The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its zero-density subseries converge. A series ${\sum }_n{x}_n$ in a topological vector space X is called ℒ-convergent if each of its lacunary subseries ${\sum }_k{x}_{n_k}$ (i.e. those with $n_{k+1}-n_k\to \infty$) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every ℒ-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence...

In this article we consider a theory of vector valued strongly singular operators. Our results include Lp, Hp and BMO continuity results. Moreover, as is well known, vector valued estimates are closely related to weighted norm inequalities. These results are developed in the first four sections of our paper. In section 5 we use our vector valued singular integrals to estimate the corresponding maximal operators. Finally in section 6 we discuss applications to weighted norm inequalities for pseudo-differential...

There exists an absolute constant $c_0$ such that for any n-dimensional Banach space E there exists a k-dimensional subspace F ⊂ E with k≤ n/2 such that $in{f}_{ellipsoid\epsilon \subset B_E}{(vol\left(B_E\right)/vol\left(\epsilon \right))}^{1/n}\le c_{0}in{f}_{zonoidZ\subset B_F}{(vol\left(B_F\right)/vol\left(Z\right))}^{1/k}$ . The concept of volume ratio with respect to ${\ell }_p$-spaces is used to prove the following distance estimate for $2\le q\le p<\infty$: $su{p}_{F\subset {\ell }_p,dimF=n}in{f}_{G\subset L_q,dimG=n}d(F,G){\sim }_{c_{pq}}{n}^{(q/2)(1/q-1/p)}$.

The geometry of random projections of centrally symmetric convex bodies in ${ℝ}^N$ is studied. It is shown that if for such a body K the Euclidean ball $B{₂}^N$ is the ellipsoid of minimal volume containing it and a random n-dimensional projection $B=P_H\left(K\right)$ is “far” from $P_H\left(B{₂}^N\right)$ then the (random) body B is as “rigid” as its “distance” to $P_H\left(B{₂}^N\right)$ permits. The result holds for the full range of dimensions 1 ≤ n ≤ λN, for arbitrary λ ∈ (0,1).

We observe that a separable Banach space $X$ is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if $ℒ(X,Y)$ is not reflexive for reflexive $X$ and $Y$ then $ℒ(X_1,Y)$ is is not reflexive for some $X_1\subset X$, $X_1$ having a basis.

It is shown that an orthonormal wavelet basis for $L^2\left(ℝ\right)$ associated with a multiresolution is an unconditional basis for $L^p\left(ℝ\right)$, 1 < p < ∞, provided the father wavelet is bounded and decays sufficiently rapidly at infinity.

We extend the classical theory of the continuous and discrete wavelet transform to functions with values in UMD spaces. As a by-product we obtain equivalent norms on Bochner spaces in terms of g-functions.

Let X be a Banach space. The property (∗) “the unit ball of X belongs to Baire(X, weak)” holds whenever the unit ball of X* is weak*-separable; on the other hand, it is also known that the validity of (∗) ensures that X* is weak*-separable. In this paper we use suitable renormings of ${\ell }^{\infty }\left(\mathbb{N}\right)$ and the Johnson-Lindenstrauss spaces to show that (∗) lies strictly between the weak*-separability of X* and that of its unit ball. As an application, we provide a negative answer to a question raised by K. Musiał....