## Differential and Integral Equations

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

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

Menita Carozza and Carlo Sbordone

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367438633

**Mathematical Reviews number (MathSciNet)**

MR1741764

**Zentralblatt MATH identifier**

0889.35027

**Subjects**

Primary: 46E39: Sobolev (and similar kinds of) spaces of functions of discrete variables

Secondary: 35J15: Second-order elliptic equations 46N20: Applications to differential and integral equations 47B38: Operators on function spaces (general)

#### Citation

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