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Hardy's inequality with weights
... One important tool in the proof of Theorem 1, to be given in Section 2, is the description due to Muckenhoupt [13] of the weights U and V such that the Hardy operators ş x 0 f ptq dt and ş 8 x f ptq dt are bounded from the Lebesgue space L s pU s , p0, 8qq to L s pV s , p0, 8qq. ...
... To prove the estimate S l pF q À }λ} q ℓ p,q for l " 1, 2 we will use the characterization, obtained by Muckenhoupt [13], of the weights U and V such that the Hardy operators ş x 0 f ptq dt and ...
... Therefore the estimate S 1 pF q À }λ} q ℓ p,q follows by [13,Theorem 2] once it is shown that ...
The purpose of this paper is to establish an atomic decomposition for functions in the weighted mixed norm space induced by a radial weight in the unit disc admitting a two-sided doubling condition. The obtained decomposition is further applied to characterize Carleson measures for , and bounded differentiation operators acting from to , induced by a positive Borel measure , on the full range of parameters .
... Different types of integral transforms require different techniques for obtaining sharp weighted norm inequalities between Lebesgue spaces. We begin by mentioning weighted Hardy-type inequalities, see, e.g., [6,16,20,22]. These inequalities usually serve as a basis for deriving corresponding weighted norm inequalities for other integral operators (in particular, the Hardy-type inequalities in Subsection 3.2 below are the main tool we use in this paper). ...
... The main tools we use in order to obtain sufficient conditions for (3) are Hardy's inequalities [6,16]. Let us introduce the following notation before. ...
... which is finite, by (16). On the other hand, by (18) and (20), ...
We obtain necessary and sufficient conditions on weights for a wide class of integral transforms to be bounded between weighted spaces, with . The kernels K(x,y) of such transforms are only assumed to satisfy upper bounds given by products of two functions, one in each variable. The obtained results are applicable to a number of transforms, some of which are included here as particular examples. Some of the new results derived here are the characterization of weights for the boundedness of the (or Struve) transform in the case , or the characterization of power weights for which the Laplace transform is bounded in the limiting cases p=1 or .
... This is achieved through an estimate for the discrete Hardy inequalities involving two general measures µ, ν on the discrete half-line. This estimate is directly adapted from Muckenhoupt [23] for the analogous continuous result and from Miclo [21] for the discrete case when p = 2. A variant of this result for rooted trees has been explored in the literature. ...
... The aim of this section is to provide a discrete form of the Muckenhoupt criterion related to the existence of a weighted p-Hardy inequality for prescribed weights (see [23]). The results presented generalize Miclo's work [21] for p = 2 extending the method to a wider range of p-values and additional relevant cases. ...
In this paper, we prove a p-Hardy inequality on the discrete half-line with weights for all real . Building on the work of Miclo for and Muckenhoupt in the continuous settings, we develop a quantitative approach for the existence of a p-Hardy inequality involving two measures and on the discrete half-line. We also investigate the comparison between sharp constants in the discrete and continuous settings and explore the stability of the inequality in the discrete case.
... For probability measures on the real line the necessary and sufficient condition for Poincaré inequality characterising the density (of the absolutely continuous part with respect to the Lebesgue measure) were established long time ago by Muckenphout, [36], ( [34]). More recently such criteria were established for other coercive inequalities (Log-Sobolev type: (LS 2 ) [7] , (LS q ) [10], for distributions with weaker tails [5],...). ...
... In case of measures on real line the following necessary and sufficient condition for Poincaré inequality to hold was provided by Muckenhoupt [36] ( [2]) which in the special case of a measure dµ ≡ ρdx can be stated as follows: Given q ∈ [1, ∞) and 1 q + 1 p = 1 ...
We study coercive inequalities on finite dimensional metric spaces with probability measures which do not have volume doubling property. This class of inequalities includes Poincar\'e and Log-Sobolev inequality. Our main result is proof of Log-Sobolev inequality on Heisenberg group equipped with either heat kernel measure or "gaussian" density build from optimal control distance. As intermediate results we prove so called U-bounds.
... Now, we present our Hardy-type inequality developed from the Hardy's inequality in [43,Theorem 1.14] or [42,Theorem 4] by Muckenhoupt (explained in Theorem C.2 within our context), over the domain of the whole real line. ...
