Journal of Fourier Analysis and Applications最新文献

In this paper we use some ideas from [12, 13] and consider the description of Hörmander type pseudo-differential operators on (mathbb {R}^d) ((dge 1)), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals’ commutator criterion, and also establish local trace-class criteria.

We introduce a family of generalized Gelfand pairs ((K_m,N_m)) where (N_m) is an (m+2)-step nilpotent Lie group and (K_m) is isomorphic to the 3-dimensional Heisenberg group. We develop the associated spherical analysis computing the set of the spherical distributions and we obtain some results on the algebra of (K_m)-invariant and left invariant differential operators on (N_m).

The ring of periodic distributions on (mathbb {R}^{texttt {d}}) with usual addition of distributions, and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring (mathcal {S}'(mathbb {Z}^{texttt {d}})) of all maps (f:mathbb {Z}^{texttt {d}}rightarrow mathbb {C}) of at most polynomial growth (that is, there exist a real number (M>0) and an integer (texttt {m}ge 0) such that ( |f(varvec{n})|le M(1+|texttt{n}_1|+cdots +|texttt {n}_{texttt {d}}|)^{texttt {m}}) for all (varvec{n}=(texttt{n}_1,cdots , texttt {n}_{texttt {d}})in mathbb {Z}^{texttt {d}})), with pointwise operations. It is shown that finitely generated ideals in (mathcal {S}'(mathbb {Z}^{texttt {d}})) are principal, and ideal membership is characterised analytically. Calling an ideal in (mathcal {S}'(mathbb {Z}^texttt{d})) fixed if there is a common index (varvec{n}in mathbb {Z}^{texttt {d}}) where each member vanishes, the fixed maximal ideals are described, and it is shown that not all maximal ideals are fixed. It is shown that finitely generated (hence principal) prime ideals in (mathcal {S}'(mathbb {Z}^{texttt {d}})) are fixed maximal ideals. The Krull dimension of (mathcal {S}'(mathbb {Z}^{texttt {d}})) is proved to be infinite, while the weak Krull dimension is shown to be equal to 1.

Abstract

Let V be a variety in (mathbb {F}_q^d) and (Esubset V) . It is known that if any line passing through the origin contains a bounded number of points from E, then (left| prod (E) right| =|{xcdot y:x, yin E}|gg q) whenever (|E|gg q^{frac{d}{2}}) . In this paper, we show that the barrier (frac{d}{2}) can be broken when V is a paraboloid in some specific dimensions. The main novelty in our approach is to link this question to the distance problem in one lower dimensional vector space, allowing us to use recent developments in this area to obtain improvements.

Let G be an additive and finite Abelian group, and p a prime number that does not divide the order of G. We show that if G has the universal spectrum property, then so does (Gtimes {mathbb {Z}}_p). This is similar to and extends a previous result for cyclic groups using the same dilation trick but on non-cyclic groups as well. Inductively applying this statement on the known list of permissible G one can replace p with any square-free number that does not divide the order of G, and produce new tiling to spectral results in finite Abelian groups generated by at most two elements.

The purpose of this paper is to prove an optimal restriction estimate for a class of flat curves in ({mathbb {R}} ^d), (dge 3). Namely, we consider the problem of determining all the pairs (p, q) for which the (L^p-L^q) estimate holds (or a suitable Lorentz norm substitute at the endpoint, where the (L^p-L^q) estimate fails) for the extension operator associated to (gamma (t) = (t, {frac{t^2}{2!}}, ldots , {frac{t^{d-1}}{(d-1)!}}, phi (t))), (0le tle 1), with respect to the affine arclength measure. In particular, we are interested in the flat case, i.e. when (phi (t)) satisfies (phi ^{(d)}(0) = 0) for all integers (dge 1). A prototypical example is given by (phi (t) = e^{-1/t}). The paper (Bak et al., J. Aust. Math. Soc. 85:1–28, 2008) addressed precisely this problem. The examples in Bak et al. (2008) are defined recursively in terms of an integral, and they represent progressively flatter curves. Although these include arbitrarily flat curves, it is not clear if they cover, for instance, the prototypical case (phi (t) = e^{-1/t}). We will show that the desired estimate does hold for that example and indeed for a class of examples satisfying some hypotheses involving a log-concavity condition.

In this paper, we consider a weighted version of one-dimensional discrete Hardy inequalities with power weights of the form (n^alpha ). We prove the inequality when (alpha ) is an even natural number with the sharp constant and remainder terms. We also find explicit constants in standard and weighted Rellich inequalities(with weights (n^alpha )) which are asymptotically sharp as (alpha rightarrow infty ). As a by-product of this work we derive a combinatorial identity using purely analytic methods, which suggests a plausible correlation between combinatorial and functional identities.

We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs. After recasting the corresponding boundary value problems as boundary integral equations, we prove that their solutions depend holomorphically upon perturbations of the arcs’ parametrizations. These results are key to prove the shape (domain) holomorphy of domain-to-solution maps associated to boundary integral equations appearing in uncertainty quantification, inverse problems and deep learning, to name a few applications.

Fix (left{ a_1, dots , a_n right} subset {mathbb {N}}), and let x be a uniformly distributed random variable on ([0,2pi ]). The probability ({mathbb {P}}(a_1,ldots ,a_n)) that (cos (a_1 x), dots , cos (a_n x)) are either all positive or all negative is non-zero since (cos (a_i x) sim 1) for x in a neighborhood of 0. We are interested in how small this probability can be. Motivated by a problem in spectral theory, Goncalves, Oliveira e Silva, and Steinerberger proved that ({mathbb {P}}(a_1,a_2) ge 1/3) with equality if and only if (left{ a_1, a_2 right} = gcd (a_1, a_2)cdot left{ 1, 3right} ). We prove ({mathbb {P}}(a_1,a_2,a_3)ge 1/9) with equality if and only if (left{ a_1, a_2, a_3 right} = gcd (a_1, a_2, a_3)cdot left{ 1, 3, 9right} ). The pattern does not continue, as (left{ 1,3,11,33right} ) achieves a smaller value than (left{ 1,3,9,27right} ). We conjecture multiples of (left{ 1,3,11,33right} ) to be optimal for (n=4), discuss implications for eigenfunctions of Schrödinger operators (-Delta + V), and give an interpretation of the problem in terms of the lonely runner problem.

In this paper we construct two new families of invariant maps that separate the orbits of the action of a finite Abelian group on a finite dimensional complex vector space. One of these families is Lipschitz continuous with respect to the quotient metric on the space of orbits, but involves computing large powers of the components of the vectors which can lead to instabilities. The other family avoids this issue by putting the powers only on the phase of the components, but in turn is not continuous. However, we show that they are Lipschitz continuous on the set of vectors with fixed support, so in particular they are Lipschitz on the set of vectors with no zero entries. Furthermore, the target dimension of these maps is small, i.e., linear in the original dimension.