Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds

Consider a Riemannian manifold M which is a Galois covering of a compact manifold, with nilpotent deck transformation group G. For the Laplace operator on M, we prove a precise estimate for the gradient of the heat kernel, and show that the Riesz transforms are bounded in L^p(M), 1 < p < . We also obtain estimates for discrete oscillations of the heat kernel, and boundedness of discrete Riesz transform operators, which are defined using the action of G on M.Mathematics Subject Classification (2000): 58J35, 35B65, 42B20in final form: 8 August 2003     

L p bounds for a maximal dyadic sum operator

The authors prove L ^p bounds in the range 1<p< for a maximal dyadic sum operator on R ⁿ . This maximal operator provides a discrete multidimensional model of Carlesons operator. Its boundedness is obtained by a simple twist of the proof of Carlesons theorem given by Lacey and Thiele [7] adapted in higher dimensions [9]. In dimension one, the L ^p boundedness of this maximal dyadic sum implies in particular an alternative proof of Hunts extension [4] of Carlesons theorem on almost everywhere convergence of Fourier integrals. Mathematics Subject Classification (2000):Primary 42A20, Secondary 42A24Grafakos is supported by the NSF. Tao is a Clay Prize Fellow and is supported by a grant from the Packard Foundation.     

Determinants and a unified approach to estimating resolvents of operators in operator ideals of Riesz type lp

An operator idealA is said to be of Riesz type l_p (0A(E), where E is a complex Banach space, is a Riesz operator with absolutely p-summable eigenvalues. The main purpose of this paper is to give a unified approach to the problem of estimating resolvents of operators T that belong to an arbitrary operator ideal of Riesz type l_p. The method used is based on a representation of the resolvent of T²ⁿ (n>p) in the weak operator topology by characteristic determinants. The same method is used to derive estimates of Fredholm minors. The results obtained extend, generalize, and simplify results of Markus, König, and Engelbrecht and Grobler, and verify a conjecture of König. Finally, the resolvent estimates are applied to establish sufficient conditions for the completeness of principal elements.     

The Dyadic Cesàro Operator on R+

The dyadic Cesàro operator C is introduced for functions in the space L ¹ := L ¹(R ₊) by means of the Walsh-Fourier transform defined by

Solution of Some Weight Problems for the Riemann–Liouville and Weyl Operators

The necessary and sufficient conditions are found for the weight function v, which provide the boundedness and compactness of the Riemann–Liouville operator R from L ^p to . The criteria are also established for the weight function w, which guarantee the boundedness and compactness of the Weyl operator W from to L ^q.     

Dirichlet operators and the positive maximum principle

Let (A, D(A)) denote the infinitesimal generator of some strongly continuous sub-Markovian contraction semigroup onL ^p (m), p1 andm not necessarily -finite. We show under mild regularity conditions thatA is a Dirichlet operator in all spacesL ^q (m), qp. It turns out that, in the limitq,A satisfies the positive maximum principle. If the test functionsC _c D(A), then the positive maximum principle implies thatA is a pseudo-differential operator associated with a negative definite symbol, i.e., a Lévy-type operator. Conversely, we provide sufficient criteria for an operator (A, D(A)) onL ^p(m) satisfying the positive maximum principle to be a Dirichlet operator. If, in particular,A onL ² (m) is a symmetric integro-differential operator associated with a negative definite symbol, thenA extends to a generator of a regular (symmetric) Dirichlet form onL ² (m) with explicitly given Beurling-Deny formula.     

The Volterra Operator is not Supercyclic

It is shown that the classical Volterra operator, which is cyclic, is not supercyclic on any of the spaces L^p[0, 1], 1 p < . This solves a question posed by Héctor Salas. This contrasts with the fact that the derivative operator, the left inverse of the Volterra operator, although unbounded, is hypercyclic on L^p[0, 1].     

Riesz transform and Riesz potentials for Dunkl transform

Analogues of Riesz potentials and Riesz transforms are defined and studied for the Dunkl transform associated with a family of weight functions that are invariant under a reflection group. The L^p$L^p$ boundedness of these operators is established in certain cases.     

Harmonic analysis in infinite dimension

We study the Hermite transform onL ²() where is a Gaussian measure on a Lusin locally convex spaceE. We are then lead to a Hilbert space () of analytic functions onE which is also a natural range for the Laplace transform. LetB be a convenient Hilbert-Schmidt operator on the Cameron-Martin spaceH of . There exists a natural sequence Cap _n of capacities onE associated toB. This implies the Kondratev-Yokoi theorem about positive linear forms on the Hida test-functions space.     

Distributional control and generalized analyticity

Let S be a strongly continuous, separation-preserving representation of a locally compact abelian group G in L^p(), where 1p<, and is an arbitrary measure. We show that S is uniformly bounded with respect to the L^p-and L-norms if and only if it satisfies a certain boundedness condition for distribution functions. These equivalent conditions facilitate the transference from L^p(G) to L^p() of the a.e. convergence for a wide class of sequences of convolution operators. The result unifies and generalizes various aspects of ergodic theory--in particular, the ergodic singular integral operators and ergodic Hardy spaces.     

