Journal of Fourier Analysis and Applications最新文献

We prove ({mathcal {M}})-regularity for a class of pseudodifferential operators in ultradifferentiable classes defined on the torus (mathbb {T}^{m+n}) which are globally (C^{infty }) hypoelliptic. The same property is also valid for certain perturbations of these operators by lower order terms.

Abstract

Let (gamma _0=frac{sqrt{5}-1}{2}=0.618ldots ) . We prove that, for any (varepsilon >0) and any trigonometric polynomial f with frequencies in the set ({n^2: N leqslant nleqslant N+N^{gamma _0-varepsilon }}) , the inequality $$begin{aligned} Vert fVert _4 ll varepsilon ^{-1/4}Vert fVert _2 end{aligned}$$ holds, which makes a progress on a conjecture of Cilleruelo and Córdoba. We also present a connection between this conjecture and the conjecture of Ruzsa which asserts that, for any (varepsilon >0) , there is (C(varepsilon )>0) such that each positive integer N has at most (C(varepsilon )) divisors in the interval ([N^{1/2}, N^{1/2}+N^{1/2-varepsilon }]) .

Hardy’s inequality on (H^p) spaces, (pin (0,1]), in the context of orthogonal expansions is investigated for general bases on a wide class of domains in (mathbb {R}^d) with Lebesgue measure. The obtained result is applied to various Hermite, Laguerre, and Jacobi expansions. For that purpose some delicate estimates of the higher order derivatives for the underlying functions and of the associated heat or Poison kernels are proved. Moreover, sharpness of studied Hardy’s inequalities is justified by a construction of an explicit counterexample, which is adjusted to all considered settings.

We study the (L^prightarrow L^q)-type uniform resolvent estimate for Laplace –Beltrami operator on the flat Euclidean cone (C(mathbb {S}_{sigma }^1)triangleq mathbb {R}_{+}times (mathbb {R}/2pi sigma mathbb {Z})) equipped with the metric (g(r,theta )=dr^2+r^2dtheta ^2) where the circle of radius (sigma >0). The key ingredient is the resolvent kernel constructed by Zhang in (J Funct Anal 282(3):109311, 2022) and the Young inequality holds under the monotonicity assumption on the flat Euclidean cone.

The fundamental solution to a pseudo-differential equation for functions defined on the d-fold product of the p-adic numbers, (mathbb {Q}_p), induces an analogue of the Wiener process in (mathbb {Q}_p^d). As in the real setting, the components are 1-dimensional p-adic Brownian motions with the same diffusion constant and exponent as the original process. Asymptotic analysis of the conditional probabilities shows that the vector components are dependent for all time. Exit time probabilities for the higher dimensional processes reveal a concrete effect of the component dependency.

In this paper we study the (L^p)-(L^q) boundedness of the Fourier multipliers in the setting where the underlying Fourier analysis is introduced with respect to the eigenfunctions of an anharmonic oscillator A. Using the notion of a global symbol that arises from this analysis, we extend a version of the Hausdorff–Young–Paley inequality that guarantees the (L^p)-(L^q) boundedness of these operators for the range (1<p le 2 le q <infty ). The boundedness results for spectral multipliers acquired, yield as particular cases Sobolev embedding theorems and time asymptotics for the (L^p)-(L^q) norms of the heat kernel associated with the anharmonic oscillator. Additionally, we consider functions f(A) of the anharmonic oscillator on modulation spaces and prove that Linskĭi’s trace formula holds true even when f(A) is simply a nuclear operator.

Inspired by the recent article Skrettingland (J. Fourier Anal. Appl. 28(2), 1–34 (2022)), this paper is devoted to the study of a suitable class of windows in the framework of bounded linear operators on (L^2({{mathbb {R}}}^{d})). We establish a natural and complete characterization for the window class such that the corresponding STFT leads to equivalent norms on modulation spaces. The positive bounded linear operators are also characterized by its Cohen’s class distributions such that the corresponding quantities form equivalent norms on modulation spaces. As a generalization, we introduce a family of operator classes corresponding to the operator-valued modulation spaces. Some applications of our main theorems to the localization operators are also concerned.

We discuss some variants of the Berry–Esseen inequality in terms of Lyapunov coefficients which may provide sharp rates of normal approximation.