Hardy's inequalities for Grushin operator and Hermite multipliers on Modulation spaces

Abstract

This thesis consists of two broad themes. First one revolves around the Hardy's inequalities for the fractional power of Grushin operator $\G$ which is chased via two different approaches. In the first approach, we first prove Hardy's inequality for the generalized sublaplacian defined on $\R\times\R^+$, then using the spherical harmonics, applying Hardy's inequality for individual components, we derive Hardy's inequality for Grushin operator. The techniques used for deriving Hardy's inequality for generalized sublaplacian are in parallel with the ones used in \cite{thangaveluroncal}. We first find Cowling-Haagerup type of formula for the fractional generalised sublaplacian and then using the modified heat kernel, we find integral representations of fractional generalized sublaplacian. Then we derive Hardy's inequality for generalized sublaplacian. In the second approach, we start with an extension problem for Grushin, with initial condition $f\in L^p(\R^{n+1})$. We derive a solution $u(\cdot,\rho)$ to that extension problem and show that solution goes to $f$ in $L^p(\R^{n+1})$ as the extension variable $\rho$ goes to $0$. Further $-\rho^{1-2s}\partial_\rho u $ goes to $B_s\G_s f$ in $L^p(\R^{n+1})$ as $\rho$ goes to $0$, thereby giving us an another way of defining fractional powers of Grushin operator $\G_s$. We also derive trace Hardy inequality for the Grushin operator with the help of extension problem. Finally we prove $L^p$-$L^q$ inequality for fractional Grushin operator, thereby deriving Hardy-Littlewood-Sobolov inequality for the Grushin operator.\\

Second theme consists of Hermite multipliers on modulation spaces $M^{p,q}(\R^n)$. We find a relation between sublaplacian multipliers $m(\tilde{\L})$ on polarised Heisenberg group $\Hb^n_{pol}$ and Hermite multipliers $m(\H)$ on modulation spaces $M^{p,q}(\R^n)$, thereby deriving the conditions on the multipliers $m$ to be Hermite multipliers on modulation spaces. We believe that the conditions on multipliers that we have found are more strict than required. We improve the results for the case the modulation spaces $M^{p,q}(\R^n)$ have $p=q$ by finding a relation between the boundedness of Hermite multipliers on $M^{p,p}$ and the boundedness of Fourier multipliers on torus $\T^n$. We also derive the conditions for boundedness of the solution of wave equation related to Hermite and the solution of Schr\"odinger equation related to Hermite on modulation spaces.