Global $$W^{1,p}$$ estimates for solutions to the linearized Monge–Ampère equations

In this paper, we investigate regularity for solutions to the linearized Monge–Ampère equations when the nonhomogeneous term has low integrability. We establish global \(W^{1,p}\) estimates for all \(p<\frac{nq}{n-q}\) for solutions to the equations with right-hand side in \(L^q\) where \(n/2<q\le n\). These estimates hold under natural assumptions on the domain, Monge–Ampère measures, and boundary data. Our estimates are affine invariant analogues of the global \(W^{1,p}\) estimates of N. Winter for fully nonlinear, uniformly elliptic equations.     

Strong extensions for q-summing o perators acting in p-convex Banach function s paces for $$1 \le p \le q$$1≤ p≤ q

Let \(1\le p\le q<\infty \) and let X be a p-convex Banach function space over a \(\sigma \)-finite measure \(\mu \). We combine the structure of the spaces \(L^p(\mu )\) and \(L^q(\xi )\) for constructing the new space \(S_{X_p}^{\,q}(\xi )\), where \(\xi \) is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-summing operator T defined on X can be continuously (strongly) extended to \(S_{X_p}^{\,q}(\xi )\). Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through \(L^q\)-spaces when \(1 \le q \le p\). Thus, our result completes the picture, showing what happens in the complementary case \(1\le p\le q\).     

$$L^{p}$$-Norms a nd Mahler’s Measure of Poly nomials o n the n-Dime nsio nal Torus

We prove Nikol’skii type inequalities that, for polynomials on the n-dimensional torus \(\mathbb {T}^n\), relate the \(L^p\)-norm with the \(L^q\)-norm (with respect to the normalized Lebesgue measure and \(0 <p <q < \infty \)). Among other things, we show that \(C=\sqrt{q/p}\) is the best constant such that \(\Vert P\Vert _{L^q}\le C^{\text {deg}(P)} \Vert P\Vert _{L^p}\) for all homogeneous polynomials P on \(\mathbb {T}^n\). We also prove an exact inequality between the \(L^p\)-norm of a polynomial P on \(\mathbb {T}^n\) and its Mahler measure M(P), which is the geometric mean of |P| with respect to the normalized Lebesgue measure on \(\mathbb {T}^n\). Using extrapolation, we transfer this estimate into a Khintchine–Kahane type inequality, which, for polynomials on \(\mathbb {T}^n\), relates a certain exponential Orlicz norm and Mahler’s measure. Applications are given, including some interpolation estimates.     

A note on regularity of the pressure in the Navier–Stokes system

In this note we improve the standard regularity of the dynamic part of the pressure in the Navier–Stokes system. Using the theory of elliptic equations with \(L^1\) right-hand side we prove that, in addition to be in \(L^2\), the dynamic pressure belongs to \(W^{1,\alpha }_{loc} \) with \(1<\alpha <\frac{n}{n-1}\), in case of Dirichlet boundary condition. For pressure boundary condition the dynamic pressure is proved to be in \(W^{1,\alpha } \). As a consequence, for the force \(\mathbf{f} \in L^q (\Omega )^n \) and \(q>n /2 \) the pressure turns out to be continuous.     

Fourier Spectrum Characterizations of $$H^{p}$$ Spaces on Tubes Over Cones for $$1\le p \le \infty $$1≤p≤∞

We give Fourier spectrum characterizations of functions in the Hardy \(H^p\) spaces on tubes for \(1\le p \le \infty .\) For \(F\in L^p(\mathbb {R}^n), \) we show that F is the non-tangential boundary limit of a function in a Hardy space, \(H^{p}(T_\Gamma ),\) where \(\Gamma \) is an open cone of \(\mathbb {R}^n\) and \(T_\Gamma \) is the related tube in \(\mathbb {C}^n,\) if and only if the classical or the distributional Fourier transform of F is supported in \(\Gamma ^*,\) where \(\Gamma ^*\) is the dual cone of \(\Gamma .\) This generalizes the results of Stein and Weiss for \(p=2\) in the same context, as well as those of Qian et al. in one complex variable for \(1\le p\le \infty .\) Furthermore, we extend the Poisson and Cauchy integral representation formulas to the \(H^p\) spaces on tubes for \(p\in [1, \infty ]\) and \(p\in [1,\infty ),\) with, respectively, the two types of representations.     

Two-Weight Norm Inequalities for the Local Maximal Function

For a local maximal function defined on a certain family of cubes lying “well inside” of \(\Omega \), a proper open subset of \({\mathbb {R}}^n\), we characterize the couple of weights (u, v) for which it is bounded from \(L^p(v)\) on \(L^q(u)\).     

