Lebesgue Space L2 Research Articles

Linear equations with infinitely many derivatives and explicit solutions to zeta nonlocal equations

We summarize our theory on existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives of the formf(∂t)ϕ=J(t),t≥0, where f is an analytic function such as the (analytic continuation of the) Riemann zeta function. We explain how to analyse initial value problems for these equations, and we prove rigorously that the functionϕ(t)=∑n≥1μ(n)nhJ(t−ln⁡(n)), in which μ is the Möbius function and J satisfies some technical conditions to be specified in Section 4, is the solution to the zeta nonlocal equationζ(∂t+h)ϕ=J(t),t≥0, in which ζ is the Riemann zeta function and h>1. We also present explicit examples of solutions to initial value problems for this equation. Our constructions can be interpreted as highlighting the importance of the cosmological daemon functions considered by Aref'eva and Volovich (2011) [1]. Our main technical tool is the Laplace transform as a unitary operator between the Lebesgue space L2 and the Hardy space H2.

Rotation number and eigenvalues of two-component modified Camassa–Holm equations

In this paper, we study the spectral problem y1y2x=−1212λm−12λn12y1y2for the two-component modified Camassa–Holm equation, where m,n are two potentials. By introducing the rotation number ρ(λ) and studying its properties, we prove that for any integer k, the periodic or anti-periodic eigenvalues are the endpoints of the interval λ∈R:ρ(λ)=−k2. Moreover, we prove that as nonlinear functionals of potentials, such eigenvalues are continuous in potentials with respect to the weak topologies in the Lebesgue space L1[0,T]. Finally, we apply the trace formula to give some estimates of the periodic eigenvalues when the L1 norms of potentials are given.

Thomson decompositions of measures in the disk

We study the classical problem of identifying the structure of P 2 ( μ ) \mathcal {P}^2(\mu ) , the closure of analytic polynomials in the Lebesgue space L 2 ( μ ) L^2(\mu ) of a compactly supported Borel measure μ \mu living in the complex plane. In his influential work, Thomson [Ann. of Math. (2) 133 (1991), pp. 477–507] showed that the space decomposes into a full L 2 L^2 -space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures μ \mu supported on the closed unit disk D ¯ \overline {\mathbb {D}} which have a part on the open disk D \mathbb {D} which is similar to the Lebesgue area measure, and a part on the unit circle T \mathbb {T} which is the restriction of the Lebesgue linear measure to a general measurable subset E E of T \mathbb {T} , we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space P 2 ( μ ) \mathcal {P}^2(\mu ) . It turns out that the space splits according to a certain natural decomposition of measurable subsets of T \mathbb {T} which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.

Matrix-valued Gabor frames over LCA groups for operators

G?vruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely ?-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator?. For a locally compact abelian groupGand a positive integer n, westudy frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space L2(G,Cn?n) , where a bounded linear operator ? on L2(G,Cn?n) controls not only lower but also the upper frame condition. We term such frames matrix-valued (?,?*)-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued (?,?*)- Gabor frames in terms of hyponormal operators. It is shown that if ? is adjointable hyponormal operator, then L2(G,Cn?n) admits a ?-tight (?,?*)-Gabor frame for every positive real number ?. A characterization of matrix-valued (?,?*)-Gabor frames is given. Finally, we show that matrix-valued (?,?*)-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.

Variational inequalities for hypersingular integrals with variable kernels

We systematically study variational inequalities for hypersingular integral operators. More precisely, we show the variational inequalities for the families 𝒯α:={Tα,𝜀}𝜀>0 of truncated hypersingular integrals with variable kernels, which are defined by Tα,𝜀f(x)=∫|x−y|>𝜀Ω(x,x−y)|x−y|n+αf(y)dy, where α≥0 and the kernel Ω belongs to L∞(ℝn)×L1(𝕊n−1). We first prove that the variation of the hypersingular integral with variable kernel is bounded from the Sobolev space L˙α2 to the Lebesgue space L2 when Ω∈L∞(ℝn)×Lq(𝕊n−1) for q>max{1,2(n−1)∕(n+2α)} and satisfies some cancellation condition in its second variable. The result is sharp in the sense that the (L˙α2,L2) boundedness of 𝒯α fails if q≤2(n−1)∕(n+2α). After strengthening the smoothness of Ω(x,z′) in its second variable, we give the weighted boundedness of the variation of the hypersingular integrals with smooth variable kernels from L˙αp(w) to Lp(w) for 1<p<∞ and w∈Ap. Finally, we extend the result to the Sobolev–Morrey spaces.

