In this paper the sharp Garding inequality is established on compact Lie groups. The positivity condition is expressed in the non-commutative phase space in terms of the full symbol, which is defined using the representations of the group. Applications are given to the $L^2$ and Sobolev boundedness of pseudo-differential operators.
In this paper we study the Cauchy problem for diffusion equations associated to a class of strongly hypoelliptic pseudo-differential operators on graded Lie groups. To do so, we develop a global complex functional calculus on graded Lie groups in order to analyse the corresponding energy estimates. One of the main aspects of this complex functional calculus is that for the $(ρ,δ)$-Euclidean Hörmander classes we recover the standard functional calculus developed by Seeley [38]. In consequence the Gårding inequality that we prove for arbitrary graded Lie groups absorbs the historical 1953's ineq…
In this paper we prove a class of second order Caffarelli-Kohn-Nirenberg inequalities which contains the sharp second order uncertainty principle recently established by Cazacu, Flynn and Lam \cite{CFL2020} as a special case. We also show the sharpness of our inequalities for several classes of parameters. Finally, we prove two stability versions of the sharp second order uncertainty principle of Cazacu, Flynn and Lam by showing that the difference of both sides of the inequality controls the distance to the set of extremal functions in $L^2$ norm of gradient of functions.
Using elementary techniques, we prove sharp anisotropic Hardy-Littlewood inequalities for positive multilinear forms. In particular, we recover an inequality proved by F. Bayart in 2018.
We prove a sharp quantitative version of the $p$-Sobolev inequality for any $1<p<n$, with a control on the strongest possible distance from the class of optimal functions. Surprisingly, the sharp exponent is constant for $p<2$, while it depends on $p$ for $p>2$.
A method of proving Hardy's type inequality for orthogonal expansions is presented in a rather general setting. Then sharp multi-dimensional Hardy's inequality associated with the Laguerre functions of convolution type is proved for type index $\al\in[-1/2,\infty)^d$. The case of the standard Laguerre functions is also investigated. Moreover, the sharp analogues of Hardy's type inequality involving $L^1$ norms in place of $H^1$ norms are obtained in both settings.
We prove a sharp integral inequality for the dyadic maximal operator due to which the evaluation of the Bellman function of this operator with respect to two variables is possible, as can be seen in [3]. Our inequality of interest is proved in this article by a simpler and more immediate way. We also study a stability result in connection with this inequality, that is we provide a necessary and sufficient condition, for a sequence of functions,under which we obtain equality in the limit. The proof of this result is based on the proof of the related inequality which we present in this article.
We prove a sharp dimension-free isoperimetric inequality, involving the volume entropy, in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
We obtain sharp estimate on $p$-spectral gaps, or equivalently optimal constant in $p$-Poincaré inequalities, for metric measure spaces satisfying measure contraction property. We also prove the rigidity for the sharp $p$-spectral gap.
We prove sharp Lieb-Thirring type inequalities for the eigenvalues of a class of one-dimensional functional difference operators associated to mirror curves. We furthermore prove that the bottom of the essential spectrum of these operators is a resonance state.
Hardy's inequality on $H^p$ spaces, $p\in(0,1]$, in the context of orthogonal expansions is investigated for general basis on a subset of $\mathbb{R}^d$ with Lebesgue measure. The obtained result is applied to various Hermite, Laguerre, and Jacobi expansions. For that purpose some delicate estimates of the higher order derivatives for the underlying functions and of the associated heat kernels are proved. Moreover, sharpness of studied Hardy's inequalities is justified by a construction of an explicit counterexample, which is adjusted to all considered settings.
In this paper we consider a generalized version of Carleman's inequality. An equivalent version of it states that $\|f\|_{A_α^{2α}}\leq\|f\|_{H^2}$, where $f$ is a holomorphic function and $α>1$. If the norms $\|f\|_{A_α^{2α}}$ are decreasing in $α$, then the inequality holds for $f$. For a dense set of functions, we calculate the derivative of the norms $\|f\|_{A_α^{2α}}$ in $α$ and give sufficient conditions for this derivative to be non-positive. As an application, we prove the inequality for linear combinations of two reproducing kernels. Some numerical evidences are also provided.
