Strong mean approximation on HP(Tn) for the Riesz means at the critical index

In this note we extend the results in [1] to high dimensions. Let f∈H^p (Tⁿ), 0<p<1, n≥1 andσ ^δ , f denote the Riesz means of f at the critical index δ=n/p−(n+1)/2. We have the following estimate: were 0<s≤2 and , is the K-functional in H^p(Tⁿ). Supported by NSFC     

Riesz means of eigenfunction expansions of elliptic differential operators on compact manifolds

Let {λ ²} and {? _λ } be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator Δ on a compact manifoldM of dimensionN. We prove that the Riesz means of order δ, defined by \(R_\Lambda ^\delta f = \sum\limits_{\lambda< \Lambda } {\left( {1 - \frac{{\lambda ^2 }}{{\Lambda ^2 }}} \right)^\delta \hat f(} \lambda ) \varphi _\lambda \) , are uniformly bounded from the Hardy spaceH ^p (M) into Weak-L ^p (M), if 0<p<1 and δ=N/p?(N+1)/2.     

Some properties of Riesz potential associated with Schr(O)dinger operator

Let (L) =-△ + V be the Schr(o)dinger operator on Rd,d ≥ 3,where △ is the Laplacian on Rd and V （≠） 0 is a nonnegative function satisfying the reverse H(O)lder inequality.In this article,the author inve...     

Riesz means of the distribution function of the eigenvalues of an elliptic operator

The asymptotic behavior for of the Riesz means of the distribution function N() of the eigenvalues of an elliptic differential operator is investigated. Under some restrictions (in particular for the Laplace operator), the second term of the asymptotics is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 143–145, 1987.     

Some characterizations of Riesz operators by means of invariant subspaces

In this paper we shall give several characterizations of Riesz operatorsT on Banach spaces by means of some closedT-invariant subspaces. Moreover we also give a characterization of these operators dual to that give by Dieudonné in [5].     

Some properties of Riesz potential associated with SchrSdinger operator

Let L = −Δ + V be the Schr?dinger operator on ℝ ^d , d ≥ 3, where Δ is the Laplacian on ℝ ^d and V ≢ 0 is a nonnegative function satisfying the reverse H?lder inequality. In this article, the author investigates some properties of the Riesz potential I _α ^L associated with L on the Campanato-type spaces Λ _L ^β and the Hardy-type spaces H _L ^p .     

Some remarks on elliptic problems with critical growth in the gradient

In this work we analyze existence, nonexistence, multiplicity and regularity of solution to problem

(1)     

Some reformulated properties of Jacobian elliptic functions

Various properties of Jacobian elliptic functions can be put in a form that remains valid under permutation of the first three of the letters c, d, n, and s that are used in pairs to name the functions. In most cases 12 formulas are thereby replaced by three: one for the three names that end in s, one for the three that begin with s, and one for the six that do not involve s. The properties thus unified in the present paper are linear relations between squared functions (16 relations being replaced by five), differential equations, and indefinite integrals of odd powers of a single function. In the last case the unification entails the elementary function RC(x,y)=RF(x,y,y), where RF(x,y,z) is the symmetric elliptic integral of the first kind. Explicit expressions in terms of RC are given for integrals of first and third powers, and alternative expressions are given with RC replaced by inverse circular, inverse hyperbolic, or logarithmic functions. Three recurrence relations for integrals of odd powers hold also for integrals of even powers.     

Some asymptotic properties of discrete means

In part 1, given n different ways of averaging n positive numbers, we iterate the resulting map in $(0,\infty {)}^n$. We prove convergence toward the diagonal, with rate estimates under smoothness assumptions. In part 2, we consider the elementary symmetric means of order p applied to the values $a_i=a(i/n),1⩽i⩽n$, of a given continuous positive function a on the normalized interval $[0,1]$ and we let $p=f(n)$. When ${\lim }_{n\to \infty }f(n)/n=0$, we prove that it admits a limit as $n\to \infty$, called the f-mean of a, which moreover coincides with ${\int }_0^{1}a(x)\mathrm{d}x$ whenever $f(n)=o(\log n)$. We record similar, quite immediate, results on the geometric side $p=n-f(n)$.     

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,μ) be an abstract Wiener space and let D_V:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V^*BV in H and in , as well as the realisations of the operators and in L^p(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:

(1) with for ;

(2) admits a bounded H^∞-functional calculus on ;

(3) with for ;

(4) admits a bounded H^∞-functional calculus on .

Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving L^p-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C₀-contraction semigroup on a Hilbert space H and let −L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L^p-domain characterisation for the operator L.

Almost everywhere convergence of Riesz means on noncompact symmetric spaceSL(3,H)/Sp(3)

LetG/K be the noncompact Riemannian symmetric spaceSL(3,H)/Sp(3). We shall prove in this paper that forf∈L ^p(SL(3,H)/Sp(3)), 1≤p≤2, the Riesz means of orderz off with respect to the eigenfunctions expansion of Laplace operator almost everywhere converge tof for Rez >δ(n,p). The critical index δ(n,p) is the same as in the classical Stein's result for Euclidean space, and as in the noncompact symmetric spaces of rank one and of complex type. Partially supported by National Natural Science Foundation of China     

Approximation on set of total measure of Bochner-Riesz means below the critical index

In this paper it has been considered convergence and approximation for functions inSobolev Spaces L_m~1(R~n)by Bochner-Riesz means below the critical index.A theorem that ofprecise approximation orders on set of total measure has been proved.     

Approximation of Bochner-Riesz means of conjugate Fourier integrals below the critical index

In this paper we study the approximation on set of full measure for functions in Sobolev spaces L _m ¹ (R ⁿ) (m∈ℕ) by Bochner-Riesz means of conjugate Fourier integrals below the critical index. A theorem concerning the precise approximation orders with relation to the number m of space L _m ¹ (R ⁿ) and the index of Bochner-Riesz means is obtained. Supported by NNSFC.     

Some results on the qualitative properties of positive solutions of quasilinear elliptic equations

We consider the Dirichlet problem in Ω with zero Dirichlet boundary conditions. We prove local summability properties of and we exploit these results to give geometric characterizations of the critical set . We extend to the case of changing sign nonlinearities some results known in the case f(s) > 0 for s > 0. Berardino Sciunzi: Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”