Let $$\Omega \subset {\mathbb {C}}^n$$ be a smooth bounded pseudoconvex domain and $$A^2 (\Omega )$$ denote its Bergman space. $$P:L^2(\Omega )\longrightarrow A^2(\Omega the projection. For measurable $$\varphi :\Omega \longrightarrow \Omega $$ , projected composition operator is defined by $$(K_\varphi f)(z) = P(f \circ \varphi )(z), z \in f\in A^2 ).$$ In 1994, Rochberg studied boundedness of...

We provide a boundedness criterion for the integral operator $$S_{\varphi }$$ on fractional Fock–Sobolev space $$F^{s,2}({{\mathbb {C}}}^n)$$ , $$s\ge 0$$ where (introduced by Zhu [18]) is given $$\begin{aligned} S_{\varphi }F(z):= \int _{{\mathbb {C}}^n} F(w) e^{z \cdot \bar{w}} \varphi (z- \bar{w}) d\lambda (w) \end{aligned}$$ with $$\varphi $$ in Fock $$F^2({{\mathbb {C}}^n})$$ and $$d\lambd...

‎In this paper‎, ‎we introduce more general contractions called $varphi $-fixed‎ ‎point point for $(F,varphi‎ ,‎alpha )_{s}$ and $(F,varphi‎ ,‎alpha )_{s}$-‎weak contractions‎. ‎We prove the existence and uniqueness of $varphi $-‎fixed point point for $(F,varphi‎ ,‎alpha )_{s}$ and $(F,varphi‎ ,‎alpha‎‎)_{s}$-weak contractions in complete $b$-metric spaces‎. ‎Some examples are‎ ‎supplied to sup...

In this paper we define $varphi$-Connes module amenability of a dual Banach algebra $mathcal{A}$ where $varphi$ is a bounded $w_{k^*}$-module homomorphism from $mathcal{A}$ to $mathcal{A}$. We are mainly concerned with the study of $varphi$-module normal virtual diagonals. We show that if $S$ is a weakly cancellative inverse semigroup with subsemigroup $E$ of idemp...

Consider the equation div$(\varphi^2 \nabla \sigma)=0$ in $\mathbb{R}^N,$ where $\varphi>0$. It is well-known that if there exists $C>0$ such $\int_{B_R}(\varphi \sigma)^2 dx\leq CR^2$ for every $R\geq 1$ then $\sigma$ necessarily constant. In this paper we prove result not true replace $R^2$ by $R^k$ $k>2$ any dimension $N$. This question related to a conjecture De Giorgi.

Let $\Omega$ be a subdomain of $\mathbb{C}$ and let $\mu$ positive Borel measure on $\Omega$. In this paper, we study the asymptotic behavior eigenvalues compact Toeplitz operators $T\_\mu$ acting Bergman spaces $(\lambda n(T\mu))$ decreasing sequence $T\_\mu$, $\rho$ an increasing function such that $\rho (n)/n^A$ is for some $A>0$. We give explicit necessary sufficient geometric condition in ...