Littlewood–Paley Theory for Triangle Buildings

For the natural two-parameter filtration \(\left( {\mathcal {F}_\lambda }: {\lambda \in P}\right) \) on the boundary of a triangle building, we define a maximal function and a square function and show their boundedness on \(L^p(\Omega _0)\) for \(p \in (1, \infty )\). At the end, we consider \(L^p(\Omega _0)\) boundedness of martingale transforms. If the building is of \({\text {GL}}(3, \mathbb {Q}_p)\), then \(\Omega _0\) can be identified with p-adic Heisenberg group.     

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.     

System Representations for the Paley–Wiener Space $$\mathcal {PW}_{\pi }^2$$

In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by convolution sums. It is possible to use either, the convolution sum where the time variable is in the argument of the bandlimited impulse response, or the convolution sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both convolution sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second convolution sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.     

Littlewood–Paley characterization of BMO and Triebel–Lizorkin spaces

We prove a generalization of the Littlewood–Paley characterisation of the BMO space where the shifts of a Schwartz function are replaced by a family of functions with suitable conditions imposed on them. We also prove that a certain family of Triebel–Lizorkin spaces can be characterized in a similar way.     

Littlewood–Paley–Stein functions for non-local Schrödinger operators

Positivity - Motivated by recent works on vertical Littlewood–Paley–Stein functions for symmetric non-local Dirichlet forms and local Schrödinger type operators respectively, we...     

Lp estimates for non-smooth bilinear Littlewood–Paley square functions on $${\mathbb{R}}$$

In this work, we study some non-smooth bilinear analogues of linear Littlewood–Paley square functions on the real line. We prove boundedness-properties in Lebesgue spaces for them. Let us consider the functions \({\phi_{n}}\) satisfying \({\widehat{\phi_n}(\xi)={\bf 1}_{[n,n+1]}(\xi)}\) and define the bilinear operator \({S_n(f,g)(x):=\int f(x+y)g(x-y) \phi_n(y) dy}\) . These bilinear operators are closely related to the bilinear Hilbert transforms. Then for exponents \({p,q,r'\in[2,\infty)}\) satisfying \({\frac{1}{p}+\frac{1}{q}=\frac{1}{r}}\) , we prove that

$\left\| \left( \sum_{n\in \mathbb{Z}}\left|S_n(f,g) \right|^2 \right)^{1/2}\right\|_{L^{r}(\mathbb{R})}\lesssim \|f\|_{L^p(\mathbb{R})}\|g\|_{L^q(\mathbb{R})}.$

     

Mixed A p -A r Inequalities for Classical Singular Integrals and Littlewood–Paley Operators

We prove mixed A _p -A _r inequalities for several basic singular integrals, Littlewood–Paley operators, and the vector-valued maximal function. Our key point is that r can be taken arbitrarily big. Hence, such inequalities are close in spirit to those obtained recently in the works by T. Hytönen and C. Pérez, and M. Lacey. On one hand, the “A _p -A _∞” constant in these works involves two independent suprema. On the other hand, the “A _p -A _r ” constant in our estimates involves a joint supremum, but of a bigger expression. We show in simple examples that both such constants are incomparable. This leads to a natural conjecture that the estimates of both types can be further improved.     

A Characterization of Some Mixed Volumes via the Brunn–Minkowski Inequality

We consider a functional $\mathcal{F}$ on the space of convex bodies in ? ⁿ of the form $$ {\mathcal{F}}(K)=\int_{\mathbb{S}^{n-1}} f(u) \mathrm{S}_{n-1}(K,du), $$ where $f\in C(\mathbb{S}^{n-1})$ is a given continuous function on the unit sphere of ? ⁿ , K is a convex body in ? ⁿ , n≥3, and S _n?1(K,?) is the area measure of K. We prove that $\mathcal{F}$ satisfies an inequality of Brunn–Minkowski type if and only if f is the support function of a convex body, i.e., $\mathcal{F}$ is a mixed volume. As a consequence, we obtain a characterization of translation invariant, continuous valuations which are homogeneous of degree n?1 and satisfy a Brunn–Minkowski type inequality.     

