• Abstract

There is a singular integral operators $S_{\varphi}$ on the Fock space $\mathcal{F}^2(\mathbb{C}),$ which originated from the unitarily equivalent version of the Hilbert transform on $L^2(\mathbb{R}).$ In this paper, we give an analytic characterization of functions $\varphi$ with finite zeros such that the integral operator $S_{\varphi}$ is bounded on $\mathcal{F}^2(\mathbb{C})$ using Hadamard’s factorization theorem. As an application, we obtain a complete characterization for such symbol functions $\varphi$ such that the Berezin transform of $S_{\varphi}$ is bounded while the operator $S_{\varphi}$ is not. Also, the corresponding problem in higher dimensions is considered.

• General Summary

• Abstract

In this paper, we consider the exterior Dirichlet problem of Hessian equations $$σ_k (λ(D^2u)) = g(x)$$ with $g$ being a perturbation of a general positive function at infinity. The existence of the viscosity solutions with generalized asymptotic behavior at infinity is established by the Perron’s method which extends the previous results for Hessian equations. By the solutions of Bernoulli ordinary differential equations, the viscosity subsolutions and supersolutions are constructed.

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This paper is a summary of the research on the mapping properties of singular integrals with rough kernels and the corresponding maximal operators. More precisely, the author presents some recent progress, interesting problems and typical methods used in the theory concerning the boundedness and continuity for the rough singular integral operators and maximal singular integral operators along certain submanifolds such as polynomial mappings, polynomial curves, homogeneous mappings, surfaces of revolution and real-analytic submanifolds on the Lebesgue spaces, Triebel–Lizorkin spaces, Besov spaces and mixed radial-angular spaces.

• General Summary

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The main aim of this paper is to derive some new summation theorems for terminating and truncated Clausen’s hypergeometric series with unit argument, when one numerator parameter and one denominator parameter are negative integers. Further, using our truncated summation theorems, we obtain the Mellin transforms of the product of exponential function and Goursat’s truncated hypergeometric function.

• General Summary

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Under some natural regularity assumptions on the exponent function, we obtain the boundedness of Marcinkiewicz integrals $\mu,$ $\mu^ρ_S$ and $\mu^{∗,ρ}_λ$ on grand variable Herz spaces. Our results enrich and improve some previous results in the literature.

• General Summary

• Abstract

Let $P$ be the classical Hardy operator on $(0, ∞)$ and $Q$ be the adjoint operator. In this paper, we get the boundedness for $P,$ $Q$ and the commutators of $P$ and $Q$ with $CMO$ functions on the weighted Herz spaces.

• General Summary

• Abstract

We study microscopic convexity properties of convex solutions of fully nonlinear parabolic equations under a structural condition introduced by Bian-Guan. We prove weak Harnack inequalities for the eigenvalues of the spatial Hessian of solutions and obtain the monotonicity of Hessian’s rank with respect to time.

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In this paper, we derive the interior gradient estimate for solutions to general prescribed curvature equations. The proof is based on a fundamental observation of Garding’s cone and some delicate inequalities under a suitably chosen coordinate chart. As an application, we obtain a Liouville type theorem.

• General Summary

• Abstract

In this paper, we establish global $C^2$ estimates to the Neumann problem for a class of fully nonlinear elliptic equations. As an application, we prove the existence and uniqueness of $k$-admissible solutions to the Neumann problems.

• General Summary