In reflexive Banach spaces with some degree of uniform convexity, we obtain estimates for Kottman's separation constant in terms of the corresponding modulus.

It is shown that maximal truncations of nonconvolution L²-bounded singular integral operators with kernels satisfying Hörmander’s condition are weak type (1,1) and $L^p$-bounded for 1 < p< ∞. Under stronger smoothness conditions, such estimates can be obtained using a generalization of Cotlar’s inequality. This inequality is not applicable here and the point of this article is to treat the boundedness of such maximal singular integral operators in an alternative way.

Let W and L be complementary subspaces of a Banach space X and let P(W,L) denote the projection on W along L. We obtain a sufficient condition for a subspace M of X to be complementary to W and we derive estimates for the norm of P(W,L) - P(W,M).

We consider generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form $L^p\left(X\right)\subseteq \gamma \left(X\right)\subseteq L^q\left(X\right)$, in terms of the type p and cotype q of the Banach space X. As an application we prove $L^p$-estimates for vector-valued Littlewood-Paley-Stein g-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions.

We investigate the Fourier transforms of functions in the Sobolev spaces $W_1^{r_1,...,r_n}$. It is proved that for any function $f\in W_1^{r_1,...,r_n}$ the Fourier transform f̂ belongs to the Lorentz space $L^{n/r,1}$, where $r=n{({\sum }_{j=1}^{n}1/r_j)}^{-1}\le n$. Furthermore, we derive from this result that for any mixed derivative $D^{s}f(f\in C_0^{\infty },s=(s_1,...,s_n))$ the weighted norm $\parallel {\left(D^{s}f\right)}^{\wedge }{\parallel }_{L^1\left(w\right)}(w\left(\xi \right)={\left|\xi \right|}^{-n})$ can be estimated by the sum of $L^1$-norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

The purpose of this paper is to continue the investigations on extremal values for inner and outer radii of the unit ball of a finite-dimensional real Banach space for the Holmes-Thompson and Busemann measures. Furthermore, we give a related new characterization of ellipsoids in ${ℝ}^d$ via codimensional cross-section measures.

The position of intermediate spaces for a Banach couple is estimated with the help of its fundamental function and co-function. We study the completeness of the collection of all such functions, and the methods of calculating and estimating them for different couples. Finally, these functions are used to compare the position of spaces obtained under the action of some interpolation functors.

For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index ${\omega }^{\alpha +1}$ which is universal for the class of separable Banach spaces with Szlenk index not exceeding ${\omega }^{\alpha }$. Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.

For each ordinal α < ω₁, we prove the existence of a separable, reflexive Banach space W with a basis so that Sz(W), $Sz(W*)\le {\omega }^{\alpha +1}$ which is universal for the class of separable, reflexive Banach spaces X satisfying Sz(X), $Sz(X*)\le {\omega }^{\alpha }$.