Conical stochastic maximal L^p-regularity for 1\leqslant p<\infty

Let \(A = -\mathrm{div} \,a(\cdot ) \nabla \) be a second order divergence form elliptic operator on \({\mathbb R}^n\) with bounded measurable real-valued coefficients and let \(W\) be a cylindrical Brownian motion in a Hilbert space \(H\) . Our main result implies that the stochastic convolution process $$\begin{aligned} u(t) = \int _0^t e^{-(t-s)A}g(s)\,dW(s), \quad t\geqslant 0, \end{aligned}$$ satisfies, for all \(1\leqslant p<\infty \) , a conical maximal \(L^p\) -regularity estimate $$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)}^p \leqslant C_p^p {\mathbb E}\Vert g \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)}^p. \end{aligned}$$ Here, \(T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)\) and \(T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)\) are the parabolic tent spaces of real-valued and \(H\) -valued functions, respectively. This contrasts with Krylov’s maximal \(L^p\) -regularity estimate $$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;{\mathbb R}^n))}^p \leqslant C^p {\mathbb E}\Vert g \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;H))}^p \end{aligned}$$ which is known to hold only for \(2\leqslant p<\infty \) , even when \(A = -\Delta \) and \(H = {\mathbb R}\) . The proof is based on an \(L^2\) -estimate and extrapolation arguments which use the fact that \(A\) satisfies suitable off-diagonal bounds. Our results are applied to obtain conical stochastic maximal \(L^p\) -regularity for a class of nonlinear SPDEs with rough initial data.     

A Frank–Wolfe type theorem for nondegenerate polynomial programs

In this paper, we study the existence of optimal solutions to a constrained polynomial optimization problem. More precisely, let \(f_0\) and \(f_1, \ldots , f_p :{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) be convenient polynomial functions, and let \(S := \{x \in {\mathbb {R}}^n \ : \ f_i(x) \le 0, i = 1, \ldots , p\} \ne \emptyset .\) Under the assumption that the map \((f_0, f_{1}, \ldots , f_{p}) :{\mathbb {R}}^n \rightarrow {\mathbb {R}}^{p + 1}\) is non-degenerate at infinity, we show that if \(f_0\) is bounded from below on \(S,\) then \(f_0\) attains its infimum on \(S.\)      

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

Randomized large distortion dimension reduction

Consider a random matrix \(H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}\) . Let \(D\ge 2\) and let \(\{W_l\}_{l=1}^{p}\) be a set of \(k\) -dimensional affine subspaces of \({\mathbb {R}}^{n}\) . We ask what is the probability that for all \(1\le l\le p\) and \(x,y\in W_l\) , $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ We show that for \(m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )\) and a variety of different classes of random matrices \(H\) , which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on \(m\) is tight in terms of \(k,p,D\) .     

Frequent Hypercyclicity of Random Entire Functions for the Differentiation Operator

We study the random entire functions defined as power series \(f(z) = \sum _{n=0}^\infty (X_n/n!) z^n\) with independent and identically distributed coefficients \((X_n)\) and show that, under very weak assumptions, they are frequently hypercyclic for the differentiation operator \(D: H({\mathbb {C}}) \rightarrow H({\mathbb {C}}),\,f \mapsto Df = f'\) . This gives a very simple probabilistic construction of \(D\) -frequently hypercyclic functions in \(H({\mathbb {C}})\) . Moreover we show that, under more restrictive assumptions on the distribution of the \((X_n)\) , these random entire functions have a growth rate that differs from the slowest growth rate possible for \(D\) -frequently hypercyclic entire functions at most by a factor of a power of a logarithm.     

Isomorphic Steiner symmetrization of p-convex sets

In this paper we show that given a \(p\) -convex set \(K \subset \mathbb{R }^n\) , there exist \(5n\) Steiner symmetrizations that transform it into an isomorphic Euclidean ball. That is, if \(|K| = |D_n| = \kappa _n\) , we may symmetrize it, using \(5n\) Steiner symmetrizations, into a set \(K'\) such that \(c_p D_n \subset K' \subset C_p D_n\) , where \(c_p\) and \(C_p\) are constants dependent on \(p\) only.     

On the Dynamics of WKB Wave Functions Whose Phase are Weak KAM Solutions of H–J Equation

In the framework of toroidal Pseudodifferential operators on the flat torus \({\mathbb {T}}^n := ({\mathbb {R}} / 2\pi {\mathbb {Z}})^n\) we begin by proving the closure under composition for the class of Weyl operators \(\mathrm {Op}^w_\hbar (b)\) with symbols \(b \in S^m (\mathbb {T}^n \times \mathbb {R}^n)\) . Subsequently, we consider \(\mathrm {Op}^w_\hbar (H)\) when \(H=\frac{1}{2} |\eta |^2 + V(x)\) where \(V \in C^\infty ({\mathbb {T}}^n)\) and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on \(\mathbb {T}^n \times {\mathbb {R}}^n\) written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space \(H^{1} (\mathbb {T}^n; {\mathbb {C}})\) with phase functions in the class of Lipschitz continuous weak KAM solutions (positive and negative type) of the Hamilton–Jacobi equation \(\frac{1}{2} |P+ \nabla _x v (P,x)|^2 + V(x) = \bar{H}(P)\) for \(P \in \ell {\mathbb {Z}}^n\) with \(\ell >0\) , and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of \(P+ \nabla _x v\) .     