Potential operators associated with Hankel and Hankel-Dunkl transforms

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q≤∞, for which the potential operators satisfy L ^p -L ^q estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, for which two-weight L ^p -L ^q estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight L ^p -L ^q bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De Nápoli, Drelichman and Durán, and Duoandikoetxea.     

L r Estimates for Degenerate Elliptic Problems

We prove L ^r estimates for the Dirichlet problem –div(a(x,u,Du))=f with f in L ^q for 1q+, where the operator satisfies (|s|)|| ^p a(x,s,), with p>1. These estimates are obtained without symmetrization and are sharp in some cases.     

On Weighted Norms of Riesz Transforms Equal to One

We consider the vector Riesz transform ^t-(t+s)/2 div^s of even order s + t in the weighted space L ₂(ⁿ;|x|^a). We establish that for t s, n >3 its norm is equal to one on some interval of values of a, while inside the interval a stronger estimate for a subordinate norm is valid.     

Criteria for the Boundedness and Compactness of Operators with Power-Logarithmic Kernels

In the present work, necessary and sufficient conditions are given in terms of a nonnegative Borel measure which ensure the boundedness and compactness of operators with power-logarithmic kernels from L ^p (0, a) to L ^p (0, a) (or to L ^q (0, a)), where 0 < a < , 1 < p, q < , > 1/p and 0.     

Gaussian kernels have only Gaussian maximizers

A Gaussian integral kernelG(x, y) onR ⁿ ×R ⁿ is the exponential of a quadratic form inx andy; the Fourier transform kernel is an example. The problem addressed here is to find the sharp bound ofG as an operator fromL ^p (R ⁿ ) toL ^p (R ⁿ ) and to prove that theL ^p (R ⁿ ) functions that saturate the bound are necessarily Gaussians. This is accomplished generally for 1<pq< and also forp>q in some special cases. Besides greatly extending previous results in this area, the proof technique is also essentially different from earlier ones. A corollary of these results is a fully multidimensional, multilinear generalization of Young's inequality.Oblatum 19-XII-1989Work partially supported by U.S. National Science Foundation grant PHY-85-15288-A03     

Non Linear Singular Drifts and Fractional Operators: when Besov meets Morrey and Campanato

In this paper, we show that, for doubling manifolds satisfiying the scaled Poincaré inequalities and \(p\in (2,\infty )\), the boundedness of the Riesz transform dΔ^?1/2 on L ^p , is essentially equivalent to the fact that \(H_{1,d}^{p}\) is equal the L ^p closure of the set of L ^p exact harmonic 1-forms. Here, \(H_{1,d}^{p}\) is a Hardy space of exact 1 ?forms, naturally associated with the Riesz transform, as defined by Auscher, McIntosh and Russ.     

Riesz Transforms Associated with Higher-Order Schrödinger Type Operators

Let L = L ₀ + V be a Schrödinger type operator, where L ₀ is a higher order elliptic operator with bounded complex coefficients in divergence form and V is a signed measurable function. Under the strongly subcritical assumption on V, we study the L ^q boundedness of Riesz transform ? ^m L ^?1/2 for q ≤ 2 based on the off-diagonal estimates of semigroup e ^{?t L} . Furthermore, the authors impose extra regularity assumptions on V to obtain the L ^q boundedness of Riesz transform ? ^m L ^?1/2 for some q > 2. In particular, these results are applied to the more interesting Schrödinger operators L = P(D) + V, where P(D) is any homogeneous positive elliptic operator with constant coefficients.     

Perturbation of orthonormal bases inL 2-spaces

The paper improves and generalizes a classical result from Paley and Wiener in their book on Fourier transforms. Paley and Wiener gave conditions on functionsh _n that imply that the sequence (1+h _n (x))e ^inx is a Riesz basis forL ²[–,]. These conditions involve theL ²-norm of the second derivativesh _n . The new results replace the differential operatoryy by more general differential operators inL ²-spaces, in particular, by the Hermite differential operator inL ²(R), ande ^inx by arbitrary orthonormal bases.     

On the Fredholm theory of Wiener-Hopf equations and the coupling method

The Fredholm properties of the Wiener-Hopf operator onL _p(⁺,^m) are investigated using the coupling method for solving operator equations. The theory applies to equations whose kernel is an element ofL ₁(,^mxm). As usual the determinant of the symbol is assumed to have no zeros on the real line. The method of analysis is independent of the realization theory for symbols that are analytic in a strip containing the real axis although in some sense closely related to it. The connection between the two methods is briefly analysed in the paper.     

On the existence of solutions of operator differential equations

We consider nonlinear equations of parabolic type in reflexive Banach spaces. We present sufficient conditions for the existence of solutions of these equations. We use methods for the investigation of problems with operators of pseudomonotone (on a subspace) type. In addition, a sufficient criterion in the Sobolev space L _p(0, T; W_p¹()L₂ (0, T; L₂()) is considered for the case where an operator introduced with the use of functional coefficients belongs to a given class. We also show that it is possible to weaken the classical condition of coerciveness.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 837–850, June, 2004.