Lebesgue Spaces Lp Research Articles

Weighted boundedness for commutators of parameterized Littlewood-Paley operators and area integrals

We establish the boundedness for commutators of parameterized Littlewood-Paley operators and area integrals on weighted Lebesgue spaces Lp(?) when 1 < p < ?, where the kernel satisfies certain logarithmic type Lipschitz condition. Moreover, the weighted endpoint estimates when p = 1 are also obtained.

Variable exponents and grand Lebesgue spaces: Some optimal results

Consider p : Ω → [1, +∞[, a measurable bounded function on a bounded set Ø with decreasing rearrangement p* : [0, |Ω|] → [1, +∞[. We construct a rearrangement invariant space with variable exponent p* denoted by [Formula: see text]. According to the growth of p*, we compare this space to the Lebesgue spaces or grand Lebesgue spaces. In particular, if p*(⋅) satisfies the log-Hölder continuity at zero, then it is contained in the grand Lebesgue space Lp*(0))(Ω). This inclusion fails to be true if we impose a slower growth as [Formula: see text] at zero. Some other results are discussed.

QUASI RIESZ TRANSFORMS, HARDY SPACES AND GENERALISED SUB-GAUSSIAN HEAT KERNEL ESTIMATES

In this thesis, we mainly study Riesz transforms and Hardy spaces associated to operators. The two subjects are closely related to volume growth and heat kernel estimates. In Chapter 1, 2 and 4, we study Riesz transforms on Riemannian manifold and on graphs. In Chapter 1, we prove that on a complete Riemannian manifold, the quasi Riesz transform is always Lp bounded on for p strictly large than 1 and no less than 2. In Chapter 2, we prove that the quasi Riesz transform is also weak L1 bounded if the manifold satisfies the doubling volume property and the sub-Gaussian heat kernel estimate. Similarly, we show in Chapter 4 the same results on graphs. In Chapter 3, we develop a Hardy space theory on metric measure spaces satisfying the doubling volume property and different local and global heat kernel estimates. Firstly we define Hardy spaces via molecules and via square functions which are adapted to the heat kernel estimates. Then we show that the two H1 spaces via molecules and via square functions are the same. Also, we compare the Hp space defined via square functions with Lp. The corresponding Hp space for p large than 1 defined via square functions is equivalent to the Lebesgue space Lp. However, it is shown that in this situation, the Hp space corresponding to Gaussian estimates does not coincide with Lp any more. Finally, as an application of this Hardy space theory, we proved that quasi Riesz transforms are bounded from H1 to L1 on fractal manifolds. In Chapter 5, we consider Vicsek graphs. We prove generalised Poincare inequalities and Sobolev inequalities on Vicsek graphs and we show that they are optimal.

On the maximal operators of Fejér means with respect to the character system of the group of 2-adic integers in hardy spaces

It was a question of Taibleson, open for a long time that the almost everywhere convergence of Fejer (or (C, 1)) means of Fourier series of integrable functions with respect the character system of the group of 2-adic integers. This question was answered by Ga t in 1997. The aim of this paper is to investigate the maximal operator of the supn |σn|. Among other things, we prove that this operator is bounded from the Hardy space Hp to the Lebesgue space Lp if and only if 1/2 < p < ∞. The two-dimensional maximal operator is also discussed.

The Hodge–de Rham Laplacian and [formula omitted]-boundedness of Riesz transforms on non-compact manifolds

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property as well as a Gaussian upper bound for the corresponding heat kernel. We study the boundedness of the Riesz transform dΔ−12 on both Hardy spaces Hp and Lebesgue spaces Lp under two different conditions on the negative part R− of the Ricci curvature. First we prove that if R− is α-subcritical for some α∈[0,1), then the Riesz transform d∗Δ⃗−12 on differential 1-forms is bounded from the associated Hardy space HΔ⃗p(Λ1T∗M) to Lp(M) for all p∈[1,2]. As a consequence, dΔ−12 is bounded on Lp for all p∈(1,p0) where p0>2 depends on α and the constant appearing in the doubling property. Second, we prove that if∫01‖|R−|12v(⋅,t)1p1‖p1dtt+∫1∞‖|R−|12v(⋅,t)1p2‖p2dtt<∞, for some p1>2 and p2>3, then the Riesz transform dΔ−12 is bounded on Lp for all 1<p<p2. Furthermore, we study the boundedness of the Riesz transform of Schrödinger operators A=Δ+V on Lp for p>2 under conditions on R− and the potential V. We prove both positive and negative results on the boundedness of dA−12 on Lp.

