Riesz Means Research Articles

Bochner–Riesz means at the critical index: weighted and sparse bounds

Bochner–Riesz means at the critical index: weighted and sparse bounds

Almost everywhere convergence of Bochner–Riesz means for the twisted Laplacian

Let L \mathcal L denote the twisted Laplacian in C d \mathbb {C}^d . We study almost everywhere convergence of the Bochner–Riesz mean S t δ ( L ) f S^\delta _{t}(\mathcal L) f of f ∈ L p ( C d ) f\in L^p(\mathbb C^d) as t → ∞ t\to \infty , which is an expansion of f f in the special Hermite functions. For 2 ≤ p ≤ ∞ 2\le p\le \infty , we obtain the sharp range of the summability indices δ \delta for which the convergence of S t δ ( L ) f S^\delta _{t}(\mathcal L) f holds for all f ∈ L p ( C d ) f\in L^p(\mathbb {C}^d) .

Sharp Semiclassical Spectral Asymptotics for Local Magnetic Schrödinger Operators on $${\mathbb {R}}^d$$ Without Full Regularity

AbstractWe consider operators acting in $$L^2({\mathbb {R}}^d)$$ L 2 ( R d ) with $$d\ge 3$$ d ≥ 3 that locally behave as a magnetic Schrödinger operator. For the magnetic Schrödinger operators, we suppose the magnetic potentials are smooth and the electric potential is five times differentiable and the fifth derivatives are Hölder continuous. Under these assumptions, we establish sharp spectral asymptotics for localised counting functions and Riesz means.

A Note on $$L^1$$-Convergence of Fourier Series with Riesz Mean

A Note on $$L^1$$-Convergence of Fourier Series with Riesz Mean

Riesz means and bilinear Riesz means on H-type groups

In this paper, we investigate the Riesz means Sδ and the bilinear Riesz means Sα associated to the sublaplacian on H-type groups. We obtain the Lp-boundedness of Sδ by using the restriction theorem on H-type groups. Our result is different from that on Heisenberg groups. We prove that Sα is bounded from Lp1×Lp2 into Lp for 1≤p1,p2≤∞ and 1/p=1/p1+1/p2 when α is larger than a suitable smoothness index α(p1,p2). Because we consider H-type groups with the center dimension larger than one, it is necessary to use some different techniques from that for Heisenberg groups.

Optimal semiclassical spectral asymptotics for differential operators with non-smooth coefficients

We consider differential operators defined as Friedrichs extensions of quadratic forms with non-smooth coefficients. We prove a two-term optimal asymptotic for the Riesz means of these operators and thereby also reprove an optimal Weyl law under certain regularity conditions. The methods used are then extended to consider more general admissible operators perturbed by a rough differential operator and to obtain optimal spectral asymptotics again under certain regularity conditions. For the Weyl law, we assume that the coefficients are differentiable with Hölder continuous derivatives, while for the Riesz means we assume that the coefficients are twice differentiable with Hölder continuous derivatives.

Limited Range Extrapolation with Quantitative Bounds and Applications

In recent years, sharp or quantitative weighted inequalities have attracted considerable attention on account of the A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_2$$\\end{document} 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 Lp(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p(w)$$\\end{document} spaces for p∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in (1, \\infty )$$\\end{document} and w∈Ap\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w \\in A_p$$\\end{document}. 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 pi∈(pi-,pi+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_i \\in (\\mathfrak {p}_i^-, \\mathfrak {p}_i^+)$$\\end{document} and weights wipi∈Api/pi-∩RH(pi+/pi)′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w_i^{p_i} \\in A_{p_i/\\mathfrak {p}_i^-} \\cap RH_{(\\mathfrak {p}_i^+/p_i)'}$$\\end{document}, i=1,…,m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1, \\ldots , m$$\\end{document}, 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.

Geometric bounds for the magnetic Neumann eigenvalues in the plane

We consider the eigenvalues of the magnetic Laplacian on a bounded domain Ω of R2 with uniform magnetic field β>0 and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy λ1 and we provide semiclassical estimates in the spirit of Kröger for the first Riesz mean of the eigenvalues. We also discuss upper bounds for the first eigenvalue for non-constant magnetic fields β=β(x) on a simply connected domain in a Riemannian surface.In particular: we prove the upper bound λ1<β for a general plane domain for a constant magnetic field, and the upper bound λ1<maxx∈Ω‾⁡|β(x)| for a variable magnetic field when Ω is simply connected.For smooth domains, we prove a lower bound of λ1 depending only on the intensity of the magnetic field β and the rolling radius of the domain.The estimates on the Riesz mean imply an upper bound for the averages of the first k eigenvalues which is sharp when k→∞ and consists of the semiclassical limit 2πk|Ω| plus an oscillating term.We also construct several examples, showing the importance of the topology: in particular we show that an arbitrarily small tubular neighborhood of a generic simple closed curve has lowest eigenvalue bounded away from zero, contrary to the case of a simply connected domain of small area, for which λ1 is always small.