... Again, using [43,Theorem 1.14] or [42,Theorem 4] by Muckenhoupt (described by Theorem C.2 within our context) withũ(−d) = 0 , since (3.49) is valid, it follows that the Hardy-type inequality (3.48) (equivalently, (3.47)) holds; furthermore, ifC L is the smallest constant for which (3.47) is satisfied, then ...
We consider the Kuramoto-Sakaguchi-Fokker-Planck equation (namely, parabolic Kuramoto-Sakaguchi, or Kuramoto-Sakaguchi equation, which is a nonlinear parabolic integro-differential equation) with inertia and white noise effects. We study the hydrodynamic limit of this Kuramoto-Sakaguchi equation. During showing this main result, as a support, we also prove a Hardy-type inequality over the whole real line.
... This section is dedicated to proving Pitt's inequality with general weights, specifically Theorem 1.4. Before we delve into the proof, we need to recall a few results by Bradley [Bra78], Calderon [Cal66], and Muckenhoupt [Muc72]. ...
... Theorem 3.2 ([Muc72,Bra78]). Let 1 ≤ p ≤ q ≤ ∞. ...
This article explores weighted inequalities for the Fourier transform in rank one Riemannian symmetric spaces of noncompact type. We establish both necessary and sufficient conditions for these inequalities to hold. To prove the weighted Fourier inequalities, we apply restriction theorems on symmetric spaces and utilize Calder{\'o}n's estimate for sublinear operators. While establishing the necessary conditions, we demonstrate that Harish-Chandra's elementary spherical functions play a crucial role in this setting. Furthermore, we apply our findings to derive Fourier inequalities with polynomial and exponential weights.
... holds. There is a lot of literature available for different forms of Hardy inequalities and it is not an easy task to mention all, but a few of the references are [6,7,8,9,14,15,18,19,21,20,23,24,27,28,29,30,31,32]. In [33], Sinnamon obtained Hardy inequalities for the case 0 < q < 1 < p < ∞. ...
... Again, in [34], Sinnamon and Stepanov established Hardy inequalities for the case 0 < q < 1 and p = 1. For the case p ≤ q, many authors Bradley [5], Talenti [35] and Muckenhoupt [23] obtained Hardy inequalities in different perspectives. In [25,26], the first author with third author obtained the Hardy inequalities on metric measure spaces for the case 1 < p ≤ q < ∞, and 0 < q < p, 1 < p < ∞. ...
... The initial problem in the theory of weighted Hardy inequalities was the one of characterizing the positive functions w, v, the weights, such that This problem was solved by Talenti [31], Muckenhoupt [23] and Bradley [4] in the case p ≤ q, by Mazja [22] when 1 ≤ q < p, Sinnamon [27,28] for 0 < q < 1 < p Communicated by Feng Dai. B P. Ortega Salvador portega@uma.es ...
... Theorem A ( [4,22,23,29,31]) Let 1 < q < ∞, 1 ≤ p < ∞ and let w, v be positive measurable functions on ( (ii) in the case q < p, Weighted weak-type inequalities for T were also studied. By a weighted weak-type ( p, q) inequality for T we mean the boundedness of T from L p (v) to L q,∞ (w), where ...
We characterize the good weights for some weighted weak-type iterated and bilinear modified Hardy inequalities to hold.
... It also includes the Riemann-Liouville integrals. The mapping properties of the general Hardy-type operators on Lebesgue spaces and extensions of Lebesgue spaces were investigated in [1,2,4,11,12,17,18,27,28,31,33,36,37,38,39]. ...
This paper extends the mapping properties of the general Hardy-type operators to local generalized Morrey spaces built on ball quasi-Banach function spaces. As applications of the main result, we establish the two weight norm inequalities of the Hardy operators to the local generalized Morrey spaces, the mapping properties of the Riemann-Liouville integrals on local generalized Morrey spaces built on rearrangement-invariant quasi-Banach function spaces, the Hardy inequalities on the local generalized Morrey spaces with variable exponents .
... To prove Proposition 2.1 we will need some classical results on doubly weighted one-dimensional Hardy inequalities. For the proof we refer to [36,43], see also [23] and references therein. ...
By the Aharonov–Casher theorem, the Pauli operator P has no zero eigenvalue when the normalized magnetic flux satisfies , but it does have a zero energy resonance. We prove that in this case a Lieb–Thirring inequality for the -th moment of the eigenvalues of P+V is valid under the optimal restrictions and . Besides the usual semiclassical integral, the right side of our inequality involves an integral where the zero energy resonance state appears explicitly. Our inequality improves earlier works that were restricted to moments of order .
