With deep grief we learned about the passing of Albrecht Pietsch.The mathematical community has lost an excellent and far-seeing researcher and dedicated university teacher.Albrecht Pietsch was one of the great German mathematicians of the separated Germany.His books enriched functional analysis and gave many researchers directions and motivation to work on.Pietsch was well known for his fundamental contributions to various fields of functional analysis.The Pietsch factorization theorem has become a highly cited fundamental result of Banach space and operator theory.His work on operator ideals and -numbers made its way into the standard tool box of this field.He was remarkably active right until the last weeks of his life returning to the subject of singular traces and providing striking new conceptual insights into the area.His diploma and PhD students remember him as a strict and determining teacher.But they also knew that if they ran into difficulties, Pietsch would be there to help.He was always ready to discuss problems and perspectives of mathematics in a very frank and sometimes inconvenient way.His students and co-workers knew him as being very authoritative, but when you had the guts to stand your grounds he was also ready to accept it without being annoyed.Another striking side of his character was accuracy.Those who wrote papers with him certainly remember how he fought for clearness and correctness of every line.In his last years, when illness marked him, still continuing mathematical research-he also thought of his own passing from the world.So, he wrote his obituary himself, giving those close colleagues of his last years the mission to publish it.We, Stefan Heinrich, Aicke Hinrichs, Thomas Khn, are in the possession of this very carefully worked out document and we want to pass it on to you.We could not have done it half as precisely.So, we include this text word by word.The latex document starts with a commented out order to the persons handling it:"Grundsatz: Nur belegbare Fakten sind erwnscht, aber KEINE Lobpreisungen." (Only verifiable facts are desired, but NO praise.)We plead guilty of not fulfilling this order, we were not able to.PIETSCH'S TEXT: Albrecht Pietsch (abbreviation: P.) was born on September 13, 1934 in Zittau (Saxony) and died on [

Abstract We study the regularity properties of Hölder continuous minimizers to non‐autonomous functionals satisfying ‐growth conditions, under Besov assumptions on the coefficients. In particular, we are able to prove higher integrability and higher differentiability results for solutions to our minimum problem.

Abstract In this paper, we discuss the boundedness of mixed Journé's class operators on weighted multi‐parameter mixed Hardy spaces via atoms decomposition. Moreover, we give a specific singular integral operator in mixed Journé's class which has better properties.

Abstract Koplienko [Sib. Mat. Zh. 25 (1984), 62–71; English transl. in Siberian Math. J. 25 (1984), 735–743] found a trace formula for perturbations of self‐adjoint operators by operators of Hilbert–Schmidt class . Later, Neidhardt introduced a similar formula in the case of pairs of unitaries via multiplicative path in [Math. Nachr. 138 (1988), 7–25]. In 2012, Potapov and Sukochev [Comm. Math. Phys. 309 (2012), no. 3, 693–702] obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon [Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107, 202; Open Question 11.2]. In this paper, we supply a new proof of the Koplienko trace formula in the case of pairs of contractions , where the initial operator is normal, via linear path by reducing the problem to a finite‐dimensional one as in the proof of Krein's trace formula by Voiculescu [Oper. Theory Adv. Appl. 24 (1987) 329–332] and Sinha and Mohapatra [Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 4, 819–853] and [Integral Equations Operator Theory 24 (1996), no. 3, 285–297]. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self‐adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko–Neidhardt trace formula for a class of pairs of contractions via multiplicative path using the finite‐dimensional approximation method.

Abstract Let denote an isometric immersion of a Kähler manifold with complex dimension into Euclidean space with codimension . We show that generic rank conditions on the second fundamental form of a non‐minimal complete real Kähler submanifold imply that is a cylinder over a real Kähler submanifold .

Abstract We consider the quasilinear diffusion problem of for an open set , , for , and any . Here, denotes an operator which may involve the distributional Riesz fractional gradient of order , with , the classical gradient or/and nonlocal derivatives , with . We show global existence results for various quasilinear diffusion systems in nondivergence form for linear elliptic operators , including classical elliptic systems, anisotropic fractional equations and systems, and anisotropic local and nonlocal operators of the following type: for coercive, invertible matrices and suitable vectorial functions .

Abstract In this paper, we extend the extrapolation theory to mixed Herz spaces and . To prove the main result, we first study the dual spaces of mixed Herz spaces, and then give the boundedness of the Hardy–Littlewood maximal operator on mixed Herz spaces. By using the extrapolation theorems, we obtain the boundedness of many integral operators on mixed Herz spaces. We also give a new characterization of via the boundedness of commutators of some operators on mixed Herz spaces.