Differential and Integral Equations

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

Menita Carozza and Carlo Sbordone

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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


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