A Note on the $$L^p$$ Integrability of a Class of Bochner–Riesz Kernels

For a general compact variety $$\Gamma $$ Γ of arbitrary codimension, one can consider the $$L^p$$ L p mapping properties of the Bochner–Riesz multiplier $$\begin{aligned} m_{\Gamma , \alpha }(\zeta ) \ = \ \mathrm{dist}(\zeta , \Gamma )^{\alpha } \phi (\zeta ) \end{aligned}$$ m Γ , α ( ζ ) = dist ( ζ , Γ ) α ϕ ( ζ ) where $$\alpha > 0$$ α > 0 and $$\phi $$ ϕ is an appropriate smooth cutoff function. Even for the sphere $$\Gamma = {{\mathbb {S}}}^{N-1}$$ Γ = S N - 1 , the exact $$L^p$$ L p boundedness range remains a central open problem in Euclidean harmonic analysis. In this paper we consider the $$L^p$$ L p integrability of the Bochner–Riesz convolution kernel for a particular class of varieties (of any codimension). For a subclass of these varieties the range of $$L^p$$ L p integrability of the kernels differs substantially from the $$L^p$$ L p boundedness range of the corresponding Bochner–Riesz multiplier operator.

Upper bounds for Ricci curvatures for submanifolds in Bochner-Kaehler manifolds

Chen established the relationship between the Ricci curvature and the squared norm of meancurvature vector for submanifolds of Riemannian space form with arbitrary codimension knownas Chen-Ricci inequality. Deng improved the inequality for Lagrangian submanifolds in complexspace form by using algebraic technique. In this paper, we establish the same inequalitiesfor different submanifolds of Bochner-Kaehler manifolds. Moreover, we obtain improvedChen-Ricci inequality for Kaehlerian slant submanifolds of Bochner-Kaehler manifolds.

Higher codimensional alpha invariants and characterization of projective spaces

We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable [Formula: see text]-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.

Ricci curvature on warped product submanifolds of complex space forms and its applications

The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality.