We derive a version of Lagrange's mean value theorem for quantum calculus. We disprove a version of Ostrowski inequality for quantum calculus appearing in the literature. We derive a correct statement and prove that our new inequality is sharp. We also derive a midpoint inequality.
In this paper we prove a sharp Ostrowski type inequality for random processes of certain classes. This inequality is later applied to a solution of the optimal recovery of the integral $\int_0^1ξ_tdt$, using the random variables $ξ_{τ_1},\dots, ξ_{τ_n}$ as an information set, where $τ_1,\dots, τ_n$ are random variables. We also consider the problem of the information set optimization.
We solve the Stechkin problem about approximation of generally speaking unbounded hypersingular integral operators by bounded ones. As a part of the proof, we also solve several related and interesting on their own problems. In particular, we obtain sharp Landau-Kolmogorov type inequalities in both additive and multiplicative forms for hypersingular integral operators and prove a sharp Ostrowski type inequality for multivatiate Sobolev classes. We also give some applications of the obtained results, in particular study the modulus of continuity of the hypersingular integral operators, and solv…
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 $p>2$.
The primary goal of this paper is to improve the operator version of Jensen inequality. As an application, we provide an improvement for the celebrated Ando's inequality. Additionally, we give a tight bound for the operator Hölder inequality.
We prove a sharp integral inequality that generalizes the well known Hardy type integral inequality for negative exponents. We also give sharp applications in two directions for Muckenhoupt weights on R. This work refines the results that appear in [9].
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.
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.
The well known duality between the Sobolev inequality and the Hardy-Littlewood-Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three norms instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the $L^2$ norm to the infimal convolution of the $L^\infty $ and $H^{-1}$ norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also y…
We establish several Poincaré--Sobolev type inequalities for the Lapalce--Beltrami operator $Δ_g$ in the hyperbolic space $\mathbb H^n$ with $n\geq 5$. These inequalities could be seen as the improved second order Poincaré inequality with remainder terms involving with the sharp Rellich inequality or sharp Sobolev inequality in $\mathbb H^n$. The novelty of these inequalities is that it combines both the sharp Poincaré inequality and the sharp Rellich inequality or the sharp Sobolev inequality for $Δ_g$ in $\mathbb H^n$. As a consequence, we obtain the Poincaré--Sobolev inequality for the seco…
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.
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.
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.
