Global LpL^{p} estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind [\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x) \partial_{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}%] where $(a_{ij})$ is symmetric uniformly positive definite on $\mathbb{R}^{p_{0}}$ ($p_{0}\leq N$), with uniformly continuous and bounded entries, and $(b_{ij})$ is a constant matrix such that the frozen operator $\mathcal{A}_{x_{0}}$ corresponding to $a_{ij}(x_{0})$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1<p<\infty$) of the kind:% [|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(\mathbb{R}% ^{N})}\leq c{|\mathcal{A}u|_{L^{p}(\mathbb{R}^{N})}+|u|_{L^{p}(\mathbb{R}% ^{N})}} for i,j=1,2,...,p_{0}.] We obtain the previous estimates as a byproduct of the following one, which is of interest in its own:% [|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(S_{T})}\leq c{|Lu|_{L^{p}(S_{T})}+|u|_{L^{p}(S_{T})}}] for any $u\in C_{0}^{\infty}(S_{T}),$ where $S_{T}$ is the strip $\mathbb{R}^{N}\times[-T,T]$, $T$ small, and $L$ is the Kolmogorov-Fokker-Planck operator% [L\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x,t) \partial_{x_{i}x_{j}}% ^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}-\partial_{t}%] with uniformly continuous and bounded $a_{ij}$'s

The sharp maximal function approach to LpL^{p} estimates for operators structured on H\"{o}rmander's vector fields

We consider a nonvariational degenerate elliptic operator structured on a system of left invariant, 1-homogeneous, H\"ormander's vector fields on a Carnot group in $R^{n}$, where the matrix of coefficients is symmetric, uniformly positive on a bounded domain of $R^{n}$ and the coefficients are bounded, measurable and locally VMO in the domain. We give a new proof of the interior $L^{p}$ estimates on the second order derivatives with respect to the vector fields, first proved by Bramanti-Brandolini in [Rend. Sem. Mat. dell'Univ. e del Politec. di Torino, Vol. 58, 4 (2000), 389-433], extending to this context Krylov' technique, introduced in [Comm. in P.D.E.s, 32 (2007), 453-475], consisting in estimating the sharp maximal function of the second order derivatives.Comment: 26 page

Interior HW^{1,p} estimates for divergence degenerate elliptic systems in Carnot groups

Let X_1,...,X_q be the basis of the space of horizontal vector fields on a homogeneous Carnot group in R^n (q<n). We consider a degenerate elliptic system of N equations, in divergence form, structured on these vector fields, where the coefficients a_{ab}^{ij} (i,j=1,2,...,q, a,b=1,2,...,N) are real valued bounded measurable functions defined in a bounded domain A of R^n, satisfying the strong Legendre condition and belonging to the space VMO_{loc}(A) (defined by the Carnot-Caratheodory distance induced by the X_i's). We prove interior HW^{1,p} estimates (2<p<\infty) for weak solutions to the system

Two characterization of BV functions on Carnot groups via the heat semigroup

In this paper we provide two different characterizations of sets with finite perimeter and functions of bounded variation in Carnot groups, analogous to those which hold in Euclidean spaces, in terms of the short-time behaviour of the heat semigroup. The second one holds under the hypothesis that the reduced boundary of a set of finite perimeter is rectifiable, a result that presently is known in Step 2 Carnot groups

L^p and Schauder estimates for nonvariational operators structured on H\"ormander vector fields with drift

We consider linear second order nonvariational partial differential operators of the kind a_{ij}X_{i}X_{j}+X_{0}, on a bounded domain of R^{n}, where the X_{i}'s (i=0,1,2,...,q, n>q+1) are real smooth vector fields satisfying H\"ormander's condition and a_{ij} (i,j=1,2,...,q) are real valued, bounded measurable functions, such that the matrix {a_{ij}} is symmetric and uniformly positive. We prove that if the coefficients a_{ij} are H\"older continuous with respect to the distance induced by the vector fields, then local Schauder estimates on X_{i}X_{j}u, X_{0}u hold; if the coefficients belong to the space VMO with respect to the distance induced by the vector fields, then local L^{p} estimates on X_{i}_{j}u, X_{0}u hold. The main novelty of the result is the presence of the drift term X_{0}, so that our class of operators covers, for instance, Kolmogorov-Fokker-Planck operators

GAMBARAN STATUS GIZI BERDASARKAN LINGKAR LENGAN PADA IBU HAMIL TRIMESTER II DAN III DI BPS SRI ISNAWATI WONOAYU SIDOARJO

Status gizi ibu hamil merupakan ukuran keberhasilan dalam pemenuhan nutrisi ibu hamil. Pengaturan gizi yang baik akan berpengaruh positif, sedangkan bila kurang baik maka pengaruhnya negatif baik pada ibu maupun janin yang dikandungnya. Penelitian ini bertujuan untuk mengetahui gambaran status gizi berdasarkan LILA (Lingkar Lengan Atas) pada ibu hamil trimester II dan III di BPS Sri Isnawati Wonoayu-Sidoarjo. Desain dalam penelitian ini adalah deskriptif. Populasi seluruh ibu hamil trimester II dan III yang berkunjung di BPS Sri Isnawati Wonoayu-Sidoarjo, besar sampel sebesar 20 responden. Pengambilan sampel menggunakan teknik Nonprobability sampling dengan Total Sampling. Variabel penelitian gambaran status gizi berdasarkan LILA (Lingkar Lengan Atas) pada ibu hamil trimester II dan III. Data diambil dengan melakukan pengukuran lingkar lengan. Data dianalisis dengan statistik deskriptif dalam bentuk persentase. Hasil penelitian menunjukkan sebagian besar (60%) ibu hamil mempunyai status gizi yang normal (LILA ≥ 23,5 cm) dan hampir setengahnya (40%) mengalami KEK (≤ 23,5 cm) . Simpulan penelitian ini menunjukkan bahwa sebagian besar ibu hamil mempunyai status gizi normal dengan ukuran lingkar lengan ≥ 23,5 cm. Saran untuk ibu hamil agar status gizinya tetap normal dengan menjaga pola makan dan selalu melaksanakan kunjungan ANC (Atenatalcare) secara rutin dan petugas kesehatan dapat memberikan PMT (Pemberian Makanan Tambahan) pada ibu hamil yang mengalami KEK (Kekurangan Energi Kronis)