On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts \(\zeta (s+i\tau )\), \(\tau \in \mathbb {R}\), of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts \(\zeta (s+ix_kh)\), \(k\in \mathbb {N}\), \(h>0\), where \(\{x_k\}\subset \mathbb {R}\) is such that the sequence \(\{ax_k\}\) with every real \(a\ne 0\) is uniformly distributed modulo 1, \(1\le x_k\le k\) for all \(k\in \mathbb {N}\) and, for \(1\le k\), \(m\le N\), \(k\ne m\), the inequality \(|x_k-x_m| \ge y^{-1}_N\) holds with \(y_N> 0\) satisfying \(y_Nx_N\ll N\).     

On the Analytic Structure of Commutative Nilmanifolds

In the classification theorems of Vinberg and Yakimova for commutative nilmanifolds, the relevant nilpotent groups have a very surprising analytic property. The manifolds are of the form \(G/K = N\rtimes K/K\) where, in all but three cases, the nilpotent group \(N\) has irreducible unitary representations whose coefficients are square integrable modulo the center \(Z\) of \(N\). Here we show that, in those three “exceptional” cases, the group \(N\) is a semidirect product \(N_{1}\rtimes \mathbb {R}\) or \(N_{1}\rtimes \mathbb {C}\) where the normal subgroup \(N_{1}\) contains the center \(Z\) of \(N\) and has irreducible unitary representations whose coefficients are square integrable modulo \(Z\). This leads directly to explicit harmonic analysis and Fourier inversion formulae for commutative nilmanifolds.     

Closures of Hardy and Hardy–Sobolev Spaces in the Bloch Type Space on the Unit Ball

For \(0<\alpha <\infty \), \(0<p<\infty \) and \(0<s<\infty \), we characterize the closures in the \(\alpha \)-Bloch norm of \(\alpha \)-Bloch functions that are in a Hardy space \(H^p\) and in a Hardy–Sobolev space \(H^p_s\) on the unit ball of \(\mathbb {C}^n\).     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

Marcinkiewicz-Type Spectral Multipliers on Hardy and Lebesgue Spaces on Product Spaces of Homogeneous Type

Let \(X_1\) and \(X_2\) be metric spaces equipped with doubling measures and let \(L_1\) and \(L_2\) be nonnegative self-adjoint operators acting on \(L^2(X_1)\) and \(L^2(X_2)\) respectively. We study multivariable spectral multipliers \(F(L_1, L_2)\) acting on the Cartesian product of \(X_1\) and \(X_2\). Under the assumptions of the finite propagation speed property and Plancherel or Stein–Tomas restriction type estimates on the operators \(L_1\) and \(L_2\), we show that if a function F satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator \(F(L_1, L_2)\) is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space \(X_1\times X_2\). We apply our results to the analysis of second-order elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner–Riesz means.     

Differential and Falsified Sampling Expansions

Differential and falsified sampling expansions \(\sum _{k\in {\mathbb {Z}}^d}c_k\varphi (M^jx+k)\), where M is a matrix dilation, are studied. In the case of differential expansions, \(c_k=Lf(M^{-j}\cdot )(-k)\), where L is an appropriate differential operator. For a large class of functions \(\varphi \), the approximation order of differential expansions was recently studied. Some smoothness of the Fourier transform of \(\varphi \) from this class is required. In the present paper, we obtain similar results for a class of band-limited functions \(\varphi \) with the discontinuous Fourier transform. In the case of falsified expansions, \(c_k\) is the mathematical expectation of random integral average of a signal f near the point \(M^{-j}k\). To estimate the approximation order of the falsified sampling expansions we compare them with the differential expansions. Error estimations in \(L_p\)-norm are given in terms of the Fourier transform of f.     

Left annihilator of generalized derivations on Lie ideals in prime rings

Let \(R\) be a prime ring, \(L\) a noncentral Lie ideal of \(R\), \(F\) a generalized derivation with associated nonzero derivation \(d\) of \(R\). If \(a\in R\) such that \(a(d(u)^{l_1} F(u)^{l_2} d(u)^{l_3} F(u)^{l_4} \ldots F(u)^{l_k})^{n}=0\) for all \(u\in L\), where \(l_1,l_2,\ldots ,l_k\) are fixed non negative integers not all are zero and \(n\) is a fixed integer, then either \(a=0\) or \(R\) satisfies \(s_4\), the standard identity in four variables.     