... Hardy's result has been generalized in various ways, of which we will mention some, which have inspired this paper. For 1 ≤ p ≤ q ≤ ∞ and non-negative measurable functions u and v on R + , Muckenhoupt ( [5]) and Bradley ([2]) gave a necessary and sufficient condition for the existence of a constant C such that ...
We characterize those non-negative, measurable functions on [0,1] and positive, continuous functions and on for which the generalized Hardy-Ces\`aro operator defines a bounded operator . Furthermore, we extend to a bounded operator on with range in . Finally, we show that the zero operator is the only weakly compact generalized Hardy-Ces\`aro operator from to .
... Proof. (14) implies that ...
Stability of the traveling wave solution to a general class of one-dimensional nonlocal evolution equations is studied in -spaces, thereby providing an alternative approach to the usual spectral analysis with respect to the supremum norm. We prove that the linearization around the traveling wave solution satisfies a Lyapunov-type stability condition in a weighted space for a naturally associated density . The result can be applied to obtain stability of the traveling wave solution under stochastic perturbations of additive or multiplicative type. For small wave speeds, we also prove an alternative Lyapunov-type stability condition in , where is the symmetrizing density for the traveling wave operator, which allows to derive a long-term stochastic stability result.
... This is a weighted norm inequality for the Hardy operator. Applying [21] (see also [23] and references therein), we see that (2.13) and therefore (2.12) holds true. It is easy to see that the latter implies (2.10) and statement (i) of the theorem. ...
We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by the J-self-adjoint linear operator L depending on a small parameter. The problem has been originated from the lubrication approximation of a viscous fluid film on the inner surface of the rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on the numerical evidence, that the set of eigenvectors of the operator L does not form a Riesz basis in \L^2 (-\pi,\pi). Our method can be applied to a wide range of the evolutional problems given by symmetric operators.
... Unaware of their independent proofs, Chisholm and Everitt [11] established Theorem 2.1 in 1971; see also [12] for a more general result in the conjugate index case 1/p + 1/q = 1. In addition, a 1972 paper by Muckenhoupt [34] has a result which contains Theorem 2.1. For further information, there is an excellent historical account of Theorem 2.1 in the book [26,Ch. ...
In 1961, Birman proved a sequence of inequalities for valid for functions in In particular, is the classical (integral) Hardy inequality and is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space of functions defined on Moreover, implies as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite these inequalities hold on the standard Sobolev space . Furthermore, in all cases, the Birman constants in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in (resp., ). We also show that these Birman constants are related to the norm of a generalized continuous Ces\`aro averaging operator whose spectral properties we determine in detail.
... The main ingredient of the proof is a weighted logarithmic Sobolev inequality, which we adopt to our setting with Dirichlet boundary conditions. Therefore, we slightly modify the arguments in [2,4,5,8] to deduce a criterion for logarithmic Sobolev inequalities on the positive half real line incorporating functions with fixed boundary conditions at 0. These kind of inequalities have there origin in the Muckenhoupt criterion [34]. Proposition 5.6. ...
This paper concerns a Fokker-Planck equation on the positive real line modeling nucleation and growth of clusters. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system. The first main result concerns the well-posedness and regularity of the Cauchy problem. The well-posedness is based on a fixed point argument, and the regularity on Schauder estimates. The first a priori estimates yield H\"older regularity of the non-local order parameter, which is improved by an iteration argument. The asymptotic behavior of solutions depends on some order parameter depending on the initial data. The system shows different behavior depending on a value , determined from the potentials and diffusion coefficient. For , there exists an equilibrium solution . If the solution converges strongly to , while if the solution converges weakly to . The excess gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the classical Becker-D\"oring equation. The system possesses a free energy, strictly decreasing along the evolution, which establishes the long time behavior. In the subcritical case the entropy method, based on suitable weighted logarithmic Sobolev inequalities and interpolation estimates, is used to obtain explicit convergence rates to the equilibrium solution. The close connection of the presented model and the Becker-D\"oring model is outlined by a family of discrete Fokker-Planck type equations interpolating between both of them. This family of models possesses a gradient flow structure, emphasizing their commonality.
... (i)⇒(iii). This part of the proof uses ideas from [9]. For r ∈ [0, 1), set 2). ...
