## Real Analysis Exchange

### On the non-compactness of maximal operators.

G. G. Oniani

#### Abstract

It is proved that if $B$ is a convex quasi-density basis and $E$ is a symmetric space on $\mathbb{R}^n$ with respect to Lebesgue measure, then there do not exist non-orthogonal weights $w$ and $v$ for which the maximal operator $M_B$ corresponding to $B$ acts compactly from the weight space $E_w$ to the weight space $E_v$.

#### Article information

Source
Real Anal. Exchange, Volume 28, Number 2 (2002), 439-446.

Dates
First available in Project Euclid: 20 July 2007

https://projecteuclid.org/euclid.rae/1184963806

Mathematical Reviews number (MathSciNet)
MR2009765

Zentralblatt MATH identifier
1080.42506

#### Citation

Oniani, G. G. On the non-compactness of maximal operators. Real Anal. Exchange 28 (2002), no. 2, 439--446. https://projecteuclid.org/euclid.rae/1184963806

#### References

• M. de Guzmán, Differentiation of integrals in $\bR^n$, Springer, 1975.
• S. Krein, Y. Petunin, E. Semenov, Interpolation of linear operators, Moscow, 1978 (Russian).
• D. E. Edmunds and A. Meskhi, On a measure of non-compactness for maximal operators, Math. Nachr., to appear.
• G. Oniani, On the integrability of strong maximal functions corresponding to different frames, Georgian Math. J., 6 (1999), 149–168.