Weighted Fourier Inequalities via Rearrangements

The method of using rearrangements to give sufficient conditions for Fourier inequalities between weighted Lebesgue spaces is revisited, a comparison between two known sufficient conditions is completed, and the method is extended to provide sufficient conditions for a new range of indices. When \(1<q<p<\infty \), simple conditions on weights ensure that the Fourier transform will map a weighted \(L^p\) space into a weighted \(L^q\) space. These are established in Theorems 1 and 4 of Benedetto and Heinig (J Fourier Anal Appl 9(1):1–37, 2003). The proofs apply when \(2<q<p\) and \(1<q<p<2\) but not in the remaining case, \(1<q<2<p\). Here, counterexamples are given to show that these simple conditions are no longer sufficient when \(1<q<2<p\). Also, various additional conditions are presented, any of which will restore sufficiency in that case.     

Gowers $$U_3$$ norm of some classes of bent Boolean functions

The Gowers \(U_3\) norm of a Boolean function is a measure of its resistance to quadratic approximations. It is known that smaller the Gowers \(U_3\) norm for a Boolean function larger is its resistance to quadratic approximations. Here, we compute Gowers \(U_3\) norms for some classes of Maiorana–McFarland bent functions. In particular, we explicitly determine the value of the Gowers \(U_3\) norm of Maiorana–McFarland bent functions obtained by using APN permutations. We prove that this value is always smaller than the Gowers \(U_3\) norms of Maiorana–McFarland bent functions obtained by using differentially \(\delta \)-uniform permutations, for all \(\delta \ge 4\). We also compute the Gowers \(U_3\) norms for a class of cubic monomial functions, not necessarily bent, and show that for \(n=6\), these norm values are less than that of Maiorana–McFarland bent functions. Further, we computationally show that there exist 6-variable functions in this class which are not bent but achieve the maximum second-order nonlinearity for 6 variables.     

Riemann Localisation on the Sphere

This paper first shows that the Riemann localisation property holds for the Fourier-Laplace series partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimension. By Riemann localisation on the sphere \(\mathbb {S}^{d}\subset \mathbb {R}^{d+1}\), \(d\ge 2\), we mean that for a suitable subset X of \(\mathbb {L}_{p}(\mathbb {S}^{d})\), \(1\le p\le \infty \), the \(\mathbb {L}_{p}\)-norm of the Fourier local convolution of \(f\in X\) converges to zero as the degree goes to infinity. The Fourier local convolution of f at \(\mathbf {x}\in \mathbb {S}^{d}\) is the Fourier convolution with a modified version of f obtained by replacing values of f by zero on a neighbourhood of \(\mathbf {x}\). The failure of Riemann localisation for \(d>2\) can be overcome by considering a filtered version: we prove that for a sphere of any dimension and sufficiently smooth filter the corresponding local convolution always has the Riemann localisation property. Key tools are asymptotic estimates of the Fourier and filtered kernels.     

CCZ equivalence of power functions

Let \(F\simeq {{\mathrm{GF}}}(p^n)\) be a finite field of characteristic p and \(p_k\) and \(p_\ell \) be power functions on F defined by \(p_k(x)=x^k\) and \(p_\ell (x)=x^\ell \) respectively. We show, that \(p_k\) and \(p_\ell \) are CCZ equivalent, if and only if there exists a positive integer \(0\le a< n\), such that \(\ell \equiv p^a k \pmod {p^n-1}\) or \(k\ell \equiv p^a \pmod {p^n-1}\).     

A Short Proof of the Large Time Energy Growth for the Boussinesq System

We give a direct proof of the fact that the \(L^p\)-norms of global solutions of the Boussinesq system in \({\mathbb {R}}^3\) grow large as \(t\rightarrow \infty \) for \(1<p<3\) and decay to zero for \(3<p\le \infty \), providing exact estimates from below and above using a suitable decomposition of the space–time space \({\mathbb {R}}^{+}\times {\mathbb {R}}^{3}\). In particular, the kinetic energy blows up as \(\Vert u(t)\Vert _2^2\sim ct^{1/2}\) for large time. This contrasts with the case of the Navier–Stokes equations.     

Elimination of extremal index zeroes from generic paths of closed $$$$-forms

Let \( \alpha \) be a Morse closed \( 1 \)-form of a smooth \( n \)-dimensional manifold \( M \). The zeroes of \( \alpha \) of index \( 0 \) or \( n \) are called centers. It is known that every non-vanishing de Rham cohomology class \( u \) contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the last statement: every generic path \( (\alpha _t)_{ t\in [0,1] }\) of closed \( 1 \)-forms in a fixed class \( u\ne 0 \) such that \( \alpha _0,\alpha _1 \) have no centers, can be modified relatively to its extremities to another such path \( (\beta _t)_{t \in [0,1]} \) having no center at all.