For a hyperbolic second-order differential operator $P$, we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and some algebraic and geometric properties of the principal symbol $p$. If the Hamilton map $F_p$ of $p$ (the linearization of the Hamilton field $H_p$ along double characteristics) has nonzero real eigenvalues at every double characteristic (the so-called effectively hyperbolic case), then it is well known that the Cauchy problem for $P$ is well posed in any Gevrey class $1\le s<+\infty$ for any lower-order term. In this paper we prove that if $p$ is noneffectively hyperbolic and, moreover, such that $Ker{F}_p^2\cap Im{F}_p^2\ne \left\{0\right\}$ on a $C^{\infty }$ double characteristic manifold $\Sigma$ of codimension $3$, assuming that there is no null bicharacteristic landing $\Sigma$ tangentially, then the Cauchy problem for $P$ is well posed in the Gevrey class $1\le s<4$ for any lower-order term (strong Gevrey well-posedness with threshold $4$), extending in particular via energy estimates a previous result of Hörmander in a model case.

Recently a notion of support and a construction of local cohomology functors for [TR5] compactly generated triangulated categories were introduced and studied by Benson, Iyengar, and Krause. Following their idea, we assign to any object of the category a new subset of $Spec\left(R\right)$, again called the (big) support. We study this support and show that it satisfies axioms such as exactness, orthogonality, and separation. Using this support, we study the behavior of the local cohomology functors and show that these triangulated functors respect boundedness. Then we restrict our study to the categories generated by only one compact object. This condition enables us to get some nice results. Our results show that one can get a satisfactory version of the local cohomology theory in the setting of triangulated categories, compatible with the known results for the local cohomology for complexes of modules.

In this paper we construct the action of Ding-Iohara and shuffle algebras on the sum of localized equivariant $K$-groups of Hilbert schemes of points on ${\mathbb{C}}^2$. We show that commutative elements $K_i$ of shuffle algebra act through vertex operators over the positive part $\{{\mathfrak{h}}_i{\}}_{i>0}$ of the Heisenberg algebra in these $K$-groups. Hence we get an action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space $\mathbb{C}[{\mathfrak{h}}_1,{\mathfrak{h}}_2,\dots ]$.

In this paper, we describe the graded canonical module of a Noetherian multisection ring of a normal projective variety. In particular, in the case of the Cox ring, we prove that the graded canonical module is a graded free module of rank one with the shift of degree $K_X$. We give two kinds of proofs. The first one utilizes the equivariant twisted inverse functor developed by the first author. The second proof is down-to-earth, which avoids the twisted inverse functor, but some additional assumptions are required in this proof.

Let $\alpha \ge 0$, and let $X$, $Y$ be Itō processes

$$d{X}_t={\varphi }_{t}d{B}_t+{\psi }_{t}dt,d{Y}_t={\zeta }_{t}d{B}_t+{\xi }_{t}dt$$

such that $\left|X_0\right|\ge \left|Y_0\right|$, $\left|\varphi \right|\ge \left|\zeta \right|$, and $\alpha \psi \ge \left|\xi \right|$. The purpose of the paper is to determine the optimal universal constant $C_{\alpha }$ in the weak-type estimate

$${sup\hspace{0.17em}}_{\lambda }\lambda \mathbb{P}\left({sup\hspace{0.17em}}_t\right|Y_t|\ge \lambda )\le C_{\alpha }{sup\hspace{0.17em}}_t\mathbb{E}\left|X_t\right|.$$

Then the inequality is extended, with unchanged constant, to the more general setting when $X$ is a submartingale and $Y$ is $\alpha$-strongly differentially subordinate to $X$. As an application, a related estimate for subharmonic functions is established. The inequalities generalize and unify the earlier results of Burkholder, Choi, and Hammack for Itō processes, stochastic integrals, and smooth functions on Euclidean domains.

The Riemann-Lebesgue lemma shows that the Vilenkin-Fourier coefficient $\mathrm{f̂}\left(n\right)$ is of $o\left(1\right)$ as $n\to \infty$ for any integrable function $f$ on Vilenkin groups. However, it is known that the Vilenkin-Fourier coefficients of integrable functions can tend to zero as slowly as we wish. The definitive result is due to B. L. Ghodadra for functions of certain classes of generalized bounded fluctuations. We prove that this is a matter only of local fluctuation for functions with the Vilenkin-Fourier series lacunary with small gaps. Our results, as in the case of trigonometric Fourier series, illustrate the interconnection between ‘localness’ of the hypothesis and type of lacunarity and allow us to interpolate the results.