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Register a new userPolynomial inequalities on the $pi/4$-circle sector
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A number of sharp inequalities are proved for the space ${mathcal P}left(^2Dleft(frac{pi}{4}right)right)$ of 2-homogeneous polynomials on ${mathbb R}^2$ endowed with the supremum norm on the sector $Dleft(frac{pi}{4}right):=left{e^{itheta}:thetain left[0,frac{pi}{4}right]right}$. Among the main results we can find sharp Bernstein and Markov inequalities and the calculation of the polarization constant and the unconditional constant of the canonical basis of the space ${mathcal P}left(^2Dleft(frac{pi}{4}right)right)$.
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In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardys original inequality. We give examples obtaining new weighted Hardy inequalities on $mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.
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For initial data in Sobolev spaces $H^s(mathbb T)$, $frac 12 < s leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t)^{3(s-frac 12) + epsilon}$, $0<epsilon ll 1$. Key to establishing this result is the discovery of a nonlinear smoothing effect for the Benjamin-Ono equation, according to which the solution to the equation satisfied by a certain gauge transform, which is widely used in the well-posedness theory of the Cauchy problem, becomes smoother once its free solution is removed.
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