Topological transitivity for cosine operator functions on groups

Let G be a locally compact group and let 1≤p<∞. We give a necessary and sufficient condition for a cosine operator function induced by a weighted translation operator on the Lebesgue space Lp(G) to be topologically transitive. By the same argument, the characterization for mixing cosine operator functions is given. We also determine when the direct sum of a sequence of cosine operator functions is transitive.

Nonlinear convolution-type equations in Lebesgue spaces

Methods of the theory of monotone operators are used to prove global theorems on the existence and uniqueness of solutions, as well as on estimates of their norms, for various classes of nonlinear integral convolution-type equations in the real Lebesgue spaces Lp(0, 1). These theorems involve nonlinear equations with potential-type kernels, including logarithmic potential-type kernels, as well as the corresponding linear integral equations within the framework of the space L2(0, 1). Corollaries illustrating the obtained results are presented.

Hardy and uncertainty inequalities on stratified Lie groups

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group G. In particular, we show that the operators Tα:f↦|⋅|−αL−α/2f, where |⋅| is a homogeneous norm, 0<α<Q/p, and L is the sub-Laplacian, are bounded on the Lebesgue space Lp(G). As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg–Pauli–Weyl inequality, relating the Lp norm of a function f to the Lq norm of |⋅|βf and the Lr norm of Lδ/2f.

Boundedness of Commutators of Marcinkiewicz Integrals on Nonhomogeneous Metric Measure Spaces

Let(X,d,μ)be a metric measure space satisfying the upper doubling condition and geometrically doubling condition in the sense of Hytönen. The aim of this paper is to establish the boundedness of commutatorMbgenerated by the Marcinkiewicz integralMand Lipschitz functionb. The authors prove thatMbis bounded from the Lebesgue spacesLp(μ)to weak Lebesgue spacesLq(μ)for1≤p&lt;n/β, from the Lebesgue spacesLp(μ)to the spacesRBMO(μ)forp=n/β, and from the Lebesgue spacesLp(μ)to the Lipschitz spacesLip(β-n/p)(μ)forn/β&lt;p≤∞. Moreover, some results in Morrey spaces and Hardy spaces are also discussed.

Boyd Indices in Generalized Grand Lebesgue Spaces and Applications

We compute the Boyd indices of generalized grand Lebesgue spaces L p),δ and give some applications.

On the weighted variable exponent amalgam space [formula omitted

In [4], a new family W (Lp(x),Lmq) of Wiener amalgam spaces was defined and investigated some properties of these spaces, where local component is a variable exponent Lebesgue space Lp(x) (ℝ) and the global component is a weighted Lebesgue space Lqm(ℝ). This present paper is a sequel to our work [4]. In Section 2, we discuss necessary and sufficient conditions for the equality W (Lp(x),Lmq)=Lq(ℝ). Later we give some characterization of Wiener amalgam space W (Lp(x),Lmq). In Section 3 we define the Wiener amalgam space W (FLp(x),Lmq) and investigate some properties of this space, where FLp(x) is the image of Lp(x) under the Fourier transform. In Section 4, we discuss boundedness of the Hardy-Littlewood maximal operator between some Wiener amalgam spaces.

Chaos for Cosine Operator Functions on Groups

Let1≤p&lt;∞andGbe a locally compact group. We characterize chaotic cosine operator functions, generated by weighted translations on the Lebesgue spaceLp(G), in terms of the weight condition. In particular, chaotic cosine operator functions and chaotic weighted translations can only occur simultaneously. We also give a necessary and sufficient condition for the direct sum of a sequence of cosine operator functions to be chaotic.

On density and π-weight of Lp(βN,R, μ)

The new topological concept of selective separability is able to give some estimates for density and π-weight of the Lebesgue space Lp(βN,R, μ) with 1 ≤ p &lt; +∞. In particular, we deduce a purely topological proof of the non-separability of such a space.

