Search

We investigate projection constants for spaces of bihomogeneous harmonic and bihomogeneous polynomials on the unit sphere in finite-dimensional complex Hilbert spaces. Using averaging techniques, we demonstrate that the minimal norm projection aligns with the natural orthogonal projection. This result enables us to establish a connection between these constants and weighted \linebreak $L_1$-norms of specific Jacobi polynomials. Consequently, we derive explicit bounds, provide practical expressions for computation, and present asymptotically sharp estimates for these constants. Our findings extend the classical Ryll and Wojtaszczyk formula for the projection constant of homogeneous polynomials in finite-dimensional complex Hilbert spaces to the bihomogeneous setting.

We investigate projection constants within classes of multivariate polynomials over finite-dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics and spaces of homogeneous polynomials as well as for spaces of polynomials of finite degree on the unit sphere. We establish a connection between these quantities and certain weighted $L_1$-norms of specific Jacobi polynomials. As a consequence, we present exact formulas, computable expressions and asymptotically accurate estimates for them. The real case we address is considerably more nuanced than its complex counterpart., Comment: Small corrections in spelling

A Christ-Kiselev maximal theorem is proved for linear operators between quasi-Banach function lattices satisfying certain lattice geometrical conditions. The result is further explored for weighted Lorentz spaces, classical Lorentz spaces, and Wiener amalgams of Lebesgue function and sequence spaces. Extensions are made to K\"othe dual operators and to operators on interpolation spaces of quasi-Banach function lattices. Several applications to maximal Fourier operators are presented.

It is shown that if the Fourier transform is a bounded map on a rearrangement-invariant space of functions on $\mathbb R^n$, modified by a weight, then the weight is bounded above and below and the space is equivalent to $L^2$. Also, if it is bounded from $L^p$ to $L^q$, each modified by the same weight, then the weight is bounded above and below and $1\le p=q'\le 2$. Applications prove the non-boundedness on these spaces of an operator related to the Schr\"odinger equation., Comment: 9 pages, no figures

The main aim of this work is to study important local Banach space constants for Boolean cube function spaces. Specifically, we focus on $\mathcal{B}_{\mathcal{S}}^N$, the finite-dimensional Banach space of all real-valued functions defined on the $N$-dimensional Boolean cube $\{-1, +1\}^N$ that have Fourier--Walsh expansions supported on a fixed~family $\mathcal{S}$ of subsets of $\{1, \ldots, N\}$. Our investigation centers on the projection, Sidon and Gordon--Lewis constants of this function space. We combine tools from different areas to derive exact formulas and asymptotic estimates of these parameters for special types of families $\mathcal{S}$ depending on the dimension $N$ of the Boolean cube and other complexity characteristics of the support set $\mathcal{S}$. Using local Banach space theory, we establish the intimate relationship among these three important constants., Comment: arXiv admin note: substantial text overlap with arXiv:2208.06467

Given a frequency sequence $\omega=(\omega_n)$ and a finite subset $J \subset \mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(\omega)$ of all Dirichlet polynomials $D(s) := \sum_{n \in J} a_n e^{-\omega_n s}, \, s \in \mathbb{C}$. The main aim is to prove asymptotically correct estimates for the projection constant $\boldsymbol{\lambda}\big(\mathcal{H}_\infty^{J}(\omega) \big)$ of the finite dimensional Banach space $\mathcal{H}_\infty^{J}(\omega)$ equipped with the norm $\|D\|= \sup_{\text{Re}\,s>0} |D(s)|$. Based on harmonic analysis on $\omega$-Dirichlet groups, we prove the formula $ \boldsymbol{\lambda}\big(\mathcal{H}_\infty^{J}(\omega) \big) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T \Big|\sum_{n \in J} e^{-i\omega_n t}\Big|\,dt\,, $ and apply it to various concrete frequencies $\omega$ and index sets $J$. To see an example, combining with a recent deep result of Harper from probabilistic analytic number theory, we for the space $\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)$ of all ordinary Dirichlet polynomials $D(s) = \sum_{n \leq x} a_n n^{-s}$ of length $x$ show the asymptotically correct order $ \boldsymbol{\lambda}\big(\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)\big) \sim \sqrt{x}/(\log \log x)^{\frac{1}{4}}. $, Comment: arXiv admin note: substantial text overlap with arXiv:2208.06467

We study the projection constant of the space of operators on $n$-dimensional Hilbert spaces, with the trace norm, $\mathcal S_1(n)$. We show an integral formula for the projection constant of $\mathcal S_1(n)$; namely $ \boldsymbol{\lambda}\big(\mathcal S_1(n)\big) = n \int_{\mathcal U_n} \vert \text{tr}(U) \vert \,dU \,, $ where the integration is with respect to the Haar probability measure on the group $\mathcal U_n$ of unitary operators. Using a probabilistic approach, we derive the limit formula $ \lim_{n\to \infty} \boldsymbol{\lambda}\big(\mathcal S_1(n)\big)/n = \sqrt{\pi}/2\,. $, Comment: arXiv admin note: substantial text overlap with arXiv:2208.06467

