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Rotational smoothing

Pedro Caro, Cristóbal J. Meroño, Ioannis Parissis

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.

Rotational smoothing

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 -vector-space Riesz potentials in with , and the convolution with fundamental solutions of elliptic constant-coefficient differential operators acting on -dimensional linear subspaces. Examples of the latter type of operators are the planar Cauchy transform in , or a solution operator for the transport equation in . The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions.

Paper Structure

This paper contains 20 sections, 22 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Rotational smoothing and inverse problems
  4. Outline
  5. More precise statements of rotational smoothing
  6. The tempered distribution qalpha-1
  7. Rotational smoothing in the general case
  8. Some preliminary notions
  9. Proof of main-th-full
  10. Proofs of claims (I)-(III)
  11. The distribution ralpha on Sn-1
  12. Proofs of Theorems loss and counter-ex
  13. Loss of derivatives when averaging in L2(SO(n))
  14. The cancellation property cancellation is necessary
  15. The functional analytical framework
  16. ...and 5 more sections

Key Result

Theorem 1

Assume $n \in \mathbb N$ with $n \geq 2$. Let $k \in \{1, \dots, n - 1\}$ and $\alpha > 0$ be given, and consider $q_\alpha \in C(\mathbb R^k)$ satisfying it:ellipticity, it:homogeneity and it:cancellation. There exists a solution operator $S_\alpha$ and a constant $C > 0$ depending on $n$, $k$, $\a with $\delta = \lceil \alpha - k \rceil$ if $\alpha - k/ 2 < \lceil \alpha - k \rceil$, and $\delta

Theorems & Definitions (44)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 2.1
  • proof : Proof of the identity \ref{['id:easy_sec']}.
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof : Proof of \ref{['lem:non_pos_integers']}
  • proof : Proof of \ref{['lem:pos_integers']}
  • ...and 34 more