The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the $(p,q)$-dual curvature measures when $p>0$ and $q<0$. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the $(p,q)$-mixed volume.

Let A1, A2 be (not necessarily unital or closed) standard operator algebras on locally convex spaces X1, X2, respectively. For k 2, define different kinds of products T1 Tk on elements in Ai, which covers the usual product T1 Tk= T1 Tk, and the Jordan triple product T1 T2= T2T1T2. Let : A1 A2 be a (not necessarily linear) map satisfying that ( (A1) (Ak))= (A1 Ak) whenever any one of Ais is of rank zero or one. It is shown that if the range of contains all rank one and rank two operators then it must be a Jordan isomorphism multiplied by a root of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.

We consider a multidimensional version of an inequality due to Leray as a substitute for Hardy’s inequality in the case $p = n ≥ 2$. In this paper we provide an optimal Sobolev-type improvement of this substitute, analogous to the corresponding improvements obtained for $p = 2 < n$ in S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (1) (2002) 186–233, and for $p > n ≥ 1$ in G. Psaradakis, An optimal Hardy-Morrey inequality, Calc. Var. Partial Differential Equations 45 (3-4) (2012) 421–441.

The geometric descriptions of the (essential) spectra of Toeplitz operators with piecewise continuous symbols are among the most beautiful results about Toeplitz operators on Hardy spaces Hp with 1<p<∞. In the Hardy space H1, the essential spectra of Toeplitz operators are known for continuous symbols and symbols in the Douglas algebra C+H∞. It is natural to ask whether the theory for piecewise continuous symbols can also be extended to H1. We answer this question in the negative and show in particular that the Toeplitz operator is never bounded on H1 if its symbol has a jump discontinuity.

It is proved that the natural embedding of a separable Banach space$X$into the corresponding Bourgain–Pisier space extends $\mathcal{L}_{\infty}$ -valued operators.