Finiteness of non-parabolic ends on submanifolds in spheres

We study a complete noncompact submanifold \(M^n\) in a sphere \(\mathbb {S}^{n+p}\) . We prove that the dimension of the space of \(L^2\) harmonic \(1\) -forms on \(M\) is finite and there are finitely many non-parabolic ends on \(M\) if the total curvature of \(M\) is finite and \(n\ge 3\) . This result is an improvement of Fu–Xu theorem on submanifolds in spheres and a generalized version of Cavalcante, Mirandola and Vitorio’s result on submanifolds in Hadamard manifolds.     

On the Action of Riesz Transforms on the Class of Bounded Functions

The paper is devoted to the \(d\) -dimensional extension of the classical identity of Stein and Weiss concerning the action of the Hilbert transform on characteristic functions. Let \((R_j)_{j=1}^d\) be the collection of Riesz transforms in \(\mathbb{R }^d\) . For \(1\le p<\infty \) , we determine the least constants \(c_{p,d}, C_{p,d}\) such that $$\begin{aligned} \int _{\mathbb{R }^d} f(x)|R_jf(x)|^p\text{ d }x&\le c_{p,d} ||f||_{L^1(\mathbb{R }^d)},\\ \int _{\mathbb{R }^d} (1-f(x))|R_jf(x)|^p\text{ d }x&\le C_{p,d} ||f||_{L^1(\mathbb{R }^d)} \end{aligned}$$ for any Borel function \(f:\mathbb{R }^d\rightarrow [0,1]\) . The proof rests on probabilistic methods and the construction of appropriate harmonic functions on \([0,1]\times \mathbb{R }\) .     

On an extension of the H^{k} mean curvature flow of closed convex hypersurfaces

In this paper we prove that the \(H^{k}\) ( \(k\) is odd and larger than \(2\) ) mean curvature flow of a closed convex hypersurface can be extended over the maximal time provided that the total \(L^{p}\) integral of the mean curvature is finite for some \(p\) .     

The \bar{\partial }-Equation on an Annulus Between Two Strictly q-Convex Domains with Smooth Boundaries

In this paper, we study the global boundary regularity of the \(\bar{\partial }\) - equation on an annulus domain \(\Omega \) between two strictly \(q\) -convex domains with smooth boundaries in \(\mathbb{C }^n\) for some bidegree. To this finish, we first show that the \(\bar{\partial }\) -operator has closed range on \(L^{2}_{r, s}(\Omega )\) and the \(\bar{\partial }\) -Neumann operator exists and is compact on \(L^{2}_{r,s}(\Omega )\) for all \(r\ge 0\) , \(q\le s\le n-q- 1\) . We also prove that the \(\bar{\partial }\) -Neumann operator and the Bergman projection operator are continuous on the Sobolev space \(W^{k}_{r,s}(\Omega )\) , \(k\ge 0\) , \(r\ge 0\) , and \(q\le s\le n-q-1\) . Consequently, the \(L^{2}\) -existence theorem for the \(\bar{\partial }\) -equation on such domain is established. As an application, we obtain a global solution for the \(\bar{\partial }\) equation with Hölder and \(L^p\) -estimates on strictly \(q\) -concave domain with smooth \(\mathcal C ^2\) boundary in \(\mathbb{C }^n\) , by using the local solutions and applying the pushing out method of Kerzman (Commun Pure Appl Math 24:301–380, 1971).     

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.     

Bent functions on partial spreads

For an arbitrary prime \(p\) we use partial spreads of \(\mathbb{F }_p^{2m}\) to construct two classes of bent functions from \(\mathbb{F }_p^{2m}\) to \(\mathbb{F }_p\) . Our constructions generalize the classes \(PS^{(-)}\) and \(PS^{(+)}\) of binary bent functions which are due to Dillon.     

On Octonionic Regular Functions and the Szegö Projection on the Octonionic Heisenberg Group

We discuss the octonionic regular functions and the octonionic regular operator on the octonionic Heisenberg group. This is the octonionic version of CR function theory in the theory of several complex variables and regular function theory on the quaternionic Heisenberg group. By identifying the octonionic algebra with \(\mathbb{R }^{8}\) , we can write the octonionic regular operator and the associated Laplacian operator as real \((8\times 8)\) -matrix differential operators. Then we use the group Fourier transform on the octonionic Heisenberg group to analyze the associated Laplacian operator and to construct its kernel. This kernel is exactly the Szegö kernel of the orthonormal projection from the space of \(L^{2}\) functions to the space of \(L^{2}\) regular functions on the octonionic Heisenberg group.     

\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.     

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 \(k,p,D\) .     