Let denote the Dirichlet type space in the unit disc induced by a radial weight v for which satisfies the doubling property In this paper, we characterize the Schatten classes of the generalized Hilbert operators \begin{equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt \end{equation*} acting on , where v satisfies the Muckenhoupt-type conditions and For , it is proved that if and only if \begin{equation*} \int_0^1 \left((1-r)\int_{-\pi}^\pi |g'(re^{i\theta})|^2\,d\theta\right)^{\frac{p}{2}}\frac{dr}{1-r} <\infty. \end{equation*}
... However, such inequalities have been known to analysts for a long time under the name of 'Hardy inequalities with weights' (see e.g. Muckenhoupt [42]). ...
Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of `second order Poincar\'e inequalities': just as ordinary Poincar\'e inequalities give variance bounds, second order Poincar\'e inequalities give central limit theorems. The proof of the main result employs Stein's method of normal approximation. A number of examples are worked out, some of which are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.
... An identical argument follows for the coarse-grained density π 0 pxq. Finally, using the fact that then the corresponding Gibbs distribution π pxq will not satisfy Poincaré's inequality for any ą 0. Following[22, Appendix A] we demonstrate this by checking that this choice of π does not satisfy the Muckenhoupt criterion[34,4] which is necessary and sufficient for the Poincaré inequality to hold, namely that sup rPR B˘prq ă 8, where ...
We study the problem of Brownian motion in a multiscale potential. The potential is assumed to have N+1 scales (i.e. N small scales and one macroscale) and to depend periodically on all the small scales. We show that for nonseparable potentials, i.e. potentials in which the microscales and the macroscale are fully coupled, the homogenized equation is an overdamped Langevin equation with multiplicative noise driven by the free energy, for which the detailed balance condition still holds. The calculation of the effective diffusion tensor requires the solution of a system of N coupled Poisson equations.
... Several general conditions for the exponential decay to equilibrium of one-dimensional diffusions have been proposed in the literature. Some of them are based on the existence of a Poincaré or Hardy-type inequalities w.r.t the invariant measure, or related variational-type integral criteria w.r.t the scale and the speed measures; see for instance [18], as well as the articles [7,60] and [17] for some particular classes of heavy tailed limiting distributions. Another more technical route is to estimate the transition densities of the process, and to find judicious Lyapunov functions as in [57], or to use coupling and transport techniques as in [19,34,35]. ...
This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These one-dimensional stochastic differential equations exhibit a quadratic drift function and a non-Lipschitz continuous diffusion function. We present a novel approach, combining tangent process techniques, Feynman-Kac path integration, and exponential change of measures, to derive sharp exponential decays to equilibrium. We also provide uniform estimates with respect to the time horizon, quantifying with some precision the fluctuations of these diffusions around a limiting deterministic Riccati differential equation. These results provide a stronger and almost sure version of the conventional central limit theorem. We illustrate these results in the context of ensemble Kalman-Bucy filtering. To the best of our knowledge, the exponential stability and the fluctuation analysis developed in this work are the first results of this kind for this class of nonlinear diffusions.
... Moreover it reveals to have strong consequences in high-dimensional analysis, in connection with other important functional inequalities, isoperimetry and concentration of measure, cf. for instance the work of Bobkov and Ledoux [7] about general convex measures including heavytailed distributions, i.e., probability distributions whose tails decay is slower than exponential. In the one-dimensional case, the analysis can be further explored either by considering Hardy-type inequalities and Sturm-Liouville equations [4,5,22] or using Stein's method as in the papers [15,29] in which the weight corresponds to the so-called Stein kernel. Recently, new theoretical guarantees have been proposed in [8,9] by means of the intertwining technique. ...
One-dimensional Poincare inequalities are used in Global Sensitivity Analysis (GSA) to provide derivative-based upper bounds and approximations of Sobol indices. We add new perspectives by investigating weighted Poincare inequalities. Our contributions are twofold. In a first part, we provide new theoretical results for weighted Poincare inequalities, guided by GSA needs. We revisit the construction of weights from monotonic functions, providing a new proof from a spectral point of view. In this approach, given a monotonic function g, the weight is built such that g is the first non-trivial eigenfunction of a convenient diffusion operator. This allows us to reconsider the linear standard, i.e. the weight associated to a linear g. In particular, we construct weights that guarantee the existence of an orthonormal basis of eigenfunctions, leading to approximation of Sobol indices with Parseval formulas. In a second part, we develop specific methods for GSA. We study the equality case of the upper bound of a total Sobol index, and link the sharpness of the inequality to the proximity of the main effect to the eigenfunction. This leads us to theoretically investigate the construction of data-driven weights from estimators of the main effects when they are monotonic, another extension of the linear standard. Finally, we illustrate the benefits of using weights on a GSA study of two toy models and a real flooding application, involving the Poincare constant and/or the whole eigenbasis.
