The $k$-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations $F_{k}[u]=0$, where $F_{k}[u]$ is the elementary symmetric function of order $k$, $1\leq k\leq n$, of the eigenvalues of the Hessian matrix $D^{2}u$. For example, $F_{1}[u]$ is the Laplacian $\Delta u$ and $F_{n}[u]$ is the real Monge-Ampère operator det $D^{2}u$, while $1$-convex functions and $n$-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative $k$-convex functions, and give several estimates for the mixed $k$-Hessian operator. Applications of these estimates to the $k$-Green functions are also established.
Let $b_1, b_2 \in {\rm BMO}(\mathbb {R}^n)$ and $T_{\sigma }$ be a bilinear Fourier multiplier operator with associated multiplier $\sigma $ satisfying the Sobolev regularity that $\sup _{\kappa \in \mathbb {Z}} \|\sigma _{\kappa }\| _{W^{s_1,s_2}(\mathbb {R}^{2n})}<\infty $ for some $s_1,s_2\in (n/2,n]$. In this paper, the behavior on $L^{p_1}(\mathbb {R}^n)\times L^{p_2}(\mathbb {R}^n)$ $(p_1,p_2\in (1,\infty ))$, on $H^1(\mathbb {R}^n)\times L^{p_2}(\mathbb {R}^n)$ $(p_2\in [2,\infty ))$, and on $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$, is considered for the commutator $T_{{\sigma }, \vec {b}} $ defined by $$ \begin {aligned} T_{\sigma ,\vec {b}} (f_1,f_2) (x)=&b_1(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(b_1f_1, f_2)(x) &+ b_2(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(f_1, b_2f_2)(x) . \end {aligned} $$ By kernel estimates of the bilinear Fourier multiplier operators and employing some techniques in the theory of bilinear singular integral operators, it is proved that these mapping properties are very similar to those of the bilinear Fourier multiplier operator which were established by Miyachi and Tomita.
The rate of growth of the energy integral of a quasiregular mapping $f\:\mathcal X\rightarrow \mathcal Y$ is estimated in terms of a special isoperimetric condition on $\mathcal Y$. The estimate leads to new Phragmén-Lindelöf type theorems.
The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space $H^{k,\infty }$; $k \in {\mathbb {N}}^*$ of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L. Baratchart and M. Zerner, On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk, J. Comput. Appl. Math. 46 (1993), 255–269 and by S. Chaabane and I. Feki, Optimal logarithmic estimates in Hardy-Sobolev spaces $H^{k,\infty }$, C. R., Math., Acad. Sci. Paris 347 (2009), 1001–1006. As an application, we prove a logarithmic stability result for the inverse problem of identifying a Robin parameter on a part of the boundary of an annular domain starting from its behavior on the complementary boundary part.
In this note we show that for a $\ast ^{n}$-module, in particular, an almost $n$-tilting module, $P$ over a ring $R$ with $A=\mathop {\mathrm End}_{R}P$ such that $P_A$ has finite flat dimension, the upper bound of the global dimension of $A$ can be estimated by the global dimension of $R$ and hence generalize the corresponding results in tilting theory and the ones in the theory of $\ast $-modules. As an application, we show that for a finitely generated projective module over a VN regular ring $R$, the global dimension of its endomorphism ring is not more than the global dimension of $R$.
The GNSS (Global Navigation Satellite System) coordinates time series are still used as a source for determining the velocities of GNSS permanent stations. These coordinates, apart from the geodynamical signals, also contain an interference signal. This paper shows the results of the comparative analysis of the GNSS coordinates time series with a deformation of the Earth's crust obtained from loading models. In the analysis, coordinates time series are used (CODE Repro2013) without loading models (Atmospheric Pressure Loading, Hydrology, Non-Tidal Ocean Loading) at the stage of the reprocessing of GNSS archival data. The analyses showed that in the case of the Up component there is a high correlation between the GNSS coordinates changes and deformations of the Earth's crust from the loading models (coefficient 0.5-0.8). Additionally, we noticed that for horizontal components (North, East) changes occur in the phase shift between coordinates, and the Earth’s crust deformations signals are accelerated or delayed each other (-150 to 200 days). This article shows new methods of iLSE (iteration Least Square Estimation) to determine periodic signals in the time series. Additionally, we compared the values of estimated amplitudes for GNSS and deformation time series. and Kaczmarek Adrian, Kontny Bernard.
Consistent estimators of the asymptotic covariance matrix of vectors of U-statistics are used in constructing asymptotic confidence regions for vectors of Kendall's correlation coefficients corresponding to various pairs of components of a random vector. The regions are products of intervals computed by means of a critical value from multivariate normal distribution. The regularity of the asymptotic covariance matrix of the vector of Kendall's sample coefficients is proved in the case of sampling from continuous multivariate distribution under mild conditions. The results are applied also to confidence intervals for the coefficient of agreement. The coverage and length of the obtained (multivariate) product of intervals are illustrated by simulation.
We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet p-Laplacian and the Navier p-biharmonic operator on a ball of radius R in R N and its asymptotics for p approaching 1 and ∞. Let p tend to ∞. There is a critical radius RC of the ball such that the principal eigenvalue goes to ∞ for 0 < R 6 RC and to 0 for R > RC . The critical radius is RC = 1 for any N ∈ N for the p-Laplacian and RC = √ 2N in the case of the p-biharmonic operator. When p approaches 1, the principal eigenvalue of the Dirichlet p-Laplacian is NR−1 × (1− (p − 1) log R(p − 1)) + o(p − 1) while the asymptotics for the principal eigenvalue of the Navier p-biharmonic operator reads 2N/R2 + O(−(p − 1) log(p − 1)).