Ω-bounds for the partial sums of some modified Dirichlet characters

Abstract We consider the problem of Ω bounds for the partial sums of a modified character, i.e., a completely multiplicative function f such that $f(p)=\chi(p)$ for all but a finite number of primes p, where χ is a primitive Dirichlet character. We prove that in some special circumstances, $\sum_{n\leq x}f(n)=\Omega((\log x)^{|S|})$, where S is the set of primes p, where $f(p)\neq \chi(p)$. This gives credence to a corrected version of a conjecture of Klurman et al., Trans. Amer. Math. Soc., 374 (11), 2021, 7967–7990. We also compute the Riesz mean of order k for large k of a modified character and show that the Diophantine properties of the irrational numbers of the form $\log p / \log q$, for primes p and q, give information on these averages.

On the Pinsky Phenomenon for B-Elliptic Operators

Necessary conditions for the summability of spectral expansions in eigenfunctions of an elliptic operator with a Bessel operator in one of the variables in an arbitrary-dimensional domain adjacent to the singularity hypersurface are obtained. It is proved that if the spectral expansion of an arbitrary function at some point of this hypersurface is summable by Riesz means, then its average value over the half-ball centered at the specified point has generalized smoothness.

Calderón–Zygmund operators, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents

We obtain the boundedness of Calderón–Zygmund operators on the entire Hardy–Morrey spaces with variable exponents. We obtain this result by refining the extrapolation theory. This refined extrapolation theory also gives the boundedness of Fourier multipliers, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents.

Almost Everywhere Convergence of Bochner–Riesz Means on Hardy–Sobolev Spaces

We investigate the convergence of Bochner–Riesz means on Hardy–Sobolev spaces. The relation between the smoothness imposed on functions and the rate of almost everywhere convergence of the generalized Bochner–Riesz means is given.

Scaling a theorem of Harald Bohr

A outstanding result of H. Bohr shows that under a certain condition on the frequency λ=(λn) (preventing the λn's from getting too close too fast) every λ-Dirichlet series D=∑ane−λns converges uniformly on all half-planes [Re>σ] with σ>0, provided D is pointwise convergent on some half-plane and has a limit function extending to a bounded holomorphic function f on [Re>0]. Riesz summability plays an important role within the classical literature of general Dirichlet series - recall that a Dirichlet series D=∑ane−λns is said to be Riesz summable of order ℓ≥0 in s∈C, whenever the sequence of all Riesz means ∑λn<xan(1−λnx)ℓe−λns converges as x tends to infinity. Extending classical work of M. Riesz, we for a given ℓ≥0 isolate and study the class of those frequencies λ, which have the following property: If the Dirichlet series D=∑ane−λns is pointwise Riesz summable for some order on some half-plane and has a limit function extending to a holomorphic function f on [Re>0] with growth of order O((1+s)ℓ), then uniformly on all half-planes [Re>σ],σ>0, we have f(s)(1+s)ℓ=1(1+s)ℓlimx→∞⁡∑λn<xan(1−λnx)ℓe−λns. Among others, we show that this holds true if λ satisfies a well-known condition isolated by E. Landau, which recovers the classical bounded case ℓ=0.

The short interval results for power moments of the Riesz mean error term

&lt;abstract&gt;&lt;p&gt;Let $ \Delta_1(x; \varphi) $ denote the error term in the classical Rankin-Selberg problem. In this paper, our main results are getting the $ k $-th $ (3\leq k\leq5) $ power moments of $ \Delta_1(x; \varphi) $ in short intervals and its asymptotic formula by using large value arguments.&lt;/p&gt;&lt;/abstract&gt;

Local properties of fourier series via deferred Riesz mean

The convergence of Fourier series of a function at a point depends upon the behaviour of the function in the neighborhood of that point, and it leads to the local property of Fourier series. In the proposed work, we introduce and study the absolute convergence of the deferred Riesz summability mean, and accordingly establish a new theorem on the local property of a factored Fourier series. We also suggest a direction for future researches on this subject, which are based upon the local properties of the Fourier series via basic notions of statistical absolute convergence.

