On differentiation of integrals with respect to bases of convex sets.

A. Stokolos

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Displaying similar documents to “On differentiation of integrals with respect to bases of convex sets.”

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We prove that if 0 < p < 1 then a normalized unconditional basis of a complemented subspace of $c_{0} (l_{p})$ must be equivalent to a permutation of a subset of the canonical unit vector basis of $c_{0} (l_{p})$ . In particular, $c_{0} (l_{p})$ has unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss, and Tzafriri have previously proved the same result for $c_{0} (l ₁)$ .

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It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^{ξ}$ generated by subsequences $(e_{l_{n}}^{ξ})$ and $(e_{m_{n}}^{ξ})$ , respectively, of the natural Schauder basis $(e_{n}^{ξ})$ of $X^{ξ}$ are isomorphic if and only if $(e_{l_{n}}^{ξ})$ and $(e_{m_{n}}^{ξ})$ are equivalent. Further, $X^{ξ}$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $(e_{n}^{ξ})$ . It is also shown that there exists a complemented subspace spanned by a block basis of $(e_{n}^{ξ})$ , which is not isomorphic to a subspace generated by a subsequence of $(e_{n}^{ζ})$ ,...

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Some usual and unusual properties of the Riemann integral for functions x : [a,b] → X where X is an F-space are investigated. In particular, a continuous integrable $l_{p}$ -valued function (0 < p < 1) with non-differentiable integral function is constructed. For some class of quasi-Banach spaces X it is proved that the set of all X-valued functions with zero derivative is dense in the space of all continuous functions, and for any two continuous functions x and y there is a sequence...

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Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^{p} (ℝ)$ -boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_{m} f (x) = p . v . \int k (x - y) m (x + y) f (y) d y$ where $k (x) = \sum_{j \in ℤ} 2^{j} φ_{j} (2^{j} x)$ for a family of functions ${φ_{j}}_{j \in ℤ}$ satisfying conditions (1.1)-(1.3) given below.

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