We study existence of invariant Einstein metrics on complex Stiefel manifolds $G/K = \mathrm{SU}(\ell+m+n)/\mathrm{SU}(n) $ and the special unitary groups $G = \mathrm{SU}(\ell+m+n)$. We decompose the Lie algebra $\frak g$ of $G$ and the tangent space $\frak p$ of $G/K$, by using the generalized flag manifolds $G/H = \mathrm{SU}(\ell+m+n)/\mathrm{S}(\mathrm{U}(\ell)\times\mathrm{U}(m)\times\mathrm{U}(n))$. We parametrize scalar products on the 2-dimensional center of the Lie algebra of $H$, and we consider $G$-invariant and left invariant metrics determined by $\mathrm{Ad}(\mathrm{S}(\mathrm{U}(\ell)\times\mathrm{U}(m)\times\mathrm{U}(n))$-invariant scalar products on $\frak g$ and $\frak p$ respectively. Then we compute their Ricci tensor for such metrics. We prove existence of $\mathrm{Ad}(\mathrm{S}(\mathrm{U}(1)\times\mathrm{U}(2)\times\mathrm{U}(2))$-invariant Einstein metrics on $V_3\mathbb{C}^{5}=\mathrm{SU}(5)/\mathrm{SU}(2)$, $\mathrm{Ad}(\mathrm{S}(\mathrm{U}(2)\times\mathrm{U}(2)\times\mathrm{U}(2))$-invariant Einstein metrics on $V_4\mathbb{C}^{6}=\mathrm{SU}(6)/\mathrm{SU}(2)$, and $\mathrm{Ad}(\mathrm{S}(\mathrm{U}(m)\times\mathrm{U}(m)\times\mathrm{U}(n))$-invariant Einstein metrics on $V_{2m}\mathbb{C}^{2m+n}=\mathrm{SU}(2m+n)/\mathrm{SU}(n)$. We also prove existence of $\mathrm{Ad}(\mathrm{S}(\mathrm{U}(1)\times\mathrm{U}(2)\times\mathrm{U}(2))$-invariant Einstein metrics on the compact Lie group $\mathrm{SU}(5)$, which are not naturally reductive. The Lie group $\mathrm{SU}(5)$ is the special unitary group of smallest rank known for the moment, admitting non naturally reductive Einstein metrics. Finally, we show that the compact Lie group $\mathrm{SU}(4+n)$ admits two non naturally reductive $\mathrm{Ad}(\mathrm{S}(\mathrm{U}(2)\times\mathrm{U}(2)\times\mathrm{U}(n)))$-invariant Einstein metrics for $ 2 \leq n \leq 25$, and four non naturally reductive Einstein metrics for $n\geq 26$. This extends previous results of K. Mori about non naturally reductive Einstein metrics on $\mathrm{SU}(4+n)$ $(n \geq 2)$.

Let $A$ be an expansive dilation on $\mathbb{R}^n$, $q\in (0,\,\infty]$ and $p (\cdot): $\mathbb{R}^n\rightarrow (0,\,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. Let $H^{p (\cdot),\, q}_A ({\mathbb{R}}^n)$ be the anisotropic variable Hardy-Lorentz space defined via the radial grand maximal function. In this paper, the authors first establish its molecular characterization via the atomic characterization of $H^{p (\cdot),\, q}_A ({\mathbb{R}}^n)$. Then, as applications, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from $H^{p (\cdot),\,q}_{A}(\mathbb{R}^n)$ to $L^{p (\cdot),\,q}(\mathbb{R}^n)$ or from $H^{p (\cdot),\,q}_{A}(\mathbb{R}^n)$ to itself. All these results are still new even in the classical isotropic setting.

We prove the strong-type and weak-type estimates for the Calderón–Zygmund singular integrals on central Morrey–Orlicz and weak central Morrey–Orlicz spaces defined in our earlier paper [26]. Next part has similar investigations between distinct central Morrey–Orlicz and weak central Morrey–Orlicz spaces. Then we define the $\lambda$-central mean oscillation–Orlicz spaces and its weak version investigating also some estimates for a modified Calderón–Zygmund singular integrals. Also, we show boundedness of a modified Calderón–Zygmund singular integrals from central Morrey–Orlicz spaces to $\lambda$-central mean oscillation–Orlicz spaces, even for different Orlicz functions, and corresponding weak version.

Based on the idea of adiabatic expansion theory, we will present a new formula for the asymptotic expansion coefficients of every derivative of the heat kernel on a compact Riemannian manifold. It will be very useful for having systematic understanding of the coefficients, and, furthermore, by using only a basic knowledge of calculus added to the formula, one can describe them explicitly up to an arbitrarily high order.

We address the uniqueness problem for mild solutions of the Boussinesq system in $\mathbb{R}^3$. We provide several uniqueness classes on the velocity and the temperature, generalizing in this way the classical $C ([0,T]; L^3(\mathbb{R}^3))$-uniqueness result for mild solutions of the Navier–Stokes equations.

In this paper, we study a $\lambda (\mu)$-ground state of the Schrödinger operator $\mathcal{H}^{\mu}$ based on recurrent relativistic $\alpha$-stable processes. For a signed measure $\mu = \mu^+ - \mu^-$ being in a suitable Kato class, we construct the $\lambda (\mu)$-ground state which is bounded and continuous. Moreover, we prove that the $\lambda (\mu)$-ground state has the mean-value property. In particular, if $\lambda (\mu) = 1$, the mean-value property means the probabilistical harmonicity of $\mathcal{H}^{\mu}$. Finally, we show that if $\alpha > d = 1$, the relativistic $\alpha$-stable process is point recurrent.

We construct a new five parameter family of constant mean curvature trinoids with two asymptotically Delaunay ends and one irregular end.