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 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 \(P+ \nabla _x v\) .     

The ring of evenly weighted points on the line

Let \(M_w = ({\mathbb {P}}^1)^n /\!/\hbox {SL}_2\) denote the geometric invariant theory quotient of \(({\mathbb {P}}^1)^n\) by the diagonal action of \(\hbox {SL}_2\) using the line bundle \(\mathcal {O}(w_1,w_2,\ldots ,w_n)\) on \(({\mathbb {P}}^1)^n\) . Let \(R_w\) be the coordinate ring of \(M_w\) . We give a closed formula for the Hilbert function of \(R_w\) , which allows us to compute the degree of \(M_w\) . The graded parts of \(R_w\) are certain Kostka numbers, so this Hilbert function computes stretched Kostka numbers. If all the weights \(w_i\) are even, we find a presentation of \(R_w\) so that the ideal \(I_w\) of this presentation has a quadratic Gröbner basis. In particular, \(R_w\) is Koszul. We obtain this result by studying the homogeneous coordinate ring of a projective toric variety arising as a degeneration of \(M_w\) .     

Characteristic functions as distributional boundary values of analytic functions in tube domains

We propose necessary and sufficient conditions for a complex-valued function f on \( {{\mathbb{R}}^n} \) to be a characteristic function of a probability measure. Certain analytic extensions of f to tubular domains in \( {{\mathbb{C}}^n} \) are studied. In order to extend the class of functions under study, we also consider the case where f is a generalized function (distribution). The main result is given in terms of completely monotonic functions on convex cones in \( {{\mathbb{R}}^n} \) .     

Representations of set-valued risk measures defined on the l-tensor product of Banach lattices

We obtain a representation for set-valued risk measures which are defined on the completed \(l\) -tensor product \(E\widetilde{\otimes }_l G\) of Banach lattices \(E\) and \(G\) . This representation extends known representations for set-valued risk measures defined on Bochner spaces \(L^p(\mathbb {P}, \mathbb {R}^d)\) of \(p\) -integrable functions with values in \(\mathbb {R}^d\) .     

Trace Formulae for Curvature of Jet Bundles over Planar Domains

For a domain \(\varOmega \) in \(\mathbb {C}\) and an operator \(T\) in \({\mathcal {B}}_n(\varOmega )\) , Cowen and Douglas construct a Hermitian holomorphic vector bundle \(E_T\) over \(\varOmega \) corresponding to \(T\) . The Hermitian holomorphic vector bundle \(E_T\) is obtained as a pull-back of the tautological bundle \(S(n,{\mathcal {H}})\) defined over \({\mathcal {G}}r(n,{\mathcal {H}})\) by a nondegenerate holomorphic map \(z\mapsto {\mathrm{ker}}(T-z),\;z\in \varOmega \) . To find the answer to the converse, Cowen and Douglas studied the jet bundle in their foundational paper. The computations in this paper for the curvature of the jet bundle are rather intricate. They have given a set of invariants to determine if two rank \(n\) Hermitian holomorphic vector bundle are equivalent. These invariants are complicated and not easy to compute. It is natural to expect that the equivalence of Hermitian holomorphic jet bundles should be easier to characterize. In fact, in the case of the Hermitian holomorphic jet bundle \({\mathcal {J}}_k({\mathcal {L}}_f)\) , we have shown that the curvature of the line bundle \({\mathcal {L}}_f\) completely determines the class of \({\mathcal {J}}_k({\mathcal {L}}_f)\) . In case of rank \(n\) Hermitian holomorphic vector bundle \(E_f\) , We have calculated the curvature of jet bundle \({\mathcal {J}}_k(E_f)\) and also obtained a trace formula for jet bundle \({\mathcal {J}}_k(E_f)\) .     

Orthogonality Principle for Bilinear Littlewood–Paley Decompositions

We explore Littlewood–Paley like decompositions of bilinear Fourier multipliers. Grafakos and Li (Am. J. Math. 128(1):91–119 2006) showed that a bilinear symbol supported in an angle in the positive quadrant is bounded from \(L^p\times L^q\) into \(L^r\) if its restrictions to dyadic annuli are bounded bilinear multipliers in the local \(L^2\) case \(p\ge 2\) , \(q\ge 2\) , \(r= 1/(p^{-1}+q^{-1})\le 2\) . We show that this range of indices is sharp and also discuss similar results for multipliers supported near axis and negative diagonal.     

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 \(\mathbb{R }^n\) .     

L^p-Integrability,Dimensions of Supports of Fourier Transforms and Applications

It is proved that there does not exist any non zero function in \(L^p({\mathbb R}^n)\) with \(1\le p\le 2n/\alpha \) if its Fourier transform is supported by a set of finite packing \(\alpha \) -measure where \(0<\alpha <n\) . It is shown that the assertion fails for \(p>2n/\alpha \) . The result is applied to prove \(L^p\) Wiener Tauberian theorems for \({\mathbb R}^n\) and \(M(2)\) .     

Selmer groups over {\mathbb {Z}}_p^d-extensions

Consider an abelian variety \(A\) defined over a global field \(K\) and let \(L/K\) be a \({\mathbb {Z}}_p^d\) -extension, unramified outside a finite set of places of \(K\) , with \({{\mathrm{Gal}}}(L/K)=\Gamma \) . Let \(\Lambda (\Gamma ):={\mathbb {Z}}_p[[\Gamma ]]\) denote the Iwasawa algebra. In this paper, we study how the characteristic ideal of the \(\Lambda (\Gamma )\) -module \(X_L\) , the dual \(p\) -primary Selmer group, varies when \(L/K\) is replaced by a strict intermediate \({\mathbb {Z}}_p^e\) -extension.     

