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1997 The distance to $L^\infty$ in some function spaces and applications
Menita Carozza, Carlo Sbordone
Differential Integral Equations 10(4): 599-607 (1997).

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.

Citation

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Menita Carozza. Carlo Sbordone. "The distance to $L^\infty$ in some function spaces and applications." Differential Integral Equations 10 (4) 599 - 607, 1997.

Information

Published: 1997
First available in Project Euclid: 1 May 2013

zbMATH: 0889.35027
MathSciNet: MR1741764

Subjects:
Primary: 46E39
Secondary: 35J15, 46N20, 47B38

Rights: Copyright © 1997 Khayyam Publishing, Inc.

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Vol.10 • No. 4 • 1997
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