Journal of Fourier Analysis and Applications最新文献

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.

The Turán problem for an open ball of radius r centered at the origin in ({mathbb {R}}^d) consists in computing the supremum of the integrals of positive definite functions compactly supported on that ball and taking the value 1 at the origin. Siegel proved, in the 1930s that this supremum is equal to (2^{-d}) mutiplied by the Lebesgue measure of the ball and is reached by a multiple of the self-convolution of the indicator function of the ball of radius r/2. Several proofs of this result are known and, in this paper, we will provide a new proof of it based on the notion of “dual Turán problem”, a related maximization problem involving positive definite distributions. We provide, in particular, an explicit construction of the Fourier transform of a maximizer for the dual Turán problem. This approach to the problem provides a direct link between certain aspects of the theory of frames in Fourier analysis and the Turán problem. In particular, as an intermediary step needed for our main result, we construct new families of Parseval frames, involving Bessel functions, on the interval [0, 1].

In recent years, sharp or quantitative weighted inequalities have attracted considerable attention on account of the (A_2) conjecture solved by Hytönen. Advances have greatly improved conceptual understanding of classical objects such as Calderón–Zygmund operators. However, plenty of operators do not fit into the class of Calderón–Zygmund operators and fail to be bounded on all (L^p(w)) spaces for (p in (1, infty )) and (w in A_p). In this paper we develop Rubio de Francia extrapolation with quantitative bounds to investigate quantitative weighted inequalities for operators beyond the (multilinear) Calderón–Zygmund theory. We mainly establish a quantitative multilinear limited range extrapolation in terms of exponents (p_i in (mathfrak {p}_i^-, mathfrak {p}_i^+)) and weights (w_i^{p_i} in A_{p_i/mathfrak {p}_i^-} cap RH_{(mathfrak {p}_i^+/p_i)'}), (i=1, ldots , m), which refines a result of Cruz-Uribe and Martell. We also present an extrapolation from multilinear operators to the corresponding commutators. Additionally, our result is quantitative and allows us to extend special quantitative estimates in the Banach space setting to the quasi-Banach space setting. Our proof is based on an off-diagonal extrapolation result with quantitative bounds. Finally, we present various applications to illustrate the utility of extrapolation by concentrating on quantitative weighted estimates for some typical multilinear operators such as bilinear Bochner–Riesz means, bilinear rough singular integrals, and multilinear Fourier multipliers. In the linear case, based on the Littlewood–Paley theory, we include weighted jump and variational inequalities for rough singular integrals.

We prove sufficient conditions for a Calderón–Zygmund operator to belong to the Schatten classes. As in the classical T1 theory, the conditions are given in terms of the smoothness of the operator kernel, and the action of both the operator and its adjoint on the function 1. To show membership to the Schatten class when (p>2) we develop new bump estimates for composed Calderón–Zygmund operators, and a new extension of Carleson’s Embedding Theorem.

This paper is concerned with the oscillatory singular integral operator (T_Q) defined by

where (Q(t)=sum _{1le ile m}a_it^{alpha _i}) is a real-valued polynomial on (mathbb {R}), (Omega ) is a homogenous function of degree zero on (mathbb {R}^n) with mean value zero on the unit sphere (S^{n-1}). Under the assumption of that (Omega in H^1(S^{n-1})), the authors show that (T_Q) is bounded on the weighted Lebesgue spaces (L^p(omega )) for (1<p<infty ) and (omega in tilde{A}_{p}^{I}(mathbb {R}_+)) with the uniform bound only depending on m, the number of monomials in polynomial Q, not on the degree of Q as in the previous results. This result is new even in the case (omega equiv 1), which can also be regarded as an improvement and generalization of the result obtained by Guo in [New York J. Math. 23 (2017), 1733-1738].

In a recent paper, Du and Zhang (Ann Math 189:837–861, 2019) proved a fractal Fourier restriction estimate and used it to establish the sharp (L^2) estimate on the Schrödinger maximal function in (mathbb R^n), (n ge 2). In this paper, we show that the Du–Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in (mathbb R^n) that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in (mathbb R^2).

The ambiguity function (AF) and Wigner distribution (WD) play an important role not only in non-stationary signal processing but also in radar and sonar systems. In this paper, we introduce modified ambiguity function and Wigner distribution associated with quadratic-phase Fourier transform (QAF, QWD). Moreover, many various useful properties of QAF and QWD are also proposed. Marginal properties and Moyal’s formulas of these distributions have elegance and simplicity comparable to those of the AF and WD. Besides, convolutions via quadratic-phase Fourier transform are also introduced. Furthermore, convolution theorems for QAF and QWD are also derived, which seem similar to those of the classical Fourier transform (FT). In addition, applications of QAF and QWD are established such as the detection of the parameters of single-component and multi-component linear frequency-modulated (LFM) signals.

Let G be a locally compact unimodular group, let (1le p<infty ), let (phi in L^infty (G)) and assume that the Fourier multiplier (M_phi ) associated with (phi ) is bounded on the noncommutative (L^p)-space (L^p(VN(G))). Then (M_phi L^p(VN(G))rightarrow L^p(VN(G))) is separating (that is, ({a^*b=ab^*=0}Rightarrow {M_phi (a)^* M_phi (b)=M_phi (a)M_phi (b)^*=0}) for any (a,bin L^p(VN(G)))) if and only if there exists (cin {mathbb {C}}) and a continuous character (psi Grightarrow {mathbb {C}}) such that (phi =cpsi ) locally almost everywhere. This provides a characterization of isometric Fourier multipliers on (L^p(VN(G))), when (pnot =2). Next, let (Omega ) be a (sigma )-finite measure space, let (phi in L^infty (Omega ^2)) and assume that the Schur multiplier associated with (phi ) is bounded on the Schatten space (S^p(L^2(Omega ))). We prove that this multiplier is separating if and only if there exist a constant (cin {mathbb {C}}) and two unitaries (alpha ,beta in L^infty (Omega )) such that (phi (s,t) =c, alpha (s)beta (t)) a.e. on (Omega ^2.) This provides a characterization of isometric Schur multipliers on (S^p(L^2(Omega ))), when (pnot =2).

For (p in [2,infty )), we consider the (L^p rightarrow L^p) boundedness of a Nikodym maximal function associated to a one-parameter family of tubes in ({mathbb {R}}^{d+1}) whose directions are determined by a non-degenerate curve (gamma ) in ({mathbb {R}}^d). These operators arise in the analysis of maximal averages over space curves. The main theorem generalises the known results for (d = 2) and (d = 3) to general dimensions. The key ingredient is an induction scheme motivated by recent work of Ko-Lee-Oh.