Necessary and sufficient conditions for the existence of Helmholtz decompositions in general domains

Consider a general domain \(\varOmega \subseteq {\mathbb {R}}^n, n\ge 2\) , and let \(1 < q <\infty \) . Our first result is based on the estimate for the gradient \(\nabla p \in G^q(\varOmega )\) in the form \(\Vert \nabla p\Vert _q \le C \,\sup |\langle \nabla p,\nabla v\rangle _{\varOmega }|/\Vert \nabla v\Vert _{q'}\) , \(\nabla v \in G^{q'}(\varOmega ), q' = \frac{q}{q-1}\) , with some constant \(C=C(\varOmega ,q)>0\) . This estimate was introduced by Simader and Sohr (Mathematical Problems Relating to the Navier–Stokes Equations. Series on Advances in Mathematics for Applied Sciences, vol. 11, pp. 1–35. World Scientific, Singapore, 1992) for smooth bounded and exterior domains. We show for general domains that the validity of this gradient estimate in \(G^q(\varOmega )\) and in \(G^{q'}(\varOmega )\) is necessary and sufficient for the validity of the Helmholtz decomposition in \(L^q(\varOmega )\) and in \(L^{q'}(\varOmega )\) . A new aspect concerns the estimate for divergence free functions \(f_0 \in L^q_{\sigma }(\varOmega )\) in the form \(\Vert f_0\Vert _q \le C \sup |\langle f_0,w\rangle _{\varOmega }|/ \Vert w\Vert _{q'}, w\in L^{q'}_{\sigma }(\varOmega )\) , for the second part of the Helmholtz decomposition. We show again for general domains that the validity of this estimate in \(L^q_{\sigma }(\varOmega )\) and in \(L^{q'}_{\sigma }(\varOmega )\) is necessary and sufficient for the validity of the Helmholtz decomposition in \(L^q(\varOmega )\) and in \(L^{q'}(\varOmega )\) .     

On homogeneous Lagrange means

Let \(f\) be a real differentiable function in an open interval \(I\) with one-to-one derivative. We observe that if the Lagrange mean \(L^{[f]}\) of a generator \(f\) is conditionally positively homogeneous, then \(f\) must be of the class \(C^{\infty }\) and the function $$\begin{aligned} g(x):=xf^{\prime }\left( x\right) -f\left( x\right) ,\quad \quad x\in I, \end{aligned}$$ is also a generator of \(L^{[f]}\) i.e. that \(L^{[g]}=L^{[f]}.\) We show that this fact and a result on equality of two Lagrange means allow easily to determine all positively homogeneous Lagrange means.     

More differentially 6-uniform power functions

In this paper, we study the differential spectra of differentially 6-uniform functions among the family of monomials \(\big \{x\mapsto x^{2^t-1},\; 1<t<n\big \}\) defined in \(\mathbb {F}_{2^{n}}\) . We show that the functions \(x\mapsto x^{2^t-1}\) when \(t=\frac{n-1}{2},\; \frac{n+3}{2}\) with odd \(n\) have a differential spectrum similar to the one of the function \(x\mapsto x^7\) which belongs to the same family. We also study the functions \(x\mapsto x^{2^t-1}\) when \(t=\frac{kn+1}{3},\frac{(3-k)n+2}{3}\) with \(kn\equiv 2\,\mathrm{mod}\,3\) which are known to be differentially 6-uniform and show that their complete differential spectrum can be provided under an assumption related to a new formulation of the Kloosterman sum. To provide the differential spectra for these functions, a recent result of Helleseth and Kholosha regarding the number of roots of polynomials of the form \(x^{2^t+1}+x+a\) is widely used in this paper. A discussion regarding the non-linearity and the algebraic degree of the vectorial functions \(x\mapsto x^{2^t-1}\) is also proposed.     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

Square Function Characterization of Weak Hardy Spaces

We obtain a new square function characterization of the weak Hardy space \(H^{p,\infty }\) for all \(p\in (0,\infty )\) . This space consists of all tempered distributions whose smooth maximal function lies in weak \(L^p\) . Our proof is based on interpolation between \(H^p\) spaces. The main difficulty we overcome is the lack of a good dense subspace of \(H^{p,\infty }\) which forces us to work with general \(H^{p,\infty }\) distributions.     

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.     

On Diophantine quintuples and D(-1)-quadruples

In this paper the known upper bound \(10^{96}\) for the number of Diophantine quintuples is reduced to \(6.8\cdot 10^{32}\) . The key ingredient for the improvement is that certain individual bounds on parameters are now combined with a more efficient counting of tuples, and estimated by sums over divisor functions. As a side effect, we also improve the known upper bound \(4\cdot 10^{70}\) for the number of \(D(-1)\) -quadruples to \(5\cdot 10^{60}\) .     

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 \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

Remark on the Helmholtz decomposition in domains with noncompact boundary

Let \(\varOmega \) be a domain in \(\mathbb {R}^{d+1}\) whose boundary is given as a uniform Lipschitz graph \(x_{d+1}=\eta (x)\) for \(x \in \mathbb {R}^d\) . For such a domain, it is known that the Helmholtz decomposition is not always valid in \(L^p(\varOmega )\) except for the energy space \(L^2 (\varOmega )\) . In this paper we show that the Helmholtz decomposition still holds in certain anisotropic spaces which include vector fields decaying slowly in the \(x_{d+1}\) variable. In particular, these classes include some infinite energy vector fields. For the purpose, we develop a new approach based on a factorization of divergence form elliptic operators whose coefficients are independent of one variable.