Continuous dependence and estimates of eigenvalues for periodic generalized Camassa-Holm equations

In this paper we are concerned with the spectral problem(y1y2)x=(−1212λm−12λm12)(y1y2) for the periodic generalized Camassa-Holm equations. The first aim is to study the structure of eigenvalues and prove the continuous dependence of eigenvalues on potentials. We prove that as nonlinear functionals of potentials, eigenvalues are continuous in potentials with respect to the weak topologies in the Lebesgue space L1[0,T]. The second aim is to find the estimates of the eigenvalues when the L1 norm of potentials is given, based on an application of trace formulas.

A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems

A Sobolev type embedding for radially symmetric functions on the unit ball B in Rn, n≥3, into the variable exponent Lebesgue space L2⋆+|x|α(B), 2⋆=2n/(n−2), α>0, is known due to J.M. do Ó, B. Ruf, and P. Ubilla, namely, the inequalitysup⁡{∫B|u(x)|2⋆+|x|αdx:u∈H0,rad1(B),‖∇u‖L2(B)=1}<+∞ holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on B, namely, the embeddingH0,radm(B)↪L2m⋆+|x|α(B) with 2≤m<n/2, 2m⁎=2n/(n−2m), and α>0 holds. Questions concerning the sharp constant for the inequality including the existence of the optimal functions are also studied. To illustrate the finding, an application to a boundary value problem on balls driven by polyharmonic operators is presented. This is the first in a set of our works concerning functional inequalities in the supercritical regime.

Commutators of Littlewood-Paley gκ∗ $g_{\kappa}^{*} $-functions on non-homogeneous metric measure spaces

Abstract The main purpose of this paper is to prove that the boundedness of the commutator $\mathcal{M}_{\kappa,b}^{*} $ generated by the Littlewood-Paley operator $\mathcal{M}_{\kappa}^{*} $ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of $\mathcal{M}_{\kappa}^{*} $ satisfies a certain Hörmander-type condition, the authors prove that $\mathcal{M}_{\kappa,b}^{*} $ is bounded on Lebesgue spaces Lp(μ) for 1 &lt; p &lt; ∞, bounded from the space L log L(μ) to the weak Lebesgue space L1,∞(μ), and is bounded from the atomic Hardy spaces H1(μ) to the weak Lebesgue spaces L1,∞(μ).

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ∂D. Perturbing the Cauchy problem for the Cauchy–Riemann system ∂u = f in D with boundary data on a closed subset S ⊂ ∂D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter e ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ∂D\S, each of them has a unique solution in some appropriate Hilbert space H +(D) densely embedded in the Lebesgue space L 2(∂D) and the Sobolev–Slobodetskiĭ space H 1/2−δ(D) for every δ > 0. The corresponding family of the solutions {u e} converges to a solution to the Cauchy problem in H +(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H +(D) is equivalent to boundedness of the family {u e} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.

Hardy-type space estimates for multilinear commutators of Calderón–Zygmund operators on nonhomogeneous metric measure spaces

Let (X,d,μ) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let T be a Calderón–Zygmund operator and let b→:=(b1,…,bm) be a finite family of \widetilde{RBMO}(μ) functions. In this article, the authors establish the boundedness of the multilinear commutator Tb→, generated by T and b→ from the atomic Hardy-type space H˜fin,b→,m,ρ1,q,m+1(μ) into the Lebesgue space L1(μ). The authors also prove that Tb→ is bounded from the atomic Hardy-type space H˜fin,b→,m,ρ1,q,m+2(μ) into the atomic Hardy space H˜1(μ) via the molecular characterization of H˜1(μ).

A general Beurling–Helson–Lowdenslager theorem on the disk

The classical Beurling–Helson–Lowdenslager theorem characterizes the shift-invariant subspaces of the Hardy space H2 and of the Lebesgue space L2. In this paper, which is self-contained, we define a very general class of norms α and define spaces Hα and Lα. We then extend the Beurling–Helson–Lowdenslager invariant subspace theorem. The idea of the proof is new and quite simple; most of the details involve extending basic well-known ‖⋅‖p-results for our more general norms.

A computable bound of the essential spectral radius of finite range Metropolis–Hastings kernels

Let π be a positive continuous target density on R. Let P be the Metropolis–Hastings operator on the Lebesgue space L2(π) corresponding to a proposal Markov kernel Q on R. When using the quasi-compactness method to estimate the spectral gap of P, a mandatory first step is to obtain an accurate bound of the essential spectral radius ress(P) of P. In this paper a computable bound of ress(P) is obtained under the following assumption on the proposal kernel: Q has a bounded continuous density q(x,y) on R2 satisfying the following finite range assumption : |u|>s⇒q(x,x+u)=0 (for some s>0). This result is illustrated with Random Walk Metropolis–Hastings kernels.

