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Register a new userOptimal constants and extremisers for some smoothing estimates
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We establish new results concerning the existence of extremisers for a broad class of smoothing estimates of the form $|psi(| abla|) exp(itphi(| abla|)f |_{L^2(w)} leq C|f|_{L^2}$, where the weight $w$ is radial and depends only on the spatial variable; such a smoothing estimate is of course equivalent to the $L^2$-boundedness of a certain oscillatory integral operator $S$ depending on $(w,psi,phi)$. Furthermore, when $w$ is homogeneous, and for certain $(psi,phi)$, we provide an explicit spectral decomposition of $S^*S$ and consequently recover an explicit formula for the optimal constant $C$ and a characterisation of extremisers. In certain well-studied cases when $w$ is inhomogeneous, we obtain new expressions for the optimal constant.
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We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds, the outgoing resolvent satisfies $|chi R(k) chi|_{L^2to L^2}leq C{k}^{-1}$ for ${k}>1$, but the constant $C$ has been little studied. We show that, for high frequencies, the constant is bounded above by $2/pi$ times the length of the longest generalized bicharacteristic of $|xi|_g^2-1$ remaining in the support of $chi.$ We show that this estimate is optimal in the case of manifolds without boundary. We then explore the implications of this result for the numerical analysis of the Helmholtz equation.
Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators regularizing in all directions. The gain of regularity is the result of rotating the directions where the corresponding operator performs the smoothing effect. In this paper we carry out a systematic study of the rotational smoothing for a class of operators that includes $k$-vector-space Riesz potentials in $mathbb{R}^n$ with $k < n$, and the convolution with fundamental solutions of elliptic constant-coefficient differential operators acting on $k$-dimensional linear subspaces. Examples of the latter type of operators are the planar Cauchy transform in $mathbb{R}^n$, or a solution operator for the transport equation in $mathbb{R}^n$. The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions.
We obtain new local smoothing estimates for the Euclidean wave equation on $mathbb{R}^{n}$, by replacing the space of initial data by a Hardy space for Fourier integral operators. This improves the bounds in the local smoothing conjecture for $pgeq 2(n+1)/(n-1)$, and complements them for $2<p<2(n+1)/(n-1)$. These estimates are invariant under application of Fourier integral operators.
We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schrodinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^infty$ estimates fail at the critical regularity in high dimensions by using stable Levy process in $R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^infty$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates.
In this article, we prove a stability estimate going from the Radon transform of a function with limited angle-distance data to the $L^p$ norm of the function itself, under some conditions on the support of the function. We apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data.
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