On One Generalization of the Hardy–Littlewood–Pólya Inequality

We generalize the Hardy–Littlewood–Pólya inequality for numerical sets to certain sets of vectors on a plane.     

A Note for one of Inequality

In this paper, we find the largest contant C such that sin2 x xtanx > 2x2 cx5tan x 0相似文献   

Morse and semistable stratifications of Kähler spacesby ${\Bbb C}^{\ast}$-actions

We prove that the Morse decomposition in the sense of Kirwan and semistable decomposition in the sense of GIT of a -K?hler manifold coincide if the moment map is proper and if the fixed points set has a finite number of connected components. For general K?hler space with holomorphic action of a complex reductive group G, if every component of the moment map is proper, the two decompositions also coincide if each semistable piece is Zariski open in its topological closure and the moment map square is minimal degenerate Morse function in the sense of Kirwan. Author’s address: Department of Mathematics, Tsinghua University, 100084 Beijing, P.R. China     

A Payne�CRayner Type Inequality for the Robin Problem on Arbitrary Minimal Surfaces in {\mathbb{R}^N}

We prove a Payne?CRayner type inequality for the first eigenfunction of the Laplacian with Robin boundary condition on any compact minimal surface with boundary in ${\mathbb{R}^N}$ . We emphasize that no topological condition is necessary on the boundary.     

Quasi–Monte Carlo rules for numerical integration over the unit sphere {\mathbb{S}^2}

We study numerical integration on the unit sphere ${\mathbb{S}^2 \subseteq\mathbb{R}^3}$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a (0, m, 2)-net given in the unit square [0, 1]² to the sphere ${\mathbb{S}^2}$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J Sci Comput 18(2):595–609, 1997]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on ${\mathbb{S}^2}$ . And finally, we prove an upper bound on the spherical cap L ₂-discrepancy of order N ^?1/2(log N)^1/2 (where N denotes the number of points). This improves upon the bound on the spherical cap L ₂-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Commun Pure Appl Math 39(S, suppl):S149–S186, 1986] by a factor of ${\sqrt{\log N}}$ . Numerical results suggest that the (0, m, 2)-nets lifted to the sphere ${\mathbb{S}^2}$ have spherical cap L ₂-discrepancy converging with the optimal order of N ^?3/4.     

Lattice-theoretic characterizations of the group classes {\mathfrak{N}^{k-1}\mathfrak{A}} and {\mathfrak{N}^k} for k?�� 2

Let ${2\leq k\in \mathbb{N}}$ . Recently, Costantini and Zacher obtained a lattice-theoretic characterization of the classes ${\mathfrak{N}^k}$ of finite soluble groups with nilpotent length at most k. It is the aim of this paper to give a lattice-theoretic characterization of the classes ${\mathfrak{N}^{k-1}\mathfrak{A}}$ of finite groups with commutator subgroup in ${\mathfrak{N}^{k-1}}$ ; in addition, our method also yields a new characterization of the classes ${\mathfrak{N}^k}$ . The main idea of our approach is to use two well-known theorems of Gaschütz on the Frattini and Fitting subgroups of finite groups.     

A Littlewood–Richardson rule for Macdonald polynomials

Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric Macdonald polynomials are a common generalization of Schur functions, Macdonald spherical functions, and Jack polynomials. We use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine Hecke algebra. From this, we obtain a product formula for Macdonald polynomials of general Lie type.     

The Gabriel–Roiter Measure for \widetilde{\mathbb{A}}_{\boldsymbol n} II

Let Q be a tame quiver of type $\widetilde{\mathbb{A}}_n$ and Rep(Q) the category of finite dimensional representations over an algebraically closed field. A representation is simply called a module. It will be shown that a regular string module has, up to isomorphism, at most two Gabriel–Roiter submodules. The quivers Q with sink-source orientations will be characterized as those, whose central parts do not contain preinjective modules. It will also be shown that there are only finitely many (central) Gabriel–Roiter measures admitting no direct predecessors. This fact will be generalized for all tame quivers.