... However, this result is also new for the classical p-Hardy inequality, and it strengthens the main results of [47] by allowing more general weights. In the one dimensional Euclidean case, weighted Hardy inequalities and characterizations thereof have appeared in [51]. ...
In this paper, we prove a self-improvement result for -fractional Hardy inequalities, in both the exponent and the regularity parameter , for bounded domains in doubling metric measure spaces. The key conceptual tool is a Caffarelli-Silvestre-type argument, which relates fractional Sobolev spaces on Z to Newton-Sobolev spaces in the hyperbolic filling of Z via trace results. Using this insight, it is shown that a fractional Hardy inequality in an open subset of Z is equivalent to a classical Hardy inequality in the filling . The main result is then obtained by applying a new weighted self-improvement result for p-Hardy inequalities. The exponent p can be self-improved by a classical Koskela-Zhong argument, but a new theory of regularizable weights is developed to obtain the self-improvement in the regularity parameter . This generalizes a result of Lehrb\"ack and Koskela on self-improvement of -weighted p-Hardy inequalities by allowing a much broader class of weights. Using the equivalence of fractional Hardy inequalities with Hardy inequalities in the fillings, we also give new examples of domains satisfying fractional Hardy inequalities.
... To begin, we note that there has recently been much interest in improving the Hardy inequality by adding positive remainder terms to the right hand side of (1.1). One of the first results in this direction was obtained by Maz'ya in [38] where it was shown that, for those smooth functions on the half-space ...
Firstly, this paper establishes useful forms of the remainder term of Hardy-type inequalities on general domains where the weights are functions of the distance to the boundary. For weakly mean convex domains we use the resulting identities to establish nonexistence of extremizers for and improve known sharp Hardy inequalities. Secondly, we establish geometrically interesting remainders for the Davies-Hardy-Tidblom inequalities for the mean distance function, as well as generalize and improve several Hardy type inequalities in the spirit of Brezis and Marcus and spectral estimates of Davies. Lastly, we apply our results to obtain Sobolev inequalities for non-regular Riemannian metrics on geometric exterior domains.
... Letting p = q > 1, and as the Lebesgue measure and d = 1/xd yields the classical Hardy inequality proved in the 1925 paper [4], which holds with best constant p/(p − 1). Muckenhoupt, in [8], showed that letting and be absolutely continuous with respect to the Lebesgue measure, the inequality holds if and only if a one-parameter supremum is finite. Bradley, in [3], extended the result for indices 1 < p q < . ...
... With the help of well-known results established in the literature [21,24,16] (see Appendix A), we then calculate the sharp constants in these inequalities. The advantage of this approach is that it gives us much more information about the constant in the resulting Hardy inequality. ...
... The cases with general weights were probably first studied by Kac and Krein [7] for p = q = 2 and v = 1, then by Beesack, see e.g. [1] for some other specific weights, by Tomaselli [20], Talenti [19] and Muckenhoupt [12] for p = q, by Bradley [3], see also (without proof) Kokilashvili [9] for p ≤ q, and literature records also unpublished papers by Artola and by Boyd and Erdős ( [15]). The case 1 ≤ q < p < ∞ was first characterized by Maz'ya and Rozin (published later in the book [11]) and Sawyer [16], the case 0 < q < 1 < p < ∞ by Sinnamon [17], and the case 0 < q < p = 1 by Sinnamon and Stepanov [18], see also [2]. ...
We establish a set of relations between several quite diverse types of weighted inequalities involving various integral operators and fairly general quasinorm-like functionals. The main result enables one to solve a specific problem by transferring it to another one for which a solution is known. The main result is formulated in a rather surprising generality, involving previously unknown cases, and it works even for some nonlinear operators such as the geometric or harmonic mean operators. Proofs use only elementary means.
... Moreover, the pairs of weights that satisfy such an estimate have been completely characterized. See for instance [28] and references therein. ⋄ As a particular case, we recover the Hardy inequality when v = w = 1. ...
