arXiv subjects
Publications and source records attributed to Toshio Horiuchi.
We study the Caffarelli-Kohn-Nirenberg type inequalities in the case of $p=1$ and generalize them adopting weight functions $w(|x|)$ on $R^n$ with $w(t)$ in ${W}(R_+)$. Here ${W}(R_+)$ is a general class of weight functions on $R_+$ including non-doubling weights like $e^{1/t}$ and $e^{-1/t}$.
In the present paper we shall study a variational problem relating the weighted Hardy inequalities with sharp missing terms. As weights we treat non-doubling functions of the distance to the boundary of bounded domain.
We establish the Caffarelli-Kohn-Nirenberg type inequalities involving{ super-logarithms (infinitely iterated logarithms).} As a result the critical Caffarelli-Kohn-Nirenberg type inequalities will be improved, and in certain cases the best constants will be discovered.
We will establish the Caffarelli-Kohn-Nirenberg type inequalities with non-doubling weights being permitted. The classical Caffarelli-Kohn-Nirenberg type inequalities are categorized into non-critical and critical cases, and it is known that there is some kind of mysterious relationship between them. Interestingly the new framework in this treatise allows them to be integrated and reveals the meaning of mysterious relationships.
In the present paper we shall establish n-dimensional Hardy's inequalities with non-doubling weight functions of the distance to the boundary, where the boundary is a $C^2$ class bounded domain of $R^N$. This work is essentially based on one dimensional weighted Hardy's inequalities with one-sided boundary condition and sharp remainders. As weights we admit rather general ones that may vanish or blow up in infinite order such as $e^{-1/t}$ or $e^{1/t}$ at $t=0$ in one dimensional case.
In the present paper we shall improve one dimensional weighted Hardy inequalities with one-sided boundary condition by adding sharp remainders. As an application, we shall establish n dimensional weighted Hardy inequalities in a bounded smooth domain with weight functions being powers of the distance function d(x) to the boundary. Our results will be applicable to variational problems in a coming paper.
Let $Ω$ be a bounded domain of $\mathbb{R}^N$ whose boundary is a $\mathbb{C}^2$ compact manifolds. In the present paper we shall study a variational problem relating the weighted Hardy inequalities with sharp missing terms. As weights we adopted powers of the distance function $δ(x)$ to the boundary $\partialΩ$.
In a bounded smooth domain of Rn, we shall establish Kato's inequalities involving measures for p-Laplace operator up to the boundary under suitable assumptions. Keywords: Kato's inequality, $p$-Laplace operator