Noether–Lefschetz theory with base locus

Let \(Z\) be a closed subscheme of a smooth complex projective variety \(Y\subseteq \mathbb {P}^N\) , with \(\dim \,Y=2r+1\ge 3\) . We describe the intermediate Néron–Severi group (i.e. the image of the cycle map \(A_r(X)\rightarrow H_{2r}(X;\mathbb {Z})\) ) of a general smooth hypersurface \(X\subset Y\) of sufficiently large degree containing \(Z\) .     

Pointwise convergence in Pringsheim’s sense of the summability of Fourier transforms on Wiener amalgam spaces

New multi-dimensional Wiener amalgam spaces \(W_c(L_p,\ell _\infty )(\mathbb{R }^d)\) are introduced by taking the usual one-dimensional spaces coordinatewise in each dimension. The strong Hardy-Littlewood maximal function is investigated on these spaces. The pointwise convergence in Pringsheim’s sense of the \(\theta \) -summability of multi-dimensional Fourier transforms is studied. It is proved that if the Fourier transform of \(\theta \) is in a suitable Herz space, then the \(\theta \) -means \(\sigma _T^\theta f\) converge to \(f\) a.e. for all \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\) . Note that \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset W_c(L_r,\ell _\infty )(\mathbb{R }^d) \supset L_r(\mathbb{R }^d)\) and \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset L_1(\log L)^{d-1}(\mathbb{R }^d)\) , where \(1 . Moreover, \(\sigma _T^\theta f(x)\) converges to \(f(x)\) at each Lebesgue point of \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\) .     

Composition operators acting on Besov spaces on the real line

We study the composition operator \(T_f(g):= f\circ g\) on Besov spaces \(B_{{p},{q}}^{s}(\mathbb{R })\) . In case \(1 < p< +\infty ,\, 0< q \le +\infty \) and \(s>1+ (1/p)\) , we will prove that the operator \(T_f\) maps \(B_{{p},{q}}^{s}(\mathbb{R })\) to itself if, and only if, \(f(0)=0\) and \(f\) belongs locally to \(B_{{p},{q}}^{s}(\mathbb{R })\) . For the case \(p=q\) , i.e., in case of Slobodeckij spaces, we can extend our results from the real line to \(\mathbb{R }^n\) .     

On Sub-determinants and the Diameter of Polyhedra

We derive a new upper bound on the diameter of a polyhedron \(P = \{x {\in } {\mathbb {R}}^n :Ax\le b\}\) , where \(A \in {\mathbb {Z}}^{m\times n}\) . The bound is polynomial in \(n\) and the largest absolute value of a sub-determinant of \(A\) , denoted by \(\Delta \) . More precisely, we show that the diameter of \(P\) is bounded by \(O(\Delta ^2 n^4\log n\Delta )\) . If \(P\) is bounded, then we show that the diameter of \(P\) is at most \(O(\Delta ^2 n^{3.5}\log n\Delta )\) . For the special case in which \(A\) is a totally unimodular matrix, the bounds are \(O(n^4\log n)\) and \(O(n^{3.5}\log n)\) respectively. This improves over the previous best bound of \(O(m^{16}n^3(\log mn)^3)\) due to Dyer and Frieze (Math Program 64:1–16, 1994).     

Badly approximable points on planar curves and a problem of Davenport

Let \({\mathcal {C}}\) be two times continuously differentiable curve in \({\mathbb {R}}^2\) with at least one point at which the curvature is non-zero. For any \(i,j \geqslant 0\) with \(i+j =1\) , let \({\mathbf {Bad}}(i,j)\) denote the set of points \((x,y) \in {\mathbb {R}}^2\) for which \( \max \{ \Vert qx\Vert ^{1/i}, \, \Vert qy\Vert ^{1/j} \} > c/q \) for all \( q \in {\mathbb {N}}\) . Here \(c = c(x,y)\) is a positive constant. Our main result implies that any finite intersection of such sets with \({\mathcal {C}}\) has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties.     

Nonlinear Schrödinger equations near an infinite well potential

The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation where the potential \(V_\lambda :\mathbb {R}^N\rightarrow \mathbb {R}\) is close to an infinite well potential \(V_\infty :\mathbb {R}^N\rightarrow \mathbb {R}\) , i. e. \(V_\infty =\infty \) on an exterior domain \(\mathbb {R}^N\setminus \Omega \) , \(V_\infty |_\Omega \in L^\infty (\Omega )\) , and \(V_\lambda \rightarrow V_\infty \) as \(\lambda \rightarrow \infty \) in a sense to be made precise. The nonlinearity may be of Gross–Pitaevskii type. A standing wave solution of \((NLS_\lambda )\) with \(\lambda =\infty \) vanishes on \(\mathbb {R}^N\setminus \Omega \) and satisfies Dirichlet boundary conditions, hence it solves We investigate when a standing wave solution \(\Phi _\infty \) of the infinite well potential \((NLS_\infty )\) gives rise to nearby solutions \(\Phi _\lambda \) of the finite well potential \((NLS_\lambda )\) with \(\lambda \gg 1\) large. Considering \((NLS_\infty )\) as a singular limit of \((NLS_\lambda )\) we prove a kind of singular continuation type results.     