On the Bilinear Bochner–Riesz Problem at Critical Index

In this paper, we study maximal and square functions associated with bilinear Bochner–Riesz means at the critical index. In particular, we prove that they satisfy weighted estimates from \(L^{p_1}(w_1)\times L^{p_2}(w_2)\rightarrow L^p(v_w)\) for bilinear weights \((w_1,w_2)\in A_{\textbf{P}}\) where \(p_1,p_2>1\) and \(\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}\). Also, we show that both the operators fail to satisfy weak-type estimates at the end-point\(\big (1,1,\frac{1}{2}\big )\).

A new approach for SPN removal: nearest value based mean filter.

In this study, a new adaptive filter is proposed to eliminate salt and pepper noise (SPN). The basis of the proposed method consists of two-stages. (1) Changing the noisy pixel value with the closest pixel value or assigning their average to the noisy pixel in case there is more than one pixel with the same distance; (2) the updating of the calculated noisy pixel values with the average filter by correlating them with the noise ratio. The method developed was named as Nearest Value Based Mean Filter (NVBMF), because of using the pixel value which the closest distance in the first stage. Results obtained with the proposed method: it has been compared with the results obtained with the Adaptive Frequency Median Filter, Adaptive Riesz Mean Filter, Improved Adaptive Weighted Mean Filter, Adaptive Switching Weight Mean Filter, Adaptive Weighted Mean Filter, Different Applied Median Filter, Iterative Mean Filter, Two-Stage Filter, Multistage Selective Convolution Filter, Different Adaptive Modified Riesz Mean Filter, Stationary Framelet Transform Based Filter and A New Type Adaptive Median Filter methods. In the comparison phase, nine different noise levels were applied to the original images. Denoised images were compared using Peak Signal-to-Noise Ratio, Image Enhancement Factor, and Structural Similarity Index Map image quality metrics. Comparisons were made using three separate image datasets and Cameraman, Airplane images. NVBMF achieved the best result in 52 out of 84 comparisons for PSNR, best in 47 out of 84 comparisons for SSIM, and best in 36 out of 84 comparisons for IEF. In addition, values nearly to the best result were obtained in comparisons where the best result could not be reached. The results obtained show that the NVBMF can be used as an effective method in denoising SPN.

Half-waves and spectral Riesz means on the 3-torus

For a full rank lattice Lambda subset mathbb {R}^d and textbf{A}in mathbb {R}^d, consider N_{d,0;Lambda ,textbf{A}}(Sigma ) = # ([Lambda +textbf{A}] cap Sigma mathbb {B}^d) = # {textbf{k}in Lambda : |textbf{k}+textbf{A}| le Sigma }. Consider the iterated integrals Nd,k+1;Λ,A(Σ)=∫0ΣNd,k;Λ,A(σ)dσ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} N_{d,k+1;\\Lambda ,\ extbf{A}}(\\Sigma ) = \\int _0^\\Sigma N_{d,k;\\Lambda ,\ extbf{A}}(\\sigma ) \\,\ extrm{d}\\sigma , \\end{aligned}$$\\end{document}for kin mathbb {N}. After an elementary derivation via the Poisson summation formula of the sharp large-Sigma asymptotics of N_{3,k;Lambda ,textbf{A}}(Sigma ) for kge 2 (these having an O(Sigma ) error term), we discuss how they are encoded in the structure of the Fourier transform mathcal {F}N_{3,k;Lambda ,textbf{A}}(tau ). The analysis is related to Hörmander’s analysis of spectral Riesz means, as the iterated integrals above are weighted spectral Riesz means for the simplest magnetic Schrödinger operator on the flat d-torus. That the N_{3,k;Lambda ,textbf{A}}(Sigma ) obey an asymptotic expansion to O(Sigma ^2) is a special case of a general result holding for all magnetic Schrödinger operators on all manifolds, and the subleading polynomial corrections can be identified in terms of the Laurent series of the half-wave trace at tau =0. The improvement to O(Sigma ) for kge 2 follows from a bound on the growth rate of the half-wave trace at late times.

Tauberian Korevaar

We focus on the Tauberian work for which Jaap Korevaar is best known, together with its connections with probability theory. We begin with a brief sketch of the field up to Beurling’s work. We follow with three sections on Beurling aspects: Beurling slow variation; the Beurling Tauberian theorem for which it was developed; Riesz means and Beurling moving averages. We then give three applications from probability theory: extremes, laws of large numbers, and large deviations. We then turn briefly to other areas of Korevaar’s work. We close with a personal postscript (whence our title).

Extension of N\"orlund and Riesz means: new estimates

Extension of N\"orlund and Riesz means: new estimates