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Thursday, August 2, 2018 - 15:35 to 16:35
SMS Conference Hall
Saswata Adhikari
Hardy and Sobolev type inequalities associated with Grushin and Dunkl operators
Abstract: The theory of Dunkl operators in the study of special functions with reflection symmetrices is very young. These operators were first introduced by C. F. Dunkl in 1989. Since Dunkl transform is generalization of classical Fourier transform, attempts were made by several authors to extend the results of classical Fourier analysis to the Dunkl setting. The classical Hardy inequality asserts that for all $u\in C_{0}^{\infty}(\mathbb{R}^{N}), N\geq 3$, \begin{eqnarray}\label{np1} \int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}dx\geq \left(\frac{N-2}{2}\right)^{2}\int\limits_{\mathbb{R}^{N}}\frac{|u|^{2}}{|x|^{2}}dx, \end{eqnarray} where the inequality is strict for $u\not\equiv 0$ and $\left(\frac{N-2}{2}\right)^{2}$ is the sharp constant. The Sobolev inequality states that for all $u\in\dot{H}^{1}(\mathbb{R}^{N}), N\geq 3$, one has \begin{eqnarray}\label{np2} \int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}dx\geq \pi N(N-2)\left(\frac{\Gamma(\frac{N}{2})}{\Gamma(N)}\right)^{\frac{2}{N}}\left(\int\limits_{\mathbb{R}^{N}}|u|^{q}dx\right)^{\frac{2}{q}}, \end{eqnarray} with $q=\frac{2N}{N-2}$. The equality holds if and only if $u(x)=ch(b(x-a))$ for some $a\in\mathbb{R}^{N}, b>0$ and $c\in\mathbb{C}$, where $h(x)=(1+x^{2})^{-\frac{N-2}{2}}$. This was proved by R. L. Frank in 2011. In this talk, we shall discuss Hardy inequality (\ref{np1}) in the half space on $\mathbb{R}^{n+m}$ associated with Grushin operator. We shall also consider analogous inequality of (\ref{np2}) in the Dunkl setting associated with the Dunkl gradient and shall show the existence of a minimizer for the Dunkl type Sobolev inequality.

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