Let $T$ be the singular integral operator with variable kernel, $T^*$ be the adjoint of $T$ and $T^{\sharp}$ be the pseudo-adjoint of $T$. Let $T_1T_2$ be the product of $T_1$ and $T_2,$ $T_1\circ T_2$ be the pseudo product of $T_1$ and $T_2.$ In this paper, we establish the boundedness for commutators of these operators and the fractional differentiation operator $D^\gamma$ on the weighted Morrey spaces.

In this paper, necessary and sufficient conditions for a closed range composition operator $C_{\phi}$ on the general family of holomorphic function spaces $F(p, q, s)$ and more generally on $\alpha$-Besov type spaces $F(p, \alpha p-2, s)$ are given. We give a Carleson measure characterization on $F(p, \alpha p-2, s)$ spaces, then we indicate how Carleson measures can be used to characterize boundedness and compactness of $C_{\phi}$ on $F(p,q,s)$ and $F(p,\alpha p-2,s)$ spaces.

In the present manuscript, we propose the modification of Jain operators which the generalization of Szász-Mirakyan operators. These new class operators are linear positive operators of discrete type depending on a real parameters. We give theorem of degree of approximation and the Voronovskaya asymptotic formula.

We consider Hardy spaces with variable exponents defined by grand maximal function on the Heisenberg group. Then we introduce some equivalent characterizations of variable Hardy spaces. By using atomic decomposition and molecular decomposition we get the boundedness of singular integral operators on variable Hardy spaces. We investigate the Littlewood-Paley characterization by virtue of the boundedness of singular integral operators.

We give an estimate of type $ \sup \times \inf $ on Riemannian manifold of dimension 4 for a Yamabe type equation.

This paper investigates the fractal dimension of the fractional integrals of a fractal function. It has been proved that there exists some linear connection between the order of Riemann-Liouville fractional integrals and the Hausdorff dimension of a fractal function.

In this paper, we give some creative characterizations of Campanato spaces via the boundedness of commutators associated with the Calderόn-Zygmund singular integral operator, fractional integrals and Hardy type operators. Furthermore, we put forward a few problems on the characterizations of Campanato type spaces via the boundedness of commutators.