The goal of this expository paper is to give a self-contained introduction to sparse domination. This is a method relying on techniques from dyadic Harmonic Analysis which has received a lot of attention in recent years. Essentially, it allows for a unified approach to proving weighted norm inequalities for a large variety of operators. In this work, we will introduce the basic ideas of dyadic Harmonic Analysis, which we use to build up to the main result we discuss on pointwise sparse domination, which is the Lerner-Ombrosi theorem. We also give applications of this theorem to some families of operators, mainly relating to singular integral operators. The text has been structured so as to motivate the introduction of new ideas through the lens of solving specific problems in Harmonic Analysis.
... If s = 1, we understand that s ′ = ∞. Theorem A [4,14,15,18,20]. Let 1 < q < ∞, 1 ≤ p < ∞ and let w, v be positive measurable function on (a, b), where −∞ ≤ a < b ≤ ∞. ...
We characterize the good weights for some weighted weak-type iterated Hardy-Copson inequalities to hold.
... Thus dµ = ρ(x)dx = e −W (x) dx is a measure on [0, ∞) with W : [0, ∞) → R smooth and convex. We will apply the Muckenhoupt criterion ( [34]; see also [2, §4.5.1]), which we state as the following lemma: -964 -Let x 0 > 0 be such that a := W ′ (x 0 ) > 0. Then W ′ (x) ⩾ a for all x > x 0 by convexity. Hence for any r, x > x 0 , ρ(r) ⩽ e −a(r−x) ρ(x) for x < r, ρ(x) ⩽ e −a(x−r) ρ(r) for x > r. ...
... We need the following result in [24, p. 44] (see the original proof in [25] ...
Inequalities are essential in pure and applied mathematics. In particular, Opial’s inequality and its generalizations have been playing an important role in the study of the existence and uniqueness of initial and boundary value problems. In this work, some new Opial-type inequalities are given and applied to generalized Riemann-Liouville-type integral operators.
... (3.2) was derived, see our book [24] and the references therein. A simple proof of the characterization Eq. (3.2) was given by B. Muckenhoupt in 1972 for p = q (see [26]) and by J.S.Bradley in 1978 for p ≤ q (see [8]). In 2002 L.E. ...
We describe some chosen ideas and results for more than 100 years prehistory and history of the remarkable development concerning Hardy-type inequalities. In particular, we present a newer convexity approach, which we believe could partly have changed this development if Hardy had discovered it. In order to emphasize the current very active interest in this subject, we finalize by presenting some examples of the recent results, which we believe have potential not only to be of interest for a broad audience from a historical perspective, but also to be useful in various applications.
An extension of Marcinkiewicz Interpolation Theorem, allowing intermediate spaces of Orlicz type, is proved. This generalization yields a necessary and sufficient condition so that every quasilinear operator, which maps the set, , of all -measurable simple functions on - finite measure space into , the class of -measurable functions on - finite measure space , and satisfies endpoint estimates of type: , , \begin{equation*} \lambda \, \nu \left( \left\lbrace y \in Y : |(Tf)(y)| > \lambda \right\rbrace \right)^{\frac{1}{p}} \leq C_{p,r} \left( \int_{\mathbb{R_+}} \mu \left( \left\lbrace x \in X : |(f)(x)| > t \right\rbrace \right)^{\frac{r}{p}} t^{r-1}dt \right)^{\frac{1}{r}}, \end{equation*} for all and ; is bounded from an Orlicz space into another.
We present factorizations of weighted Lebesgue, Ce\-s\` aro and Copson spaces, for weights satisfying the conditions which assure the boundedness of the Hardy's integral operator between weighted Lebesgue spaces. Our results enhance, among other, the best known forms of weighted Hardy inequalities.
We summarize some results as well as we prove some new results about the Orlicz–Lorentz–Karamata spaces and martingale Hardy Orlicz–Lorentz–Karamata spaces. More precisely, Doob's maximal inequality for submartingales and Burkholder–Davis–Gundy inequality are presented. We also show some fundamental martingale inequalities and modular inequalities. Additionally, based on atomic decompositions, duality theorems and fractional integral operators are discussed. As applications in Fourier analysis, we consider the Walsh–Fourier series on Orlicz–Lorentz–Karamata spaces. The dyadic maximal operators on martingale Hardy Orlicz–Lorentz–Karamata spaces are presented. The boundedness of maximal Fejér operator is proved, which further implies some convergence results of the Fejér means.
