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Register a new userFunction spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs
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We develop a functional framework suitable for the treatment of partial differential equations and variational problems posed on evolving families of Banach spaces. We propose a definition for the weak time derivative which does not rely on the availability of an inner product or Hilbertian structure and explore conditions under which the spaces of weakly differentiable functions (with values in an evolving Banach space) relate to the classical Sobolev--Bochner spaces. An Aubin--Lions compactness result in this setting is also proved. We then analyse several concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev--Bochner spaces. We conclude with the formulation and proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising in particular the evolutionary $p$-Laplace equation on a moving domain or surface) and identify some additional evolutionary problems that can be appropriately formulated with the abstract setting developed in this work.
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Let (X j , d j , $mu$ j), j = 0, 1,. .. , m be metric measure spaces. Given 0 < p $kappa$ $le$ $infty$ for $kappa$ = 1,. .. , m and an analytic family of multilinear operators T z : L p 1 (X 1) x $bullet$ $bullet$ $bullet$ L p m (X m) $rightarrow$ L 1 loc (X 0), for z in the complex unit strip, we prove a theorem in the spirit of Steins complex interpolation for analytic families. Analyticity and our admissibility condition are defined in the weak (integral) sense and relax the pointwise definitions given in [9]. Continuous functions with compact support are natural dense subspaces of Lebesgue spaces over metric measure spaces and we assume the operators T z are initially defined on them. Our main lemma concerns the approximation of continuous functions with compact support by similar functions that depend analytically in an auxiliary parameter z. An application of the main theorem concerning bilinear estimates for Schr{o}dinger operators on L p is included.
This paper is devoted to the study of the relatively compact sets in Quasi-Banach function spaces, providing an important improvement of the known results. As an application, we take the final step in establishing a relative compactness criteria for function spaces with any weight without any assumption.
In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator $T$ are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying similar growth conditions. The main results extend the results of Xia and Zheng to the case of the Bergman space when $1 < p < infty$, and in the weighted Bargmann-Fock space setting, our results provide new, more general conditions that imply the work of Xia and Zheng via a more familiar approach that can also handle the $1 < p < infty$ case.
We show that the Gurarij space $mathbb{G}$ has extremely amenable automorphism group. This answers a question of Melleray and Tsankov. We also compute the universal minimal flow of the automorphism group of the Poulsen simplex $mathbb{P}$ and we prove that it consists of the canonical action on $mathbb{P}$ itself. This answers a question of Conley and T{o}rnquist. We show that the pointwise stabilizer of any closed proper face of $mathbb{P}$ is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simplex) is extremely amenable. These results are obtained via several Kechris-Pestov-Todorcevic correspondences, by establishing the approximate Ramsey property for several classes of finite-dimensional Banach spaces and function systems and thei
In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent $pin (0,1]$. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic partial differential equations.
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