arXiv subjects
Publications and source records attributed to László Székelyhidi.
We confirm a conjecture posed by T. Iwaniec in 1982: the $L^p$ norm of the Beurling--Ahlfors transform is $p-1$ if $p\geq 2$, and $(p-1)^{-1}$ if $1<p\leq 2$.
In an earlier paper we solved a long-standing problem which goes back to Laurent Schwartz's work on mean-periodic functions. Namely, we completely characterised those locally compact Abelian groups having spectral synthesis. The method is based on the localisation concept. In this paper we show that localisation can be used to prove another basic result in spectral synthesis: the principal ideal theorem.
In his classical paper, Laurent Schwartz proved that on the real line, in every linear translation invariant space of continuous complex valued functions, which is closed under compact convergence the exponential monomials span a dense subspace. He studied so-called local ideals in the space of Fourier transforms, and his proof based on the observation that, on the one hand, these local ideals are completely determined by the exponential monomials in the space, and, on the other hand, these local ideals completely determine the space itself. On the other hand, Dimitri Gurevich gave counterexamples for Schwartz's theorem in higher dimension. In this paper we show that the ideas of localisation can be extended to general locally compact Abelian groups using abstract derivations on the Fourier algebra of compactly supported measures. Based on this method we present necessary and sufficient conditions for spectral synthesis for varieties on locally compact Abelian groups. Using localisation, in \cite{MR4789359} we proved that spectral synthesis holds on a locally compact Abelian group $G$ if and only if it holds on $G/B$, where $B$ is the closed subgroup of compact elements. This may lead to a complete characterisation of locally compact Abelian groups having spectral synthesis.
In this paper we solve a long-standing problem which goes back to Laurent Schwartz's work on mean periodic functions. Namely, we completely characterise those locally compact Abelian groups having spectral synthesis. So far a characterisation theorem was available for discrete Abelian groups only. Here we use a kind of localisation concept for the ideals of the Fourier algebra of the underlying group. We show that localisability of ideals is equivalent to synthesizability. Based on this equivalence we show that if spectral synthesis holds on each extension of it by a locally compact Abelian group consisting of compact elements, and also on any extension to a direct sum with a copy of the integers. Then, using Schwartz's result and Gurevich's counterexamples we apply the structure theory of locally compact Abelian groups.
Harmonic synthesis describes translation invariant linear spaces of continuous complex valued functions on locally compact abelian groups. The basic result due to L. Schwartz states that such spaces on the reals are topologically generated by the exponential monomials in the space -- in other words the locally compact abelian group of the reals is synthesizable. This result does not hold for continuous functions in several real variables as it was shown by D.I. Gurevich's counterexamples. On the other hand, if two discrete abelian groups have this synthesizability property, then so does their direct sum, as well. In this paper we show that if two locally compact abelian groups have this synthesizability property and at least one of them is discrete, then their direct sum is synthesizable. In fact, more generally, we show that any extension of a synthesizable locally compact abelian group by a synthesizable discrete abelian group is synthesizable. This is an important step toward the complete characterisation of synthesizable locally compact abelian groups.
In this paper we show that spectral analysis implies spectral synthesis for arbitrary varieties on locally compact Abelian groups, which have no discrete subgroup of infinite torsion free rank..
Recently we introduced the concept of localisability of ideals in the Fourier algebra of locally compact Abelian groups. It turns out that localisability can be used to characterise synthesisability of varieties. Based on this we show that spectral synthesis holds on continuous images of varieties which have spectral synthesis.
In our former paper we introduced the concept of localisation of ideals in the Fourier algebra of a locally compact Abelian group. It turns out that localisability of a closed ideal in the Fourier algebra is equivalent to the synthesisability of the annihilator of that closed ideal which corresponds to this ideal in the measure algebra. This equivalence provides an effective tool to prove synthesisability of varieties on locally compact Abelian groups. In this paper we utilise this tool to show that when investigating synthesisability of a variety, roughly speaking compact elements of the group can be neglected. Our main result is that spectral synthesis holds on a locally compact Abelian group $G$ if and only if it holds on $G/B$, where $B$ is the closed subgroup of all compact elements. In particular, spectral synthesis holds on compact Abelian groups. Also we obtain a simple proof for the characterisation theorem of spectral synthesis on discrete Abelian groups.
