The distance to \(L^{\infty }\) in some function spaces and applications. (English) Zbl 0889.35027

Summary: \(L^{\infty }\) is not dense in some function spaces like: the space \(EXP\) of exponentially integrable functions; the Marcinkiewicz space \(L^{q,\infty } = \text{weak-}L^q\); the Orlicz space \(L^A\) when the convex continuously increasing function \(A\) does not satisfy the so-called \(\Delta _2\)-condition. We find formulas for the distance to \(L^{\infty }\) in these spaces. Using the simple observation that if a bounded linear operator \(T:L^q\rightarrow W\) satisfies \(T(L^{\infty })\subset L^{\infty }\), then dist\(_W(Tf,L^{\infty })=0\), \(\forall f \in L^q\), we give some applications of previous results to integrability properties of the Riesz potential and of solutions to linear elliptic equations.


35J25 Boundary value problems for second-order elliptic equations
47G30 Pseudodifferential operators