Character varieties of abelian groups

We prove that for every reductive group $G$ with a maximal torus ${\mathbb {T}}$ and the Weyl group $W,\, {\mathbb {T}}^N/W$ is the normalization of the irreducible component, $X_G^0({\mathbb {Z}}^N)$ , of the $G$ -character variety $X_G({\mathbb {Z}}^N)$ of ${\mathbb {Z}}^N$ containing the trivial representation. We also prove that $X_G^0({\mathbb {Z}}^N)={\mathbb {T}}^N/W$ for all classical groups. Additionally, we prove that even though there are no irreducible representations in $X_G^0({\mathbb {Z}}^N)$ for non-abelian $G$ , the tangent spaces to $X_G^0({\mathbb {Z}}^N)$ coincide with $H^1({\mathbb {Z}}^N, Ad\, \rho )$ . Consequently, $X_G^0({\mathbb {Z}}^2)$ , has the “Goldman” symplectic form for which the combinatorial formulas for Goldman bracket hold.     

\ell ^p-Linear Independence of the System of Integer Translates

Various properties of the system \({\mathcal {B}}_{\psi }\) of integer translates of a square integrable function \(\psi \in L^2({\mathbb {R}})\) can be completely described in terms of the periodization function \(p_{\psi }(\xi )=\sum _{k\in {\mathbb {Z}}}|\widehat{\psi }(\xi +k)|^2\) . In this paper, we consider the problem of \(\ell ^p\) -linear independence, where \(p>2\) . The results we present include the method of construction for one type of counterexamples to several naturally taken conjectures, a new sufficient condition for \(\ell ^p\) -linear independence and a characterization theorem having an additional assumption on \({\mathcal {B}}_{\psi }\) . In the latter, we obtain the characterization in terms of the sets of multiplicity of Lebesgue measure zero.     

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.     

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 \(S.\)      

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 \(k,p,D\) .     

Spectral Measures Associated with the Factorization of the Lebesgue Measure on a Set via Convolution

Let \(Q\) be a fundamental domain of some full-rank lattice in \({\mathbb {R}}^d\) and let \(\mu \) and \(\nu \) be two positive Borel measures on \({\mathbb {R}}^d\) such that the convolution \(\mu *\nu \) is a multiple of \(\chi _Q\) . We consider the problem as to whether or not both measures must be spectral (i.e. each of their respective associated \(L^2\) space admits an orthogonal basis of exponentials) and we show that this is the case when \(Q = [0,1]^d\) . This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral measures. In addition, we propose a generalized Fuglede’s Conjecture for spectral measures on \({\mathbb {R}}^1\) and we show that it implies the classical Fuglede’s Conjecture on \({\mathbb {R}}^1\) .     

On a linearized system arising in the study of viscous surface flow down an inclined plane

We consider the linearized problem describing motion of a viscous incompressible fluid flow down an inclined plane under the effect of gravity. We formulate the problem for downward periodic disturbances from the laminar steady flow as an evolution equation in the product of Sobolev spaces \(H^{\frac{3}{2}}_0({\mathbb {T}}) \times {\mathbb {P}}^0H^0(\Omega ) \) . It is crucial to study qualitative behavior of solutions that the operator arising in the linearized problem has compact resolvent operators and generates an analytic semigroup in \(H^{\frac{3}{2}}_0({\mathbb {T}}) \times {\mathbb {P}}^0H^0(\Omega ) \) .     

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.     

{\mathfrak {sl}}_3-Web bases,intermediate crystal bases and categorification

We give an explicit graded cellular basis of the \({\mathfrak {sl}}_3\) -web algebra \(K_S\) . In order to do this, we identify Kuperberg’s basis for the \({\mathfrak {sl}}_3\) -web space \(W_S\) with a version of Leclerc–Toffin’s intermediate crystal basis and we identify Brundan, Kleshchev and Wang’s degree of tableaux with the weight of flows on webs and the \(q\) -degree of foams. We use these observations to give a “foamy” version of Hu and Mathas graded cellular basis of the cyclotomic Hecke algebra which turns out to be a graded cellular basis of the \({\mathfrak {sl}}_3\) -web algebra. We restrict ourselves to the \({\mathfrak {sl}}_3\) case over \(\mathbb {C}\) here, but our approach should, up to the combinatorics of \({\mathfrak {sl}}_N\) -webs, work for all \(N>1\) or over \(\mathbb {Z}\) .     

Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions

We consider the conormal bundle of a Schubert variety \(S_I\) in the cotangent bundle \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) of the Grassmannian \({{\mathrm{\mathrm {Gr}}}}\) of \(k\) -planes in \({{\mathrm{\mathbb {C}}}}^n\) . This conormal bundle has a fundamental class \({\kappa _I}\) in the equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\) . Here \({{\mathrm{\mathbb T}}}=({{\mathrm{\mathbb {C}}}}^*)^n\times {{\mathrm{\mathbb {C}}}}^*\) . The torus \(({{\mathrm{\mathbb {C}}}}^*)^n\) acts on \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) in the standard way and the last factor \({{\mathrm{\mathbb {C}}}}^*\) acts by multiplication on fibers of the bundle. We express this fundamental class as a sum \(Y_I\) of the Yangian \(Y(\mathfrak {gl}_2)\) weight functions \((W_J)_J\) . We describe a relation of \(Y_I\) with the double Schur polynomial \([S_I]\) . A modified version of the \(\kappa _I\) classes, named \(\kappa '_I\) , satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) . This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on \({{\mathrm{\mathrm {Gr}}}}\) . The classes \((\kappa '_I)_I\) form a basis in the suitably localized equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\) . This basis depends on the choice of the coordinate flag in \({{\mathrm{\mathbb {C}}}}^n\) . We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.