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$1$ AND DOUBLE PHASE FUNCTIONALS
Our aim in this paper is to deal with integrability of maximal functions for Herz–Morrey spaces on the unit ball with variable exponent 
$p_{1}(\cdot )$ approaching 
$1$ and for double phase functionals 
$\unicode[STIX]{x1D6F7}_{d}(x,t)=t^{p_{1}(x)}+a(x)t^{p_{2}}$, where 
$a(x)^{1/p_{2}}$ is nonnegative, bounded and Hölder continuous of order 
$\unicode[STIX]{x1D703}\in (0,1]$ and 
$1/p_{2}=1-\unicode[STIX]{x1D703}/N>0$. We also establish Sobolev type inequality for Riesz potentials on the unit ball.
$p>0$
In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic 
$p>0$. Using my characterization, I show that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion-free. I also show that a conjecture of KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic 
$p>0$ is true for smooth projective surfaces. I give a new proof of a result by Li and prove a refinement of it. Based on my characterization of abelian varieties, I propose modifications of Li’s conjecture, which I expect to be true.
The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme 
$\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter 
$m-2n$ to the category of tensor representations of 
$\text{OSp}(m|2n)$. This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra 
${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of 
$V^{\otimes r}$ is an isomorphism if and only if 
$r<(m+1)(n+1)$.
We describe the irreducible components of the jet schemes with origin in the singular locus of a two-dimensional quasi-ordinary hypersurface singularity. A weighted graph is associated with these components and with their embedding dimensions and their codimensions in the jet schemes of the ambient space. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (equivalent to a divisorial valuation on 
$\mathbb{A}^{3}$), that computes the log-canonical threshold of the singularity embedded in 
$\mathbb{A}^{3}$. This provides us with pairs 
$X\subset \mathbb{A}^{3}$ whose log-canonical thresholds are not computed by monomial divisorial valuations. Note that for a pair 
$C\subset \mathbb{A}^{2}$, where 
$C$ is a plane curve, the log-canonical threshold is always computed by a monomial divisorial valuation (in suitable coordinates of 
$\mathbb{A}^{2}$).
We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is 
$F$-finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama’s description of the restricted base locus to klt or strongly 
$F$-regular varieties over arbitrary fields.