Fix an integer n >= 2, an exponent 1 < p < infinity, and a domain ohm subset of R (n) . Let ohm & lowast; , ohm \ {x}, where x is an element of ohm. Under some further conditions, we construct optimal Hardy-weights for the Finsler p-Dirichlet integral Q(0)[phi; ohm(& lowast;) ] sic , integral(ohm & lowast;) H(x, del phi) (p) dx on C (infinity)(c) (ohm & lowast; ), and the Finsler p-Dirichlet integral with a potential Q(V) [phi; ohm] sic , ohm (H(x, del phi) (p )+ V |phi| (p) ) dx on C (infinity)(c) (ohm), where H(x, & centerdot;) is a family of norms on R n parameterized by x is an element of ohm & lowast; or x is an element of ohm, respectively, and the potential V lies in a subspace M-<^>q (loc)(p; ohm) of a local Morrey space M-loc(q)(p; ohm)
We show almost sure wellposedness of mild solution to the cubic nonlinear wave equation in a weighted Besov space over R2. To achieve this, we show that any weak limit of phi 4 measures on increasing tori is invariant under the equation. We review and slightly simplify the periodic theory and the construction of the weak limit measure, and then use finite speed of propagation to reduce the infinite-volume case to the previous setup. Our argument also gives a weaker invariance result on the nonlinear Schro & uml;dinger equation in the same setting.
We introduce the property (W-infinity) and the weakly infinity-compact approximation property (WICAP) of a Banach space X. We establish a characterization of the property (W-infinity) and relationships of the property (W-infinity), the approximate identities for the algebra W-infinity(X) and the WICAP. As a consequence, we obtain that both & ell;(p)(1 < p < infinity) and c(0) fail property (W-infinity). It is also shown that the WICAP is strictly stronger than the weakly compact approximation property.
We identify isometric isomorphisms of the space of Kurzweil-Henstock integrable functions as bi-absolutely-continuous changes of variable.
For every pair of non-constant holomorphic maps beta i : Si -+ S0 between noncompact Riemann surfaces, where i E {1, 2}, there exists an associated fiber product S1 & times;(beta 1,beta 2)S2 that has the structure of a singular Riemann surface, endowed with a canonical map beta to S0 satisfying beta io pi i = beta, where pi i is coordinate projection onto Si. This paper explores the relationship between the space of ends of this fiber product and the space of ends of its normal fiber product. In addition, we establish conditions on the maps beta 1 and beta 2 that ensure connectivity in the fiber product. Upon examination of these conditions, we establish a connection between the space of ends of the fiber product and the topological characteristics of the Riemann surfaces S1 and S2. Finally, we investigate the fiber product of infinite superelliptic curves by analyzing its connectedness and the space of ends.