We investigate the local properties , including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: { ∂ t u ¯ − y − a ∇ ⋅ ( y a ∇ u ¯ ) = 0 a m p ; in B 1 + × ( − 1 , 0 ) − ∂ y a u ¯ = q ( x , t ) u a m p ; on B 1 × { 0 } × ( − 1 , 0 ) , \begin{equation*} \begin {cases} \partial _t \overline {u} - y^{-a} \nabla \cdot (y^a \nabla \overline {u}) = 0 \quad &\text { in } \mathbb {B}_1^+ \times (-1,0) \\ -\partial _y^a \overline {u} = q(x,t)u \quad &\text { on } B_1 \times \{0\} \times (-1,0), \end{cases} \end{equation*} where a ∈ ( − 1 , 1 ) a\in (-1,1) is a fixed parameter, B 1 + ⊂ R N + 1 \mathbb {B}_1^+\subset \mathbb {R}^{N+1} is the upper unit half ball and B 1 B_1 is the unit ball in R N \mathbb {R}^N . Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator H s u ( x , t ) = 1 | Γ ( − s ) | ∫ − ∞ t ∫ R N [ u ( x , t ) − u ( z , τ ) ] G N ( x − z , t − τ ) ( t − τ ) 1 + s d z d τ . \begin{equation*} H^su(x,t) = \frac {1}{|\Gamma (-s)|} \int _{-\infty }^t \int _{\mathbb {R}^N} \left [u(x,t) - u(z,\tau )\right ] \frac {G_N(x-z,t-\tau )}{(t-\tau )^{1+s}} dzd\tau . \end{equation*}
We characterise the possible blow-ups and we examine the structure of the nodal set of solutions vanishing with a finite order . More precisely, we prove that the nodal set has at least parabolic Hausdorff codimension one in R N × R \mathbb {R}^N\times \mathbb {R} , and can be written as the union of a locally smooth part and a singular part, which turns out to possess remarkable stratification properties. Moreover, the asymptotic behaviour of general solutions near their nodal points is classified in terms of a class of explicit polynomials of Hermite and Laguerre type, obtained as eigenfunctions to an Ornstein-Uhlenbeck type operator. Our main results are obtained through a fine blow-up analysis which relies on the monotonicity of an Almgren-Poon type quotient and some new Liouville type results for parabolic equations, combined with more classical results including Federer’s reduction principle and the parabolic Whitney’s extension.
Generalized indefinite strings provide a canonical model for self-adjoint operators with simple spectrum (other classical models are Jacobi matrices, Krein strings and 2×2 canonical systems). We prove a number of Szegő-type theorems for generalized indefinite strings and related spectral problems (including Krein strings, canonical systems and Dirac operators). More specifically, for several classes of coefficients (that can be regarded as Hilbert–Schmidt perturbations of model problems), we provide a complete characterization of the corresponding set of spectral measures. In particular, our results also apply to the isospectral Lax operator for the conservative Camassa–Holm flow and allow us to establish existence of global weak solutions with various step-like initial conditions of low regularity via the inverse spectral transform.
We establish coanalytic models for a broad class of Hardy-type operators on L2[0, 1]. In particular, we show that the logarithmic Hardy operator is unitarily equivalent to the difference between the identity operator and the backward shift on a Bergman-type space. This result leads to several applications related to zero sets and invariant subspaces in weighted Bergman spaces. Additionally, we study logarithmic Hardy operators on Lp[0, 1] and obtain results concerning their boundedness, operator norms, and spectra.
Fractional spline wavelet systems are considered. Together with the Battle–Lemarié scaling and wavelet functions of natural orders, they are used to find conditions that ensure certain inequalities between the norms of images and pre-images of fractional integration operators in Besov spaces with Muckenhoupt weights on R \mathbb {R} .
We study the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. For this aim, we establish decay estimates of the fractional heat semigroup in several uniformly local Zygumnd spaces. Furthermore, we apply the real interpolation method in uniformly local Zygmund spaces to obtain sharp integral estimates on the inhomogeneous term and the nonlinear term. This enables us to find sharp sufficient conditions for the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term.
We study weighted Sobolev inequalities on open convex cones endowed with -homogeneous weights satisfying a certain concavity condition. We establish a so-called reduction principle for these inequalities and characterize optimal rearrangement-invariant function spaces for these weighted Sobolev inequalities. Both optimal target and optimal domain spaces are characterized. Abstract results are accompanied by general yet concrete examples of optimal function spaces. For these examples, the class of so-called Lorentz--Karamata spaces, which contains in particular Lebesgue spaces, Lorentz spaces, and some Orlicz spaces, is used.