The general problem we address is to develop new methods in the study of projection constants of Banach spaces of multivariate polynomials. The relative projection constant $\boldsymbol{\lambda}(X,Y)$ of a subspace $X$ of a Banach $Y$ is the smallest norm among all possible projections on $Y$ onto $X$, and the projection constant $\boldsymbol{\lambda}(X)$ is the supremum of all relative projection constants of $X$ taken with respect to all possible super spaces $Y$. This is one of the most significant notions of modern Banach space theory and has been intensively studied since the birth of abstract operator theory. We focus on projection constants of Banach spaces of multivariate polynomials formed either by trigonometric polynomials $f(g)=\sum_{\gamma \in E} \hat{f}(\gamma) \gamma(g)$ defined on a compact topological group $G$, which have Fourier coefficients $\hat{f}(\gamma)$ supported in a finite set $E$ of characters; or analytic polynomials $P(z)=\sum_{\alpha\in J}c_\alpha(P)\,z^\alpha$, which are defined on a Banach space $X_n = (\mathbb{C}^n, \|\cdot\|)$ and have monomial coefficients $c_\alpha(P)$ supported in a finite set $J \subset \mathbb{N}_0^n$ of multi indices. Depending on the underlying structure (of the group, Banach space or index set), the goal is to prove precise formulas or asymptotically optimal estimates. Our general setting is flexible enough to handle a wide variety of Banach spaces of polynomials, including analytic polynomials on polydiscs, Dirichlet polynomials on the complex plane, and polynomials on Boolean cubes $\{-1,+1\}^n$. Moreover, we get an explicit formula for the projection constant of the space of trace class operators. The methods developed here enable us to prove new estimates for important invariants such as the unconditional basis constant and the Gordon-Lewis constant for Banach spaces of multivariate polynomials., Comment: 181 pages

We prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislaykov and the Kahane-Salem-Zygmund inequality. As a by-product we show various multiplier theorems for spaces of trigonometric polynomials on the $n$-dimensional torus $\mathbb{T}^n$ or Boolean cubes $\{-1,1\}^N$. Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application we show that various recent $\ell_1$-multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane-Salem-Zygmund inequality., Comment: 37 pages

Given a bounded measurable function $\sigma$ on $\mathbb{R}^n$, we let $T_\sigma $ be the operator obtained by multiplication on the Fourier transform by $\sigma $. Let $0

The main aim of this work is to give a general approach to the celebrated Kahane-Salem-Zygmund inequalities. We prove estimates for exponential Orlicz norms of averages $\sup_{1\le j \leq N} \big |\sum_{1 \leq i \leq K}\gamma_i(\cdot) a_{i,j}\big|$ where $(a_{i,j})$ denotes a matrix of scalars and the $(\gamma_i)$ a sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get novel Kahane-Salem-Zygmund type inequalities -- in particular, for spaces of subgaussian random polynomials and multilinear forms on finite dimensional Banach spaces as well as subgaussian random Dirichlet polynomials. Finally, we use abstract interpolation theory to widen our approach considerably., Comment: 47 pages

We prove the stability of isomorphisms between Banach spaces generated by interpolation methods introduced by Cwikel-Kalton-Milman-Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also the so-called $\pm$ or $G_1$ and $G_2$ methods defined by Peetre and Gustavsson-Peetre. This result is used to show the existence of solution of certain operator analytic equation. A by product of these results is a more general variant of the Albrecht-M\"uller result which states that the interpolated isomorphisms satisfy uniqueness-of-inverses between interpolation spaces. We show applications for positive operators between Calder\'on function lattices. We also derive connections between the spectrum of interpolated operators., Comment: 25 pages

We investigate the stability of compactness of bilinear operators acting on the product of interpolation of Banach spaces. We develop a general framework for such results and our method applies to abstract methods of interpolation in the senseof Aronszajn and Gagliardo. A key step is to show an one-sided bilinear interpolation theorem on compactness for bilinear operators on couples satisfying an approximation property. We show applications to general cases, including Peetre's method and the general real interpolation methods., Comment: 21 pages

We study the asymptotic decay of the Fourier spectrum of real functions $f\colon \{-1,1\}^N \rightarrow \mathbb{R}$ in the spirit of Bohr's phenomenon from complex analysis. Every such function admits a canonical representation through its Fourier-Walsh expansion $f(x) = \sum_{S\subset \{1,\ldots,N\}}\widehat{f}(S) x^S \,,$ where $x^S = \prod_{k \in S} x_k$. Given a class $\mathcal{F}$ of functions on the Boolean cube $\{-1, 1\}^{N} $, the Boolean radius of $\mathcal{F}$ is defined to be the largest $\rho \geq 0$ such that $\sum_{S}{|\widehat{f}(S)| \rho^{|S|}} \leq \|f\|_{\infty}$ for every $f \in \mathcal{F}$. We give the precise asymptotic behaviour of the Boolean radius of several natural subclasses of functions on finite Boolean cubes, as e.g. the class of all real functions on $\{-1, 1\}^{N}$, the subclass made of all homogeneous functions or certain threshold functions. Compared with the classical complex situation subtle differences as well as striking parallels occur., Comment: 26 pages

