### Analysis on the unit ball and on the simplex.

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The Littlewood-Paley theory is extended to weighted spaces of distributions on [-1,1] with Jacobi weights $w\left(t\right)={(1-t)}^{\alpha}{(1+t)}^{\beta}$. Almost exponentially localized polynomial elements (needlets) ${\phi}_{\xi}$, ${\psi}_{\xi}$ are constructed and, in complete analogy with the classical case on ℝⁿ, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $\u27e8f,{\phi}_{\xi}\u27e9$ in respective sequence spaces.

By using averaging techniques, some oscillation criteria for quasilinear elliptic differential equations of second order ${\sum}_{i,j=1}^{N}{D}_{i}[{A}_{ij}{\left(x\right)\left|Dy\right|}^{p-2}{D}_{j}y]+p\left(x\right)f\left(y\right)=0$ are obtained. These results extend and generalize the criteria for linear differential equations due to Kamenev, Philos and Wong.

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