## Structure of Cesàro function spaces.(English)Zbl 1200.46027

Let $$1\leq p \leq \infty$$. The Cesàro sequence space $$\text{ces}_p$$ is the set of real sequences $$x = \{x_k\}$$ such that
$\|x\|_{c(p)} = \left[\sum_{n=1}^\infty\left(\frac{1}{n} \sum_{k=1}^n|x_k|\right)^p\right]^{1/p} < \infty,\quad 1\leq p < \infty,$
and
$\|x\|_{c(\infty)} = \sup_{n\in\mathbb{N}}\frac{1}{n} \sum_{n=1}^n |x_k| < \infty, \quad p=\infty.$
The Cesàro function spaces Ces$$_p = \text{Ces}_p(I)$$ are classes of Lebesgue measurable real functions $$f$$ on $$I =[0,1]$$ or $$I=[0,\infty)$$ such that
$\|f\|_{c(p)} = \left[\int_I\left(\frac{1}{x} \int_{0}^x|f(t)|\,dt\right)^p \,dx\right]^{1/p} < \infty,\quad 1\leq p < \infty,$
and
$\|f\|_{c(\infty)} = \sup_{x\in I, x>0}\frac{1}{x} \int_{0}^x |f(t)|\,dt < \infty, \quad p=\infty.$
The authors investigate several isomorphic structure properties of Cesàro sequence and function spaces. They find their dual spaces up to equivalence of norm, with different description for $$[0,1]$$ and $$[0,\infty)$$. They prove that Ces$$_p$$, $$1<p<\infty$$, is strictly convex but not reflexive. It is also shown that these spaces are not isomorphic to any space $$L^q$$ with $$1\leq q\leq \infty$$, and do not enjoy the Dunford-Pettis property while they have the weak Banach-Saks property. It is characterized when the Cesàro spaces contain isomorphic and complemented copies of $$\ell^p$$, and a description when Ces$$_p[0,1]$$ contains an isomorphic subspace of $$\ell^q$$ is given. Finally, the authors prove that Ces$$_p[0,1]$$ and Ces$$_p[0,\infty)$$ are isomorphic for $$1<p\leq\infty$$.

### MSC:

 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B20 Geometry and structure of normed linear spaces 46B42 Banach lattices
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