Weak lower semicontinuity for polyconvex integrals in the limit case

We prove a lower semicontinuity result for polyconvex functionals of the Calculus of Variations along sequences of maps \(u:\Omega \subset \mathbb{R }^n\rightarrow \mathbb{R }^m\) in \(W^{1,m}\) , \(2\le m\le n\) , bounded in \(W^{1,m-1}\) and convergent in \(L^1\) under mild technical conditions but without any extra coercivity assumption on the integrand.     

A doubling subset of L_p for p>2 that is inherently infinite dimensional

It is shown that for every \(p\in (2,\infty )\) there exists a doubling subset of \(L_p\) that does not admit a bi-Lipschitz embedding into \(\mathbb R^k\) for any \(k\in \mathbb N\) .     

\mathcal {A}_{p, {\mathbb {E}}} Weights,Maximal Operators,and Hardy Spaces Associated with a Family of General Sets

Suppose that \({\mathbb {E}}:=\{E_r(x)\}_{r\in {\mathcal {I}}, x\in X}\) is a family of open subsets of a topological space \(X\) endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. We establish an \(\mathcal {A}_{{\mathbb {E}}, p}\) weights theory with respect to \({\mathbb {E}}\) and get the characterization of weighted weak type (1,1) and strong type \((p,p)\) , \(1<p\le \infty \) , for the maximal operator \({\mathcal {M}}_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\) . As applications, we introduce the weighted atomic Hardy space \(H^1_{{\mathbb {E}}, w}\) and its dual \(BMO_{{\mathbb {E}},w}\) , and give a maximal function characterization of \(H^1_{{\mathbb {E}},w}\) . Our results generalize several well-known results.     

A nonlinear Stein theorem

For vector valued solutions \(u\) to the \(p\) -Laplacian system \(-\triangle _p u=F\) in a domain of \({\mathbb {R}}^n,\,p>1,\,n \ge 2,\) if \(F\) belongs to the limiting Lorentz space \(L(n,1),\) then \(Du\) is continuous.     

Two-dimensional slices of nonpseudoconvex domains with rough boundary

For a domain \(D\subset {\mathbb C}^n,\; n\ge 3\) , the set \(E\) is defined as the set of all points \(z\in {\mathbb C}^n\) for which the intersection of \(D\) with every complex \(2\) -plane through \(z\) is pseudoconvex. For \(D\) nonpseudoconvex, it is shown that \(E\) is contained in an affine subspace of codimension \(2\) . This results solves a problem raised by Nikolov and Pflug.     

Fixed points and homology of superelliptic Jacobians

Let \(\eta : C_{f,N}\rightarrow \mathbb {P}^1\) be a cyclic cover of \(\mathbb {P}^1\) of degree \(N\) which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic covering group \({{\mathrm{Aut}}}(\eta )\simeq \mathbb {Z}/ N \mathbb {Z}\) acting on the Jacobian \(J_N:={{\mathrm{Jac}}}(C_{f,N})\) . For each prime \(\ell \) distinct from the characteristic of the base field, the Tate module \(T_\ell J_N\) is shown to be a free module over the ring \(\mathbb {Z}_\ell [T]/(\sum _{i=0}^{N-1}T^i)\) . We also study the subvarieties of \(J_N\) and calculate the degree of the induced polarization on the new part \(J_N^\mathrm {new}\) of the Jacobian.     

Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let \(K\subset \mathbb R ^N\) be a convex body containing the origin. A measurable set \(G\subset \mathbb R ^N\) with positive Lebesgue measure is said to be uniformly \(K\) -dense if, for any fixed \(r>0\) , the measure of \(G\cap (x+r K)\) is constant when \(x\) varies on the boundary of \(G\) (here, \(x+r K\) denotes a translation of a dilation of \(K\) ). We first prove that \(G\) must always be strictly convex and at least \(C^{1,1}\) -regular; also, if \(K\) is centrally symmetric, \(K\) must be strictly convex, \(C^{1,1}\) -regular and such that \(K=G-G\) up to homotheties; this implies in turn that \(G\) must be \(C^{2,1}\) -regular. Then for \(N=2\) , we prove that \(G\) is uniformly \(K\) -dense if and only if \(K\) and \(G\) are homothetic to the same ellipse. This result was already proven by Amar et al. in 2008 . However, our proof removes their regularity assumptions on \(K\) and \(G\) , and more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski’s inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near \(r=0\) for the measure of \(G\cap (x+r\,K)\) (needed in 2008).