M(a, fi) S M'(ah ft)Jf^(a,, ft). This is M. Riesz's fundamental theorem. (See M. Riesz [5] and a different proof in Paley [4] ; see also a generalization of the theorem in L. C. Young [ i l ] . ) M. Riesz's argument proved the convexity only in the triangle O ^ a ^ l , 0 ^ / 3 ^ 1 , a+f3*zl. The extension to the whole quadrant is due to Thorin [9]. We shall not give the proof of the theorem her...

Introduction. The db complex is well defined on any smooth CR manifold M, and once a metric is fixed, so is the complex Laplace-Beltrami operator Db on forms of type (p, q). For compact M without boundary, the bb cohomology of M may be studied via Ub [6], and thus local smoothness of solutions to Dbu = ƒ is important. In its own right, Ub is a prototype of doubly characteristic operators. Under...

In a recent paper, the authors have proved results characterizing convexity-preserving maps deened on a subset of a not-necessarily nite dimensional real vector space as projective maps. The purpose of this note is threefold. First, we state a theorem characterizing continuous, injective, convexity-preserving maps from a relatively open, connected subset of an aane subspace of R m into R n as p...

We consider two different notions of convexity of metric spaces, namely (strict/uniform) ball convexity and (strict/uniform) distance convexity. Our main theorem states that (strict/uniform) distance convexity is preserved under a fairly general product construction, whereas we provide an example which shows that the same does not hold for (strict/uniform) ball convexity, not even when consider...

We investigate the norm of sums of independent vector-valued random variables in noncommutative Lp spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class S q in Sp(lmq ) with optimal order m ∼ n. Using these embeddings we show the surprising fact that the sharp type (cotype) index in the sense of operator spaces for Lp[0, 1] is min(p, p) (max(p, p)). Simi...

We provide finite sample properties of general regularized statistical criteria in the presence pseudo-observations. Under restricted strong convexity assumption unpenalized loss function and regularity conditions on penalty, we derive non-asymptotic error bounds M-estimator. This penalized framework with pseudo-observations is then applied to M-estimation some usual copula-based models. These ...

Problem of defining convexity of a digital region is considered. Definition of DL− (digital line) convexity is proposed, and it is shown to be stronger than the other two definitions, T−(triangle) convexity and L−(line) convexity. In attempt to connect the convexity of digital sets, to the fact that digital set can be a fuzzy set, the notion of convexity of the membership function is introduced...

On a time-oriented Lorentzian manifold (M, g) with non-empty boundary satisfying a convexity assumption, we show that the topological, differentiable, and conformal structure of suitable subsets S ⊂ M of sources is uniquely determined by measurements of the intersection of future light cones from points in S with a fixed open subset of the boundary of M ; here, light rays are reflected at ∂M ac...