In this paper we connect Calderón and Zygmund’s notion of Lp-differentiability (Calderón and Zygmund, Proc Natl Acad Sci USA 46:1385–1389, 1960) with some recent characterizations of Sobolev spaces via the asymptotics of non-local functionals due to Bourgain, Brezis, and Mironescu (Optimal Control and Partial Differential Equations, pp. 439–455, 2001). We show how the results of the former can be generalized to the setting of the latter, while the latter results can be strengthened in the spirit of the former. As a consequence of these results we give several new characterizations of Sobolev spaces, a novel condition for whether a function of bounded variation is in the Sobolev space W1,1, and complete the proof of a characterization of the Sobolev spaces recently claimed in (Leoni and Spector, J Funct Anal 261:2926–2958, 2011; Leoni and Spector, J Funct Anal 266:1106–1114, 2014).

We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of [CO]. Related <i>A</i>-formulas hold for a transversal choice of chains. Two different computations are provided: a direct calculation using the classification of holomorphic discs by Oh and the author in [CO], and another method by using an <i>analogue of divisor equation</i> in Gromov-Witten invariants to the case of discs. Floer cohomology rings are shown to be isomorphic to Clifford algebras, whose quadratic forms are given by the Hessians of functions <i>W</i>, which turn out to be the superpotentials of Landau-Ginzburg mirrors. In the case of , this proves the prediction made by Hori, Kapustin and Li by B-model calculations via physical arguments. The latter method also provides correspondence between higher derivatives of the superpotential of LG mirror with

Let k be the algebraic closure of a finite field of odd characteristic p, and X a smooth projective scheme over the Witt ring W (k) which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham flow1 and prove that the category of periodic Higgs-de Rham flows over X/W(k) is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the e ́tale fundamental group π_1(X_K ) of the generic fiber of X, after Fontaine-Laffaille and Faltings, where K is the fraction field of W(k). Moreover, we prove that every semistable Higgs bundle over the special fiber X_k of X of rank ≤ p initiates a semistable Higgs–de Rham flow and thus those of rank ≤ p − 1 with trivial Chern classes induce k-representations of π_1(X_K ). A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic p, it was constructed by Ogus–Vologodsky in the nonabelian Hodge theory in positive characteristic; in the affine local case, our construction is related to the local Ogus–Vologodsky correspondence of Shiho.

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