Let $$(Lv)(t)=sum^{n} _{i,j=1} (-1)^{j} d_{j} left( s^{2alpha}(t) b_{ij}(t) mu(t) d_{i}v(t)right),$$ be a non-selfadjoint differential operator on the Hilbert space $L_{2}(Omega)$ with Dirichlet-type boundary conditions. In continuing of papers [10-12], let the conditions made on the operator $ L$ be sufficiently more general than [11] and [12] as defined in Section $1$. In this paper, we estim...

Let $G$ be a locally compact abelian metric group with Haar measure $\lambda $ and $\hat{G}$ its dual $\mu ,$ ( G) is finite. Assume that$~1<p_{i}<\infty $, $p_{i}^{\prime }=\frac{ p_{i}}{p_{i}-1}$, $( i=1,2,3) $\theta \geq 0$. L^{(p_{i}^{\prime },\theta }( small Lebesgue spaces. A bounded measurable function $m( \xi ,\eta ) defined on $\hat{G}\times \hat{G}$ said to bilinear multiplier of type...

For any non-negative integer v we construct explicitly ⌊v2⌋+1 independent covariant bilinear differential operators from Jk,m × Jk′,m′ to Jk+k′+v,m+m′ . As an application we construct a covariant bilinear differential operator mapping S (2) k ×S (2) k′ to S (2) k+k′+v. Here Jk,m denotes the space of Jacobi forms of weight k and index m and S (2) k the space of Siegel modular forms of degree 2 a...

Classically, there are many interesting connections between differential operators and the theory of elliptic modular forms and many interesting results have been explored. In particular, it has been known for some time how to obtain an elliptic modular form from the derivatives ofN elliptic modular forms, which has already been studied in detail by R. Rankin in [9] and [10]. When N = 2, as a s...

Distributional estimates for the Carleson operator acting on characteristic functions of measurable sets of finite measure were obtained by Hunt [12]. In this article we describe a simple method that yields such estimates for general operators acting on one or more functions. As an application we discuss how distributional estimates are obtained for the linear and bilinear Hilbert transform. Th...

We study an extension of the classical Paley-Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and FourierNeumann type series as special cases. Concerning applications, several new results are obtained. From the Dunkl analogue of Gegenbauer’s expansion of the plane wave, we ...

The aim of this paper is to study L-fuzzy closure operator in Lfuzzy topological spaces. We introduce two kinds of L-fuzzy closure operators from different point view and prove that both L-TFCS–the category of topological L-fuzzy closure spaces–and L-PTFCS–the category of topological pointwise L-fuzzy closure spaces–are isomorphic to L-FCTOP.

A new symmetrizability criterion for linear matrix spaces is proposed, with applications to the theory of first order conservation laws. Let L ⊂ Mat(n, k) be a real linear subspace of the space of (n × n)-matrices with coefficients from the field k = R or C. Definition 1. The family L is said to be hyperbolic if (1) A ∈ L for all A ∈ L, and all matrices in L have a simple real spectrum (i.e., t...

We establish L × L to L estimates for some general paraproducts, which arise in the study of the bilinear Hilbert transform along curves.

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight-32 homogeneous subspaces. Among the main results, it proved that if such a superalgebra V simple, then V(2) has canonical commutative associative algebra structure equipped with non-degenerate symmetric bilinear form V(32) naturally V(2)-module V(2)-valued (C-valued) form, satisfying s...