We study the numerical radius \(\omega(T)\) of a bounded linear operator \(T\) on a complex Hilbert space. Write \(T=A+iB\), where \(A\) and \(B\) are the real and imaginary parts of \(T\). A central result is the exact formula \[ \omega(T)=\frac{\pi}{2}\sup_{\|x\|=1}\int_0^1 \left|\left\langle (A\cos \pi t+B\sin \pi t)x,x\right\rangle\right|\,dt . \] For every \(n\in\mathbb N\), we also prove the power form \[ \omega^{2n}(T)=\frac{4^n}{\binom{2n}{n}} \sup_{\|x\|=1}\int_0^1 \left\langle (A\cos \pi t+B\sin \pi t)x,x\right\rangle^{2n}\,dt . \] This yields sharp upper bounds; for \(n=1\), together with the matching integral lower bound, it recovers Kittaneh's classical two-sided inequality \[ \frac14\|T^*T+TT^*\|\le \omega^2(T)\le \frac12\|T^*T+TT^*\|. \] Furthermore, we obtain exact numerical radius formulas for some block operator matrices, including a \(3\times3\) nilpotent matrix and longer alternating paths. The factor \(\pi/2\) is best possible.
Let \(\mathcal{H}\) be a complex Hilbert space, and let \(T=H+iK\in\mathcal{B}(\mathcal{H})\) be the Cartesian decomposition of \(T\), where \(H=H^{*}\) and \(K=K^{*}\). We introduce the \(L^2\)-Cartesian numerical radius \[ \omega_2(T)=\left(\frac{1}{2\pi}\int_0^{2\pi}\|\cos\theta H+\sin\theta K\|^2\,d\theta\right)^{1...
We consider the elliptic equation $$ -\Delta u=d(x)^\alpha u^p,\quad x\in \Omega\quad (\text{or } x\in\mathbb{R}^N\setminus \overline{\Omega}), $$ where \(\alpha, p \in \mathbb{R}\), \(d(x) = \text{dist} (x,\partial\Omega)\) and \(\Omega\subset\mathbb{R}^N\) \((N\geq 3)\) is a bounded smooth domain. We establish estima...
Xi-You Cheng, Hui-Juan Shao, Lei Wei et al.· Electronic Journal of Differ...· 0 citations
We study the boundary regularity of solutions for the tempered fractional Laplacian \[ (-\Delta)^s_\lambda u=f \quad \text{in }\Omega, \qquad u=0 \quad \text{in }\mathbb{R}^n\setminus\Omega, \] where \(00\), and \(\Omega\subset\mathbb{R}^n\) is a bounded Lipschitz domain satisfying the exterior ball condition. The temp...
Let \(\varphi\) be an analytic self-map of the open unit disk \(\mathbb{D}\) in the complex plane. For any function \(\psi\) analytic in the unit disk \(\mathbb{D}\), the weighted composition operator is given by \(W_{\varphi,\psi}f(z)=\psi(z) f(\varphi(z))\) for \(z\in \mathbb{D}\) and \(f\) analytic on \(\mathbb{D}\)...
D. Borgohain· Serdica Mathematical Journal· 0 citations
For any \(n\geq 1\), let \((M,\theta)\) be a compact strictly pseudoconvex hypersurface type CR manifold with dimension \(2n+1\), \(Q=2n+2\) be its homogeneous dimension. We prove the sharp Folland--Stein Sobolev inequality on \(M\): if \(Q^*=2Q/(Q-2)\), then there exists a constant \(A>0\), depending only on \((M,\the...
We establish a nonasymptotic Gaussian approximation bound over centered Euclidean balls for sums of independent, not necessarily identically distributed, random vectors in \(\mathbb R^p\). Let \(S_n\) denote the sum, let \(\Sigma=\operatorname{Var}(S_n)\), and let \(G\sim N(0,\Sigma)\). Define the effective rank and th...
A. Naumov, Sergey Samsonov, Vladimir Spokoiny et al.· 0 citations
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