In a former paper we introduced the concept of localisation of ideals in the Fourier algebra of a locally compact Abelian group. It turns out that localisability of a closed ideal in the Fourier algebra is equivalent to the synthesisability of the annihilator of that closed ideal which corresponds to this ideal in the measure algebra. This equivalence provides an effective tool to prove synthesisability of varieties on locally compact Abelian groups. In another paper we used this method to show that when investigating synthesisability of a variety, roughly speaking, compact elements of the group can be neglected. Using these results, in this paper we completely characterise those locally compact Abelian groups on which spectral synthesis holds.
In one of our former papers {\it Endomorphisms of the measure algebra of commutative hypergroups arXiv:2204.07499 we considered exponential monomials on hypergroups and higher order derivations of the corresponding measure algebra. Continuing with this, we are now looking for the connection between the generalized exponential polynomials of a commutative hypergroup and the higher order derivations of the corresponding measure algebra.
Endomorphisms of the measure algebra of commutative hypergroups are investigated. We focus on derivations and higher order derivations which are closely related to moment function sequences of higher rank. We describe the exact connection between those higher order derivations which are endomorphisms of the measure algebra if it is considered as a module over the ring of continuous functions.
In this paper we consider generalized moment functions of higher order. These functions are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. In our former paper we investigated the properties of generalized moment functions of higher order on commutative groups. In particular, we proved the characterization of generalized moment functions on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and a sequence additive functions. In the present paper we continue the study of generalized moment function sequences of higher order in the more abstract setting, namely we consider functions defined on a hypergroup. We characterize these functions on the polynomial hypergroup in one variable by means of partial derivatives of a composition of polynomials generating the polynomial hypergroup and an analytic function. As an example, we give an explicit formula for moment generating functions of rank at most two on the Tchebyshev hypergroup.
We apply the technique of convex integration to obtain non-uniqueness and existence results for power-law fluids, in dimension $d\ge 2$. For the power index $q$ below the compactness threshold, i.e. $q \in (1, \frac{2d}{d+2})$, we show ill-posedness of Leray-Hopf solutions. For a wider class of indices $q \in (1, \frac{3d+2}{d+2})$ we show ill-posedness of distributional (non-Leray-Hopf) solutions, extending the seminal paper of Buckmaster and Vicol. In this wider class we also construct non-unique solutions for every datum in $L^2$.
The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its variety is one dimensional, then this exponential monomial is a linear combination of generalized moment functions.
In this paper we continue the discussion about relations between exponential polynomials and generalized moment generating functions on a commutative hypergroup. We are interested in the following problem: is it true that every finite dimensional variety is spanned by moment functions? Let $m$ be an exponential on $X$. In our former paper we have proved that if the linear space of all $m$-sine functions in the variety of an $m$-exponential monomial is (at most) one dimensional, then this variety is spanned by moment functions generated by $m$. In this paper we show that this may happen also in cases where the $m$-sine functions span a more than one dimensional subspace in the variety. We recall the notion of a polynomial hypergroup in $d$ variables, describe exponentials on it and give the characterization of the so called $m$-sine functions. Next we show that the Fourier algebra of a polynomial hypergroup in $d$ variables is the polynomial ring in $d$ variables. Finally, using Ehrenpreis--Palamodov Theorem we show that every exponential polynomial on the polynomial hypergroup in $d$ variables is a linear combination of moment functions contained in its variety.
The main purpose of this work is to prove characterization theorems for generalized moment functions on groups. According one of the main results these are exponential polynomials that can be described with the aid of complete (exponential) Bell polynomials. These characterizations will be immediate consequences of our main result about the characterization of generalized moment functions of higher rank.
In this paper we make an attempt to extend L. Schwartz's classical result on spectral synthesis to several dimensions. Due to counterexamples of D. I. Gurevich this is impossible for translation invariant varieties. Our idea is to replace translations by proper euclidean motions in higher dimensions. For this purpose we introduce the basic concepts of spectral analysis and synthesis in the non-commutative setting based on Gelfand pairs, where "translation invariance" will be replaced by invariance with respect to a compact group of automorphisms. The role of exponential functions will be played by spherical functions. As an application we obtain the extension of L. Schwartz's fundamental result.
In a recent paper we introduced sine functions on commutative hypergroups. These functions are natural generalizations of those functions on groups which are products of additive and multiplicative homomorphisms. In this paper we describe sine functions on different types of hypergroups, including polynomial hypergroups, Sturm--Liouville hypergroups, etc. A non-commutative hypergroup is also considered.