Searcharxiv⌕ Search

arXiv subjects

Jon Johnsen

Publications and source records attributed to Jon Johnsen.

29 records · Page 2Linked to original sources
arXiv2017

A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin--Triebel spaces with mixed norms

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general Lizorkin--Triebel spaces based on mixed $L_p$-norms. In this context a Nikol'skij--Plancherel--Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to spaces of the quasi-homogeneous type.

math.AP↗
arXiv2017

On the trace problem for Triebel--Lizorkin spaces with mixed norms

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by the first and last coordinates are applied to functions belonging to quasi-homogeneous, mixed-norm Lizorkin--Triebel spaces; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised except for the borderline cases; these are also covered in case of the first variable. With respect to the first variable the trace spaces are proved to be mixed-norm Lizorkin--Triebel spaces with a specific sum exponent. For the last variable they are similarly defined Besov spaces. The treatment includes continuous right-inverses and higher order traces. The results rely on a sequence version of Nikolskij's inequality, Marschall's inequality for pseudo-differential operators (and Fourier multiplier assertions), as well as dyadic ball criteria.

math.AP↗
arXiv2017

Type 1,1-operators defined by vanishing frequency modulation

This paper presents a general definition of pseudo-differential operators of type $1,1$; the definition is shown to be the largest one that is both compatible with negligible operators and stable under vanishing frequency modulation. Elaborating counter-examples of Ching, Hörmander and Parenti--Rodino, type $1,1$-operators with unclosable graphs are proved to exist; others are shown to lack the microlocal property as they flip the wavefront set of an almost nowhere differentiable function. In contrast the definition is shown to imply the pseudo-local property, so type 1,1-operators cannot create singularities but only change their nature. The familiar rule that the support of the argument is transported by the support of the distribution kernel is generalised to arbitrary type $1,1$-operators. A similar spectral support rule is also proved. As no restrictions appear for classical type $1,0$-operators, this is a new result which in many cases makes it unnecessary to reduce to elementary symbols. As an important tool, a convergent sequence of distributions is said to converge regularly if it moreover converges as smooth functions outside the singular support of the limit. This notion is shown to allow limit processes in extended versions of the formula relating operators and kernels.

math.AP↗
arXiv2016

Parametrices and exact paralinearisation of semi-linear boundary problems

The subject is parametrices for semi-linear problems, based on parametrices for linear boundary problems and on non-linearities that decompose into solution-dependent linear operators acting on the solutions. Non-linearities of product type are shown to admit this via exact paralinearisation. The parametrices give regularity properties under weak conditions; improvements in subdomains result from pseudo-locality of type $1,1$-operators. The framework encompasses a broad class of boundary problems in Hölder and $L_p$-Sobolev spaces (and also Besov and Lizorkin--Triebel spaces). The Besov analyses of homogeneous distributions, tensor products and halfspace extensions have been revised. Examples include the von Karman equation.

math.AP↗
arXiv2016

Simple proofs of nowhere-differentiability for Weierstrass's function and cases of slow growth

Using a few basics from integration theory, a short proof of nowhere-differentiability of Weierstrass functions is given. Restated in terms of the Fourier transformation, the method consists in principle of a second microlocalisation, which is used to derive two general results on existence of nowhere differentiable functions. Examples are given in which the frequencies are of polynomial growth and of almost quadratic growth as a borderline case.

math.CA↗
arXiv2016

$L_p$-Theory of Type $1,1$-Operators

This is a continuation of recent work on the general definition of pseudo-differential operators of type $1,1$, in Hörmander's sense. Continuity in $L_p$-Sobolev spaces and Hölder--Zygmund spaces, and more generally in Besov and Lizorkin--Triebel spaces, is proved for positive smoothness, with extension to arbitrary smoothness for operators in the self-adjoint subclass. As a main tool the paradifferential decomposition is used for type $1,1$-operators in combination with the Spectral Support Rule for pseudo-differential operators and pointwise estimates in terms of maximal functions of Peetre--Fefferman--Stein type.

math.AP↗
arXiv2016

Pointwise estimates of pseudo-differential operators

As a new technique it is shown how general pseudo-differential operators can be estimated at arbitrary points in Euclidean space when acting on functions $u$ with compact spectra. The estimate is a factorisation inequality, in which one factor is the Peetre--Fefferman--Stein maximal function of $u$, whilst the other is a symbol factor carrying the whole information on the symbol. The symbol factor is estimated in terms of the spectral radius of $u$, so that the framework is well suited for Littlewood--Paley analysis. It is also shown how it gives easy access to results on polynomial bounds and estimates in $L_p$, including a new result for type $1,1$-operators that they are always bounded on $L_p$-functions with compact spectra.

math.AP↗
arXiv2016

Characterisation by Local Means of Anisotropic Lizorkin--Triebel Spaces with Mixed Norms

This is a contribution to the theory of Lizorkin--Triebel spaces having mixed Lebesgue norms and quasi-homogeneous smoothness. We discuss their characterisation in terms of general quasi-norms based on convolutions. In particular, this covers the case of local means, in Triebel's terminology. The main step is an extension of some crucial inequalities due to Rychkov to the case with mixed norms.

math.FA↗
arXiv2016

Anisotropic Lizorkin--Triebel Spaces with Mixed Norms --- Traces on Smooth Boundaries

This article deals with trace operators on anisotropic Lizorkin--Triebel spaces with mixed norms over cylindrical domains with smooth boundary. As a preparation we include a rather self-contained exposition of Lizorkin--Triebel spaces on manifolds and extend these results to mixed-norm Lizorkin--Triebel spaces on cylinders in Euclidean space. In addition Rychkov's universal extension operator for a half space is shown to be bounded with respect to the mixed norms, and a support preserving right-inverse of the trace is given explicitly and proved to be continuous in the scale of mixed-norm Lizorkin--Triebel spaces. As an application, the heat equation is considered in these spaces, and the necessary compatibility conditions on the data are deduced.

math.AP↗
arXiv2016

Fundamental Results for Pseudo-Differential Operators of Type $\mathbf{1},\mathbf{1}$

This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type $1,1$ in Hörmander's sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such operators. It is shown that type $1,1$-operators are defined and continuous on the full space of temperate distributions, if they fulfil Hörmander's twisted diagonal condition, or more generally if they belong to the self-adjoint subclass; and that they are always defined on the temperate smooth functions. As a main tool the paradifferential decomposition is derived for type $1,1$-operators, and to confirm a natural hypothesis the symmetric term is shown to cause the domain restrictions; whereas the other terms are shown to define nice type $1,1$-operators fulfilling the twisted diagonal condition. The decomposition is analysed in the type $1,1$-context by combining the Spectral Support Rule and the factorisation inequality, which gives pointwise estimates of pseudo-differential operators in terms of maximal functions.

math.AP↗