Unified treatment of fractional integral inequalities via linear functionals

In the paper we prove several inequalities involving two isotonic linear functionals. We consider inequalities for functions with variable bounds, for Lipschitz and H\" older type functions etc. These results give us an elegant method for obtaining a number of inequalities for various kinds of fractional integral operators such as for the Riemann-Liouville fractional integral operator, the Hadamard fractional integral operator, fractional hyperqeometric integral and corresponding q-integrals

Paley-Littlewood decomposition for sectorial operators and interpolation spaces

We prove Paley-Littlewood decompositions for the scales of fractional powers of $0$-sectorial operators $A$ on a Banach space which correspond to Triebel-Lizorkin spaces and the scale of Besov spaces if $A$ is the classical Laplace operator on $L^p(\mathbb{R}^n).$We use the $H^\infty$-calculus, spectral multiplier theorems and generalized square functions on Banach spaces and apply our results to Laplace-type operators on manifolds and graphs, Schr\"odinger operators and Hermite expansion.We also give variants of these results for bisectorial operators and for generators of groups with a bounded $H^\infty$-calculus on strips.Comment: 2nd version to appear in Mathematische Nachrichten, Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag, 201

A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces

In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem.Comment: 27 pages, 4 figure