Recent zbMATH articles in MSC 47Fhttps://www.zbmath.org/atom/cc/47F2021-04-16T16:22:00+00:00WerkzeugOn the attainability of the best constant in fractional Hardy-Sobolev inequalities involving the spectral Dirichlet Laplacian.https://www.zbmath.org/1456.460352021-04-16T16:22:00+00:00"Ustinov, N. S."https://www.zbmath.org/authors/?q=ai:ustinov.n-sSummary: We prove the attainability of the best constant in the fractional Hardy-Sobolev inequality with a boundary singularity for the spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin. A similar result has been proved earlier for the conventional Hardy-Sobolev inequality [\textit{A. V. Dem'yanov} and \textit{A. I. Nazarov}, Zap. Nauchn. Semin. POMI 336, 25--45 (2006; Zbl 1136.35088); translation in J. Math. Sci., New York 143, No. 2, 2857--2868 (2007)].The dimensional Brunn-Minkowski inequality in Gauss space.https://www.zbmath.org/1456.520112021-04-16T16:22:00+00:00"Eskenazis, Alexandros"https://www.zbmath.org/authors/?q=ai:eskenazis.alexandros"Moschidis, Georgios"https://www.zbmath.org/authors/?q=ai:moschidis.georgiosThe authors prove the Gaussian analogue of the classical Brunn-Minkowski inequality for the Lebesgue measure, thus settling a problem raised in [\textit{R. J. Gardner} and \textit{A. Zvavitch}, Trans. Am. Math. Soc. 362, No. 10, 5333--5353 (2010; Zbl 1205.52002)]. They also settle the case when equality holds.
Reviewer: George Stoica (Saint John)Semigroup maximal functions, Riesz transforms, and Morrey spaces associated with Schrödinger operators on the Heisenberg groups.https://www.zbmath.org/1456.420252021-04-16T16:22:00+00:00"Wang, Hua"https://www.zbmath.org/authors/?q=ai:wang.hua|wang.hua.1|wang.hua.2Summary: Let \(\mathcal{L}=- \Delta_{\mathbb{H}^n}+V\) be a Schrödinger operator on the Heisenberg group \(\mathbb{H}^n\), where \(\Delta_{\mathbb{H}^n}\) is the sub-Laplacian on \(\mathbb{H}^n\) and the nonnegative potential \(V\) belongs to the reverse Hölder class \(\mathcal{B}_q\) with \(q\in [Q/2,\infty)\). Here, \(Q=2n+2\) is the homogeneous dimension of \(\mathbb{H}^n\). Assume that \(\{ e^{- t \mathcal{L}}\}_{t>0}\) is the heat semigroup generated by \(\mathcal{L}\). The semigroup maximal function related to the Schrödinger operator \(\mathcal{L}\) is defined by \(\mathcal{T}_{\mathcal{L}}^\ast (f)(u) := \sup_{t>0} | e^{- t \mathcal{L}} f (u)|\). The Riesz transform associated with the operator \(\mathcal{L}\) is defined by \(\mathcal{R}_{\mathcal{L}}= \nabla_{\mathbb{H}^n} \mathcal{L}^{-1/2}\), and the dual Riesz transform is defined by \(\mathcal{R}_{\mathcal{L}}^\ast= \mathcal{L}^{-1/2} \nabla_{\mathbb{H}^n} \), where \(\nabla_{\mathbb{H}^n}\) is the gradient operator on \(\mathbb{H}^n\). In this paper, the author first introduces a class of Morrey spaces associated with the Schrödinger operator \(\mathcal{L}\) on \(\mathbb{H}^n\). Then, by using some pointwise estimates of the kernels related to the nonnegative potential, the author establishes the boundedness properties of these operators \(\mathcal{T}_{\mathcal{L}}^\ast\), \(\mathcal{R}_{\mathcal{L}}\), and \(\mathcal{R}_{\mathcal{L}}^\ast\) acting on the Morrey spaces. In addition, it is shown that the Riesz transform \(\mathcal{R}_{\mathcal{L}}= \nabla_{\mathbb{H}^n} \mathcal{L}^{-1/2}\) is of weak-type \((1,1)\). It can be shown that the same conclusions are also true for these operators on generalized Morrey spaces.