Operators In Lp Research Articles

Bounds for Maximal Functions Associated with Rotational Invariant Measures in High Dimensions

ABSTRACT. In recent articles, it was proved that when mu is a finite, radial measure in Rn with a bounded, radially decreasing density, the Lp(mu) norm of the associated maximal operator grows to infinity with the dimension for a small range of values of p near 1. We prove that when mu is Lebesgue measure restricted to the unit ball and p < 2, the Lp operator norms of the maximal operator are unbounded in dimension, even when the action is restricted to radially decreasing functions. In spite of this, this maximal operator admits dimension-free Lp bounds for every p > 2, when restricted to radially decreasing functions. On the other hand, when mu is the Gaussian measure, the Lp operator norms of the maximal operator grow to infinity with the dimension for any finite p > 1, even in the subspace of radially decreasing functions.

An extrapolation theorem with applications to weighted estimates for singular integrals

We prove an extrapolation theorem saying that the weighted weak type (1,1) inequality for A1 weights implies the strong Lp(w) bound in terms of the Lp(w) operator norm of the maximal operator M. The weak Muckenhoupt–Wheeden conjecture along with this result allows us to conjecture that the following estimate holds for a Calderón–Zygmund operator T for any p>1:‖T‖Lp(w)⩽c‖M‖Lp(w)p. The latter conjecture would yield the sharp estimates for ‖T‖Lp(w) in terms of the Aq characteristic of w for any 1<q<p. In this paper we get a weaker inequality‖T‖Lp(w)⩽c‖M‖Lp(w)plog(1+‖M‖Lp(w)) with the corresponding estimates for ‖w‖Aq when 1<q<p.

Convergence order of wavelet thresholding estimator for differential operators on Besov spaces

The almost everywhere convergence of wavelet series is important in wavelet analysis [S. Kelly, M.A. Kon, L.A. Raphael, Local convergence for wavelet expansions, J. Funct. Anal. 126 (1994) 102–138]. Since the thresholding method plays fundamental roles in data compression, signal processing and other areas, the pointwise convergence of resulting wavelet series was investigated by T. Tao and B. Vidakovic in 1996 and 2000. Chen and Meng study the same problem for differential operators in Lp(R) setting [Di-Rong Chen, Hong-Tao Meng, Convergence of wavelet thresholding estimators of differential operators, Appl. Comput. Harmon. Anal. 25 (2008) 266–275]. This paper deals with the uniform and norm convergence of wavelet thresholding estimator of differential operators on Besov spaces and Sobolev spaces with integer exponents. In particular, the convergence order is focused. To do that, we first study the corresponding problems for non-standard forms of those operators.

Holomorphic families of Schrödinger operators in Lp

Given two m-sectorial operators in a reflexive Banach space X, a sufficient condition is presented for the family {T+κA;κ∈Σc} to be holomorphic of type (A), where Σ is a closed convex region in the κ-plane such that ∂Σ is the left branch of a hyperbola. The abstract formulation is almost identical with Kato’s when X is a Hilbert space. However, ∂Σ is typically reduced to a parabola when X is a Hilbert space. The m-sectoriality is a generalized notion of the nonnegative selfadjointness so that Schrödinger operators with singular potentials in the Lp-spaces (1<p<∞) are typical examples of the abstract theory. In this connection a detailed analysis is given in the case of −Δ+κ|x|−2. This application is an Lp-generalization of Kato’s.

Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces

Let L be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with L, such as the heat semigroup and Riesz transform, are not, in general, of CalderonZygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in Lp, Sobolev, and some new Hardy spaces naturally associated to L. First, we show that the known ranges of boundedness in Lp for the heat semigroup and Riesz transform of L, are sharp. In particular, the heat semigroup e−tL need not be bounded in Lp if p < [2n/(n+2), 2n/(n−2)]. Then we provide a complete description of all Sobolev spaces in which L admits a bounded functional calculus, in particular, where e−tL is bounded. Secondly, we develop a comprehensive theory of Hardy and Lipschitz spaces associated to L, that serves the range of p beyond [2n/(n + 2), 2n/(n − 2)]. It includes, in particular, characterizations by the sharp maximal function and the Riesz transform (for certain ranges of p), as well as the molecular decomposition and duality and interpolation theorems.

Compactness characterization of commutators for Littlewood-Paley operators

In this paper, the authors prove that the commutator [b,L] is a compact operator in Lp(Rn) if and only if b $\in$ VMO(Rn), where L denotes the Littlewood-Paley operators, such as the Littlewood-Paley g-function, Lusin area integral and Littlewood-Paley gλ* function.