The work is devoted to the study of boundedness properties of integration and differentiation operators of Riemann–Liouville type on the real axis and semi–axis. The operators are acting in smoothness function spaces of Besov type with Muckenhoupt weight functions (weights) of standard and local type. The problem posed is solved by decomposing elements of the function spaces with respect to spline wavelet systems, which are the main solution tools. The article presents in detail a scheme for constructing such systems. In their terms, the corresponding decomposition theorems are established in the paper. The main results of the study are conditions on weights for the fulfilment of inequalities connecting the norms of images and pre–images of Riemann–Liouville integration operators.
Let (u,v) be a solution to the Cauchy problem for a semilinear parabolic system \mathrm{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2\Delta v+u^q\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ (u(\cdot,0),v(\cdot,0))=(\mu,\nu) & $\quad\mbox{in}\quad{\mathbb{R}}^N,$ } where , , , , with , and is a pair of nonnegative Radon measures or locally integrable nonnegative functions in . In this paper we establish sharp sufficient conditions on the initial data for the existence of solutions to problem~(P) using uniformly local Morrey spaces and uniformly local weak Zygmund type spaces.
We derive and analyze a fully computable discrete scheme for fractional partial differential equations posed on the full space . Based on a reformulation using the well-known Caffarelli–Silvestre extension, we study a modified variational formulation to obtain well-posedness. Our scheme is obtained by combining a diagonalization procedure with a reformulation using boundary integral equations and a coupling of finite elements and boundary elements. For our discrete method we present a-priori estimates as well as numerical examples.
В работе получены верхние и нижние оценки на аппроксимативные числа двумерного оператора Харди прямоугольного интегрирования, действующего в весовых пространствах Лебега на . Библиография: 22 названия.
We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl. Acad. Sci. USA, 2008 & Math. Ann., 2011). This concept enables us to give very short/elegant proofs of a number of celebrated functional inequalities on Riemannian manifolds with sectional curvature bounded from above by simply solving a Riccati-type ODE. Among others, we provide alternative proofs for Caccioppoli inequalities, Hardy-type inequalities and their improvements, spectral gap estimates, interpolation inequalities, and Ghoussoub-Moradifam-type weighted inequalities. Concerning the multiplicative form, we prove sharp uncertainty principles on Cartan-Hadamard manifolds, i.e., Heisenberg-Pauli-Weyl uncertainty principles, Hydrogen uncertainty principles and Caffarelli-Kohn-Nirenberg inequalities. Some sharpness and rigidity phenomena are also discussed.
Let 0 < q ≤ ∞ , b be a slowly varying function and Φ : [ 0 , ∞ ) ⟶ [ 0 , ∞ ) be an increasing function with Φ ( 0 ) = 0 and lim r → ∞ Φ ( r ) = ∞ . In this paper, we introduce a new class of function spaces L Φ , q , b which unify and generalize the Lorentz-Karamata spaces with Φ ( t ) = t p and the Orlicz-Lorentz spaces with b ≡ 1 . Based on the new spaces, we introduce five new Hardy spaces containing martingales, the so-called Orlicz-Lorentz-Karamata Hardy martingale spaces and then develop a theory of these martingale Hardy spaces. To be precise, we first investigate several properties of Orlicz-Lorentz-Karamata spaces and then present Doob’s maximal inequalities by using Hardy’s inequalities. The characterization of these Hardy martingale spaces are constructed via the atomic decompositions. As applications of the atomic decompositions, martingale inequalities and the relation of the different martingale Hardy spaces are presented. The dual theorems and a new John-Nirenberg type inequality for the new framework are also established. Moreover, we study the boundedness of fractional integral operators on Orlicz-Lorentz-Karamata Hardy martingale spaces. The results obtained here generalize the previous results for Lorentz-Karamata Hardy martingale spaces as well as for Orlicz-Lorentz Hardy martingales spaces. Especially, we remove the condition that b is non-decreasing as in [38, 39] and the condition q Φ - 1 < 1 / q in [24], respectively.
In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case p = 1 {p=1} and 1 ≤ q < ∞ {1\leq q<\infty} . This result complements the Hardy inequalities obtained in [M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc. Roy. Soc. A. 475 2019, 2223, Article ID 20180310] in the case 1 < p ≤ q < ∞ {1<p\leq q<\infty} . The case p = 1 {p=1} requires a different argument and does not follow as the limit of known inequalities for p > 1 {p>1} . As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan–Hadamard manifolds for the case p = 1 {p=1} and 1 ≤ q < ∞ {1\leq q<\infty} .
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