On stochastic evolution equations for nonlinear bipolar fluids: Well-posedness and some properties of the solution

We investigate the stochastic evolution equations describing the motion of a non-Newtonian fluids excited by multiplicative noise of Lévy type. We show that the system we consider has a unique global strong solution. We also give some results concerning the properties of the solution. We mainly prove that the unique solution satisfies the Markov–Feller property. This enables us to prove by means of some results from ergodic theory that the semigroup associated to the unique solution admits at least an invariant measure which is ergodic and tight on a subspace of the Lebesgue space L2.

Estimates for Parameter Littlewood-Paleygκ⁎Functions on Nonhomogeneous Metric Measure Spaces

Let(X,d,μ)be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measure conditions. In this paper, the authors prove that, under the assumption that the kernel ofMκ⁎satisfies a certain Hörmander-type condition,Mκ⁎,ρis bounded from Lebesgue spacesLp(μ)to Lebesgue spacesLp(μ)forp≥2and is bounded fromL1(μ)intoL1,∞(μ). As a corollary,Mκ⁎,ρis bounded onLp(μ)for1&lt;p&lt;2. In addition, the authors also obtain thatMκ⁎,ρis bounded from the atomic Hardy spaceH1(μ)into the Lebesgue spaceL1(μ).

A sharp error estimate of the best approximation by algebraic polynomials in the weighted space L 2(−1, 1)

We obtain an exact error estimate of the best approximation by algebraic polynomials in the Lebesgue space L 2(−1, 1) with the weight 1 − x 2 of degree λ > −1.

STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES

AbstractLetXbe a locally compact space, and 𝔏∞0(X,ι) be the space of all essentially boundedι-measurable functionsfonXvanishing at infinity. We introduce and study a locally convex topologyβ1(X,ι) on the Lebesgue space 𝔏1(X,ι) such that the strong dual of (𝔏1(X,ι),β1(X,ι)) can be identified with$({\frak L}_0^\infty (X,\iota ),\|\cdot \|_\infty )$. Next, by showing thatβ1(X,ι) can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove thatL1(G) , the group algebra of a locally compact Hausdorff topological groupG, equipped with the convolution multiplication is a complete semitopological algebra under theβ1(G) topology.

Walrasian Equilibrium Problem with Memory Term

The aim of this paper is to study the Walrasian equilibrium problem when the data are time-dependent. In order to have a more realistic model, the excess demand function depends on the current price and on previous events of the market. Hence, a memory term is introduced; it describes the precedent states of the equilibrium. This model is reformulated as an evolutionary variational inequality in the Lebesgue space L2([0,T],ℝ), and, thanks to this characterization, existence and qualitative results on equilibrium solution are given.

Sharp estimates for the commutators of the Hilbert, Riesz transforms and the Beurling-Ahlfors operator on weighted Lebesgue spaces

We prove that the operator norm on weighted Lebesgue space L2(w) of the commutators of the Hilbert, Riesz and Beurling transforms with a BMO function b depends quadratically on the A2-characteristic of the weight, as opposed to the linear dependence known to hold for the operators themselves. It is known that the operator norms of these commutators can be controlled by the norm of the commutator with appropriate Haar shift operators, and we prove the estimate for these commutators. For the shift operator corresponding to the Hilbert transform we use Bellman function methods, however there is now a general theorem for a class of Haar shift operators that can be used instead to deduce similar results. We invoke this general theorem to obtain the corresponding result for the Riesz transforms and the Beurling-Ahlfors operator. We can then extrapolate to Lp(w), and the results are sharp for 1 < p < 1.

A Remark on the Stokes Problem with Initial Data in L 1

In this note we study the initial boundary value problem of the Stokes system. We assume the initial data belonging to the Lebesgue space L 1. We develop a suitable approach to discuss the existence, uniqueness and continuous dependence on the data. As matter of course, we also consider the case of the Lebesgue space L p , $${p\in(1,\infty)}$$ .

The Cauchy problem for operators with injective symbol in the Lebesgue space L 2 in a domain

Let D be a bounded domain in ℝn (n ≥ 2) with infinitely smooth boundary ∂D. We give some necessary and sufficient conditions for the Cauchy problem to be solvable in the Lebesgue space L2(D) in D for an arbitrary differential operator A having an injective principal symbol. Furthermore, using bases with double orthogonality, we construct Carleman’s formula that restores a (vector-)function in L2(D) from the Cauchy data given on a relatively open connected set Γ ⊂ ∂D and the values Au in D whenever the data belong to L2(Γ) and L2(D) respectively.