Suppose $\vec{b}=(b_1,\cdots,b_m)\in (BMO)^m,$ $ I_{\alpha,m}^{{\Pi b}}$ is the iterated commutator of $\vec{b}$ and the $m$-linear multilinear fractional integral operator $I_{\alpha,m}$. The purpose of this paper is to discuss the boundedness properties of $I_{\alpha,m}$ and $I_{\alpha,m}^{{\Pi b}}$ on generalized Herz spaces with general Muckenhoupt weights.

We obtain the expressions for sectional curvature, holomorphic sectional curvature and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite nearly Kähler manifold and obtain characterization theorems for holomorphic sectional and holomorphic bisectional curvature. We also establish a condition for a GCR-lightlike submanifold of an indefinite complex space form to be a null holomorphically flat.

In this article, we obtain the $L^p$-boundedness of commutators of Lipschitz functions and singular integrals with non-smooth kernels on Euclidean spaces.

Let $\Omega\in L^2(\mathrm{S}^{n-1})$ be homogeneous function of degree zero and $b$ be BMO functions. In this paper, we obtain some boundedness of the Littlewood-Paley Operators and their higher-order commutators on Herz spaces with variable exponent.

Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedure for the construction of tight wavelet frames generated by the Walsh polynomials using Extension Principles was recently considered by Shah in [Tight wavelet frames generated by the Walsh polynomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11(6) (2013), 1350042]. In this paper, we establish a complete characterization of tight wavelet frames generated by the Walsh polynomials in terms of the polyphase matrices formed by the polyphase components of the Walsh polynomials.

We know that the Box dimension of $f(x)\in C^1[0,1]$ is 1. In this paper, we prove that the Box dimension of continuous functions with bounded variation is still 1. Furthermore, Box dimension of Weyl fractional integral of above function is also 1.

Let $P(z)$ be a polynomial of degree $n$ which does not vanish in $|z|< k $, $k\geq 1$. It is known that for each $0\leq s< n$ and $1\leq R\leq k$, $$M\big(P^{(s)},R\big)\leq \Big(\frac{1}{R^{s}+k^{s}}\Big)\Big[\Big\{\frac{d^{(s)}}{dx^{(s)}}(1+x^{n})\Big\}_{x=1}\Big]\Big(\frac{R+k}{1+k}\Big)^{n}M(P,1).$$ In this paper, we obtain certain extensions and refinements of this inequality by involving binomial coefficients and some of the coefficients of the polynomial $P(z)$.

In this paper, we propose the $q$ analogue of modified Baskakov-Beta operators. The Voronovskaja type theorem and some direct results for the above operators are discussed. The rate of convergence and weighted approximation by the operators are studied.