Strong extrapolation spaces and interpolation

We introduce a new class of rearrangement invariant spaces on the segment [0, 1] which contains the most common extrapolation spaces with respect to the Lp-scale as p → ∞. We characterize the class and demonstrate under certain conditions that the Peetre Open image in new window -functional has extrapolatory description in the couple (E, L∞) if and only if E belongs to the class. By way of application, we establish a new extrapolation theorem for the bounded operators in Lp.

Ergodic theorem for the algebra of operators in 𝕃p

AbstractA measure-preserving isomorphism α of a probability space (Ω,ℱ,μ) generates an algebraic automorphism $\tilde \alpha $ of the algebra of bounded linear operators in 𝕃p(Ω,ℱ,μ). An ergodic theorem for $\tilde \alpha $ is proved.

Dominated Compactness Theorem in Banach Function Spaces and its Applications

A famous dominated compactness theorem due to Krasnosel’skiĭ states that compactness of a regular linear integral operator in Lp follows from that of a majorant operator. This theorem is extended to the case of the spaces \(L^{p(\cdot)}(\Omega, \mu, \varrho), \mu \Omega < \infty\), with variable exponent p(·), where we also admit power type weights \(\varrho\). This extension is obtained as a corollary to a more general similar dominated compactness theorem for arbitrary Banach function spaces for which the dual and associate spaces coincide. The result on compactness in the spaces \(L^{p(\cdot)}(\Omega, \mu, \varrho)\) is applied to fractional integral operators over bounded open sets.

On a new form of the ergodic theorem for the unit sphere with application to spectral theory

AbstractLet 1 ≤ p &lt; ∞ and let T be an ergodic measure‐preserving transformation of the finite measure space (X, μ). The classical Lp ergodic theorem of von Neumann asserts that for any f ϵ Lp (X, μ), equation image When X = 𝕊n (the unit sphere in ℝn +1) and μ is the standard area measure of 𝕊n , we establish a new form of the ergodic theorem. That is, we can replace the sequence of finite subsets of O(n + 1) (the orthogonal transformation group of ℝn +1) with a sequence of some fixed finite subsets of O(n + 1) which are independent of any measure‐preserving transformation, such that for any f ϵ Lp (𝕊n , μ), equation image As an application, we can completely determine the point spectrum of the Laplace operator in Lp (ℝn ) (1 ≤ p &lt; ∞, n ≥ 1) spaces. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Polynomial approximation on the sphere using scattered data

AbstractWe consider the problem of approximately reconstructing a function f defined on the surface of the unit sphere in the Euclidean space ℝq +1 by using samples of f at scattered sites. A central role is played by the construction of a new operator for polynomial approximation, which is a uniformly bounded quasi‐projection in the de la Vallée Poussin style, i.e. it reproduces spherical polynomials up to a certain degree and has uniformly bounded Lp operator norm for 1 ≤ p ≤ ∞. Using certain positive quadrature rules for scattered sites due to Mhaskar, Narcowich and Ward, we discretize this operator obtaining a polynomial approximation of the target function which can be computed from scattered data and provides the same approximation degree of the best polynomial approximation. To establish the error estimates we use Marcinkiewicz–Zygmund inequalities, which we derive from our continuous approximating operator. We give concrete bounds for all constants in the Marcinkiewicz–Zygmund inequalities as well as in the error estimates. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Об аппроксимации свермками и баэисах иэ сдвигов функции

For 2π-periodic functions f ∈ Lp(\( \mathbb{T} \)), 1 ≤ p < ∞, σ ∈ V (\( \mathbb{T} \)) and g ∈ L(\( \mathbb{T} \)), we consider the convolutions $$ (f*d\sigma )_T (x) = \int_0^{2\pi } {f(x - t)d\sigma (t), } (f*g)_T (x) = \int_0^{2\pi } {f(x - t)g(t)dt.} $$ For fixed functions σ ∈ V (\( \mathbb{T} \)) and g ∈ L(\( \mathbb{T} \)), necessary and sufficient conditions are obtained for the density of the ranges of these operators in Lp. Similar result is proved for the dyadic convolution $$ (f*g)_2 (x) = \int_0^1 {f(x \oplus t)g(t)dt,} $$ where ⊕ is the operation of dyadic addition on [0, 1).

On Lp boundedness of wave operators for 4-dimensional Schrödinger operators with threshold singularities

Let H=−Δ+V(x) be a Schrödinger operator on L2(R4), H0=−Δ. Assume that |V(x)|+|∇ V(x)|⩽C 〈 x 〉−δ for some δ>8. Let W ± = s − l i m t → ± ∞ e i t H e − i t H 0 be the wave operators. It is known that W± extend to bounded operators in Lp(R4) for all 1⩽p⩽∞, if 0 is neither an eigenvalue nor a resonance of H. We show that if 0 is an eigenvalue, but not a resonance of H, then the W± are still bounded in Lp(R4) for all p such that 4/3<p<4.

