## Differential and Integral Equations

### The distance to $L^\infty$ in some function spaces and applications

#### Abstract

$\scriptstyle L^\infty$ is not dense in some function spaces like: the space $\scriptstyle EXP$ of exponentially integrable functions; the Marcinkiewicz space $\scriptstyle L^{q,\infty}=\text{weak-}L^q$; the Orlicz space $\scriptstyle L^A$ when the convex continuously increasing function $\scriptstyle A$, does not satisfy the so-called $\scriptstyle \Delta _2$-condition. We find formulas for the distance to $\scriptstyle L^{\infty}$ in these spaces. Using the simple observation that if a bounded linear operator $\scriptstyle T:L^q\to W$ satisfies $\scriptstyle T(L^{\infty})\subset L^{\infty}$, then $\scriptstyle \text{dist}_W(Tf,L^{\infty})=0,\quad \forall f\in L^q,$ we give some applications of previous results (see Section 5) to integrability properties of Riesz potential and of solutions to linear elliptic equations.

#### Article information

Source
Differential Integral Equations, Volume 10, Number 4 (1997), 599-607.

Dates
First available in Project Euclid: 1 May 2013

Carozza, Menita; Sbordone, Carlo. The distance to $L^\infty$ in some function spaces and applications. Differential Integral Equations 10 (1997), no. 4, 599--607. https://projecteuclid.org/euclid.die/1367438633