G$, we prove that $K$-closedness is equivalent to a uniform logarithmic bound on the oscillation of the exponents. The same bound characterizes the existence of a common bounded projection and interpolation equality at a single interior parameter. If it fails, then for each $0<θ<1$ there is one function belonging to all ambient interpolation spaces $(X_0,X_1)_{θ,q}$, $0<q\leq\infty$, and to none of the corresponding spaces $(Y_0,Y_1)_{θ,q}$. Each endpoint subspace still admits a bounded positive projection. The exponents can be chosen arbitrarily close to prescribed constants $1<a<b<\infty$, with constant sum and smoothness away from zero. We also give a finite-profile criterion in terms of the endpoint maxima, compute the associated $K$-functional loss, and obtain a counterexample for adapted sequence spaces.