The main purpose of this paper is the study of a~new class of summing multilinear operators acting from the product of Banach lattices with some nontrivial lattice convexity. A~mixed Pietsch-Maurey-Rosenthal type factorization theorem for these operators is proved under weaker convexity requirements than the ones that are needed in the Maurey-Rosenthal factorization through products of $L^q$-spaces. A~by-product of our factorization is an extension of multilinear operators defined by a~$q$-concavity type property to a~product of special Banach function lattices which inherit some lattice-geometric properties of the domain spaces, as order continuity and $p$-convexity. Factorization through Fremlin's tensor products is also analyzed. Applications are presented to study a~special class of linear operators between Banach function lattices that can be characterized by a strong version of $q$-concavity. This class contains $q$-dominated operators, and so the obtained results provide a~new factorization theorem for operators from this class.

Let $f$ be a real-valued, degree-$d$ Boolean function defined on the $n$-dimensional Boolean cube $\{\pm 1\}^{n}$, and $f(x) = \sum_{S \subset \{1,\ldots,d\}} \widehat{f}(S) \prod_{k \in S} x_k$ its Fourier-Walsh expansion. The main result states that there is an absolute constant $C >0$ such that the $\ell_{2d/(d+1)}$-sum of the Fourier coefficients of $f:\{\pm 1\}^{n} \rightarrow [-1,1]$ is bounded by $\leq C^{\sqrt{d \log d}}$. It was recently proved that a similar result holds for complex-valued polynomials on the $n$-dimensional poly torus $\mathbb{T}^n$, but that in contrast to this a replacement of the $n$-dimensional torus $\mathbb{T}^n$ by $n$-dimensional cube $[-1, 1]^n$ leads to a substantially weaker estimate. This in the Boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case. We indicate how our result is linked with several questions in quantum information theory., Comment: 25 pages

A number of Calder\'on-Mityagin couples and relative Calder\'on-Mityagin pairs are identified among Banach function spaces defined in terms of the least decreasing majorant construction on the half line. The interpolation structure of such spaces is shown to closely parallel that of the rearrangement invariant spaces, and it is proved that a couple of these spaces is a Calder\'on-Mityagin couple if and only if the corresponding couple of rearrangement invariant spaces is a Calder\'on-Mityagin couple. Consequently, the class of all interpolation spaces for any couple of spaces of this type admits a complete description by the K-method if and only if the class of all interpolation spaces for the corresponding couple of rearrangement invariant spaces does. Analogous results are proved for spaces defined in terms of the level function construction. In the main, the conclusions for both types of spaces remain valid when Lebesgue measure on the half line is replaced by a general Borel measure on R. However, for certain measures the class of interpolation spaces of these new spaces may be degenerate, reducing the "if and only if" of the main results to a single implication., Comment: 19 pages

We prove Nikol'skii type inequalities which for polynomials on the $n$-dimensional torus $\mathbb{T}^n$ relate the $L^p$-with the $L^q$-norm (with respect to the normalized Lebesgue measure and $0

We prove that the Lorentz sequence space $\ell_{\frac{2m}{m+1},1}$ is, in a~precise sense, optimal among all symmetric Banach sequence spaces satisfying a Bohnenblust-Hille type inequality for $m$-linear forms or $m$-homogeneous polynomials on $\mathbb{C}^n$. Motivated by this result we develop methods for dealing with subtle Bohnenblust-Hille type inequalities in the setting of Lorentz spaces. Based on an interpolation approach and the Blei-Fournier inequalities involving mixed type spaces, we prove multilinear and polynomial Bohnenblust-Hille type inequalities in Lorentz spaces with subpolynomial and subexponential constants. Improving a remarkable result of Balasubramanian-Calado-Queff\'elec, we show an application to the theory of Dirichlet series.

We derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in particular, we derive necessary and sufficient conditions guaranteeing the trace type inequality for this operator. We also establish the Fefferman-Stein type inequality, and obtain one-weight criteria when a weight function is of product type. As a consequence, appropriate results for multilinear Riesz potential operator with product kernels follow., Comment: 16 pages

We prove a substantial extension of a well-known result due to Bennett and Carl: The inclusion of a 2-concave symmetric Banach sequence space E into l_2 is (E,1)-summing, i.e. for every unconditionally summable sequence (x_n) in E the scalar sequence (||x_n||_2) is contained in E. Various applications are given, e.g. to the theory of eigenvalue distribution of compact operators and approximation theory., Comment: 22 pages