The principal frequency of [formula omitted] as a limit of Rayleigh quotients involving Luxemburg norms

The asymptotic behavior of Rayleigh quotients involving both Luxemburg norms and modulars in the variable exponent Lebesgue space Lp(⋅) is studied as p(⋅)→∞. In a particular case, we recover a well-known result of Juutinen, Lindqvist and Manfredi regarding the limit, as p→∞ of the minima of Rayleigh quotients associated to the eigenvalue problem for the p-Laplacian.

Cesàro Operators on the Hardy Spaces of the Half-Plane

AbstractIn this article we study the Cesàro operatorand its companion operator 𝒯 on Hardy spaces of the upper half plane. We identify 𝒞 and 𝒯 as resolvents for appropriate semigroups of composition operators and we find the norm and the spectrum in each case. The relation of 𝒞 and 𝒯 with the corresponding Cesàro operators on Lebesgue spaces Lp(ℝ) of the boundary line is also discussed.

Grand Sobolev spaces and their applications in geometric function theory and PDEs

Function spaces that are slightly larger than the Lebesgue Lp(Ω) spaces (even larger than the Marcinkiewicz Lp,∞(Ω) spaces) have been introduced by Iwaniec and Sbordone [Arch. Ration. Mech. Anal. 119 (1992), 129–143] in connection with integrability properties of the Jacobian. These are the grand Lebesgue spaces Lp)(Ω). In this survey we collect a number of results which prove that these spaces are useful in various classical settings of geometric function theory and partial differential equations (PDEs).

Characterization of the Weak-Type Boundedness of the Hilbert Transform on Weighted Lorentz Spaces

We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces \(\varLambda^{p}_{u}(w)\), with p>0, in terms of some geometric conditions on the weights u and w and the weak-type boundedness of the Hardy–Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of H on weighted Lebesgue spaces Lp(u) and Muckenhoupt weights Ap, and the theory on classical Lorentz spaces Λp(w) and Arino-Muckenhoupt weights Bp.

Grand Lebesgue spaces with respect to measurable functions

Let 1<p<∞. Given Ω⊂Rn a measurable set of finite Lebesgue measure, the norm of the grand Lebesgue spaces Lp)(Ω) is given by |f|Lp)(Ω)=sup0<ε<p−1ε1p−ε(1|Ω|∫Ω|f|p−εdx)1p−ε. In this paper we consider the norm |f|Lp),δ(Ω) obtained replacing ε1p−ε by a generic nonnegative measurable function δ(ε). We find necessary and sufficient conditions on δ in order to get a functional equivalent to a Banach function norm, and we determine the “interesting” class Bp of functions δ, with the property that every generalized function norm is equivalent to a function norm built with δ∈Bp. We then define the Lp),δ(Ω) spaces, prove some embedding results and conclude with the proof of the generalized Hardy inequality.

L p -L q Estimates for Bergman Projections in Bounded Symmetric Domains of Tube Type

Let D be an irreducible bounded symmetric domain of tube type in ℂ n . The class of Bloch functions is well known in this context, in connection with Hankel operators or duality of Bergman spaces. Contrary to what happens in the unit ball, Bloch functions do not belong to all Lebesgue spaces L p (D) for p<∞ in higher rank. We give here both necessary and sufficient conditions on p for such an embedding. This question is equivalent to local boundedness properties of the Bergman projection in the tube domain over a symmetric cone that is conformally equivalent to D. We are linked to consider L ∞–L q inequalities on symmetric cones, which may be of independent interest, and study more systematically estimates with loss for the Bergman projection. The proofs are based on a very precise estimate on an integral related to the Gamma function of a symmetric cone.

On the factorization of bounded measurable functions in the classes of Cauchy type integrals with density from Lp(·)(Γ;ω)

Let Γ be a simple rectifiable closed Carleson curve and be its measurable bounded nonzero function on Γ. It is proved in the paper that the factorization of such a function in the class of Cauchy type integrals with density from the Lebesgue space Lp(·)(Γ;ω) with variable exponent is equivalent to the property of the singular integral operator , where is the operator generated by a Cauchy singular operator, to be Noetherian in the space Lp(·)(Γ;ω). This result generalizes a theorem of I. B. Simonenko.