Let A be a general expansive matrix and X a ball quasi-Banach function space on ${\mathbb{R}}^n$, which supports both a Fefferman–Stein vector-valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_X^A({\mathbb{R}}^n)$, associated with both A and X, via the nontangential grand maximal function, and then establish its various equivalent characterizations, respectively, in terms of radial and nontangential maximal functions, (finite) atoms, and molecules. As an application, the authors obtain the boundedness of anisotropic Calderón–Zygmund operators from $H_X^A({\mathbb{R}}^n)$ to X or to $H_X^A({\mathbb{R}}^n)$ itself via first establishing some boundedness criteria of linear operators on $H_X^A({\mathbb{R}}^n)$. All these results have a wide range of generality and, particularly, even when they are applied to the Morrey space and the Orlicz-slice space, the obtained results are also new. The novelties of this article exist in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of quasi-norm $\Vert \cdot {\Vert }_X$, the authors embed X into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of the anisotropic weighted Lebesgue space.

Let X be an irreducible $2n$-dimensional holomorphic symplectic manifold. A reflexive sheaf F is very modular if its Azumaya algebra $\mathcal{E}nd(F)$ deforms with X to every Kähler deformation of X. We show that if F is a slope-stable reflexive sheaf of positive rank and the obstruction map $H{H}^2(X)\to {\mathrm{Ext}}^2(F,F)$ has rank 1, then F is very modular. We associate to such a sheaf a vector in the Looijenga–Lunts–Verbitsky lattice of rank $b_2(X)+2$. Three sources of examples of such modular sheaves emerge. The first source consists of slope-stable reflexive sheaves F of positive rank that are isomorphic to the image $\mathrm{\Phi }({\mathcal{O}}_X)$ of the structure sheaf via an equivalence $\mathrm{\Phi }:D^b(X)\to D^b(Y)$ of the derived categories of two irreducible holomorphic symplectic manifolds. The second source consists of such F, which are isomorphic to the image of a skyscraper sheaf via a derived equivalence. The third source consists of images $\mathrm{\Phi }(L)$ of torsion sheaves L supported as line bundles on holomorphic lagrangian submanifolds Z such that Z deforms with X in codimension 1 in moduli, and L is a rational power of the canonical line bundle of Z.

An example of the first source is constructed using a stable and rigid vector bundle G on a $K3$ surface X to get the very modular vector bundle F on the Hilbert scheme $X^{[n]}$ associated to the equivariant vector bundle $G⊠\cdots ⊠G$ on $X^n$ via the Bridgeland–King–Reid (BKR) correspondence. This builds upon and partially generalizes results of O’Grady for $n=2$. A construction of the second source associates to a set ${\{G_i\}}_{i=1}^n$ of n distinct stable vector bundles in the same two-dimensional moduli space of vector bundles on a $K3$ surface X the very modular vector bundle F on $X^{[n]}$ corresponding to the equivariant bundle ${\oplus }_{\mathit{\sigma }\in {\mathfrak{S}}_n}[G_{\mathit{\sigma }(1)}⊠\cdots ⊠G_{\mathit{\sigma }(n)}]$ on $X^n$.