Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRotational smoothing
129
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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.
Several results on constrained spline smoothing are obtained. In particular, we establish a general result, showing how one can constructively smooth any monotone or convex piecewise polynomial function (ppf) (or any $q$-monotone ppf, $qgeq 3$, with one additional degree of smoothness) to be of minimal defect while keeping it close to the original function in the ${mathbb L}_p$-(quasi)norm. It is well known that approximating a function by ppfs of minimal defect (splines) avoids introduction of artifacts which may be unrelated to the original function, thus it is always preferable. On the other hand, it is usually easier to construct constrained ppfs with as little requirements on smoothness as possible. Our results allow to obtain shape-preserving splines of minimal defect with equidistant or Chebyshev knots. The validity of the corresponding Jackson-type estimates for shape-preserving spline approximation is summarized, in particular we show, that the ${mathbb L}_p$-estimates, $pge1$, can be immediately derived from the ${mathbb L}_infty$-estimates.
In this work we consider an example of a linear time-degenerate Schrodinger operator. We show that with the appropriate assumptions the operator satisfies a Kato smoothing effect. We also show that the solutions to the nonlinear initial value problems involving this operator and polynomial derivative nonlinearities are locally well-posed and their solutions also satisfy the same smoothing estimates as the linear solutions.
We consider a family of spherically symmetric, asymptotically Euclidean manifolds with two trapped sets, one which is unstable and one which is semi-stable. The phase space structure is that of an inflection transmission set. We prove a sharp local smoothing estimate for the linear Schrodinger equation with a loss which depends on how flat the manifold is near each of the trapped sets. The result interpolates between the family of similar estimates in cite{ChWu-lsm}. As a consequence of the techniques of proof, we also show a sharp high energy resolvent estimate with a polynomial loss depending on how flat the manifold is near each of the trapped sets.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