Semigroups for Generalized Birth-and-Death Equations in lp Spaces

We prove the existence of C0 semigroups related to some birth-and-death type infinite systems of ODEs with possibly unbounded coefficients, in the scale of spaces lp, $1\leq p<\infty.$ For some particular cases we also provide a characterization of the spectra of their generators. For the proof of the generation theorem in the case p > 1 we extend the Chernoff perturbation result ([9]) on relatively bounded perturbations of generators. The results presented here have been used in [5] and they play important role for analysing chaoticity of dynamical systems considered there. As a by-product of our approach we obtain a result related to the classical Shubin theorem [20]. We show that this theorem, saying that for a class of bounded infinite matrices the spectrum of the corresponding maximal operator in lp is independent on $p\in [1,\infty),$ cannot be extended to unbounded matrices.

Time Regularity for Random Walks on Locally Compact Groups

Let G be a compactly generated, locally compact group, and let T be the operator of convolution with a probability measure μ on G. Our main results give sufficient conditions on μ for the operator T to be analytic in Lp(G), 1 < p < ∞, where analyticity means that one has an estimate of form \(\Vert (I-T)T^n \Vert \leq cn^{-1}\) for all n = 1, 2, ... in Lp operator norm. Counterexamples show that analyticity may not hold if some of the conditions are not satisfied.

Singular Integrals and Commutators in Generalized Morrey Spaces

We consider a singular an integral operator \({\fancyscript K}\) with a variable Calderon–Zygmund type kernel k(x; ξ), x ∈ ℝn, ξ ∈ ℝn\{0}, satisfying a mixed homogeneity condition of the form \( k{\left( {x;\mu ^{{\alpha _{1} }} \xi _{1} , \ldots ,\mu ^{{\alpha _{n} }} \xi _{n} } \right)} = \mu ^{{ - {\sum\nolimits_{i = 1}^n {\alpha _{i} } }}} k{\left( {x;\xi } \right)},\alpha _{i} \geqslant 1 \) and μ > 0. The continuity of this operator in Lp(ℝn) is well studied by Fabes and Riviere. Our goal is to extend their result to generalized Morrey spaces Lp,ω(ℝn), p ∈ (1,∞) with a weight ω satisfying suitable dabbling and integral conditions. A special attention is paid to the commutator \({\fancyscript C}\) [a, k] = \({\fancyscript K}\)a − a\({\fancyscript K}\) with the operator of multiplication by BMO functions.

Maximal properties of the normalized Cauchy transform

We study the normalized Cauchy transform in the unit disk. Our goal is to find an analog of the classical theorem by M. Riesz for the case of arbitrary weights. Let μ \mu be a positive finite measure on the unit circle of the complex plane and f ∈ L 1 ( μ ) f\in L^{1}(\mu ) . Denote by K μ K\mu and K f μ Kf\mu the Cauchy integrals of the measures μ \mu and f μ f\mu , respectively. The normalized Cauchy transform is defined as C μ : f ↦ K f μ K μ C_{\mu }: f\mapsto \frac {Kf\mu }{K\mu } . We prove that C μ C_{\mu } is bounded as an operator in L p ( μ ) L^{p}(\mu ) for 1 &gt; p ≤ 2 1&gt;p\leq 2 but is unbounded (in general) for p &gt; 2 p&gt;2 . The associated maximal non-tangential operator is bounded for 1 &gt; p &gt; 2 1&gt;p&gt;2 and has weak type ( 2 , 2 ) (2,2) but is unbounded for p &gt; 2 p&gt;2 .

C0-groups with Polynomial Growth

We consider some conditions guaranteeing that an unbounded operator is a generator of some strongly continuous group satisfying a polynomial growth condition. The result is applied to a Friedrichs model given by T f (x) = xf(x)+ ψ (x) and considered as an operator in Lp (R). Some sufficient condition on φ and ψ are found such that iT generates an example of a group with polynomial growth.

Long time behavior of solutions of Davey-Stewartson equations

In the present paper we study the long time behavior of solutions to the Davey-Stewartson system in the Banach spaces. We make use of the properties of the semigroup generated by the linear principal operator inLp (Ω) and prove that the Davey-Stewartson system possesses a compact global attractorAp inLP (Ω). Furthermore, one show that the attractor is in fact independent ofp and prove the attractor has finite Hausdorff and fractal dimensions.

Compactness Conditions for the Green Operator in Lp(ℝ) Corresponding to a General Sturm–Liouville Operator

Compactness Conditions for the Green Operator in Lp(ℝ) Corresponding to a General Sturm–Liouville Operator