Reverse Holder Inequality Research Articles

Reverse Hölder inequality for a class of suitable weak solutions to the Navier–Stokes equations

In the paper, a special class of suitable weak solutions to the three-dimensional nonstationary Navier–Stokes equations is considered, and a reverse Holder inequality for them is proved. An interesting feature of this class is that it contains solutions having majorants invariant to the Navier-Stokes scaling. Bibliography: titles.

On Beckenbach–Radó type integral inequalities

Beckenbach and Rado characterized logarithmically subharmonic functions in the plane in terms of integral inequalities involving spherical averages. In this work we generalize this result to higher dimensions and thus answer to a question raised by Beckenbach and Rado. We also consider generalizations of integral inequalities suggested by Beckenbach and Rado and discuss connections to reverse Holder inequalities and Muckenhoupt weights.

Endpoint Properties of Localized Riesz Transforms and Fractional Integrals Associated to Schrödinger Operators

Let \({\mathcal L}\equiv-\Delta+V\) be the Schrodinger operator in \({{\mathbb R}^n}\), where V is a nonnegative function satisfying the reverse Holder inequality. Let ρ be an admissible function modeled on the known auxiliary function determined by V. In this paper, the authors characterize the localized Hardy spaces \(H^1_\rho({{\mathbb R}^n})\) in terms of localized Riesz transforms and establish the boundedness on the BMO-type space \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) of these operators as well as the boundedness from \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) to \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman–Stein decomposition of \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) via localized Riesz transforms. When ρ is the known auxiliary function determined by V, \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) is just the known space \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\), and \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) in this case is correspondingly denoted by \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\). As applications, when n ≥ 3, the authors further obtain the boundedness on \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) of Riesz transforms \(\nabla{\mathcal L}^{-1/2}\) and their adjoint operators, as well as the boundedness from \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) to \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\) of their maximal operators. Also, some endpoint estimates of fractional integrals associated to \({\mathcal L}\) are presented.

Mutual estimates of L p -norms and the Bellman function

In this paper, we describe the range of the Lp-norm of a function under fixed Lp-norms with two other different exponents p and under a natural multiplicative restriction of the type of the Muckenhoupt condition. Particular cases of such results are simple inequalities as the interpolation inequality between two Lp-norms as well as such nontrivial inequalities as the Gehring inequality or the reverse Holder inequality for Mackenhoupt weights. The basic method of our paper is the search for the exact Bellman function of the corresponding extremal problem. Bibliography: 5

Lp ESTIMATES FOR WEAK SOLUTIONS TO NONLINEAR SUB-ELLIPTIC SYSTEMS RELATED TO HÖRMANDER’S VECTOR FIELDS

This paper is concerned with the regularity of weak solutions to the general nonlinear sub-elliptic systems related to Hormander’s vector fields. Based on the reverse Holder inequality established by Gianazza, the Lp estimates for weak solutions to the systems under the super-quadratic controllable growth condition and the natural growth condition are established, respectively. Then we obtain the Holder continuity for weak solutions.

Harmonic analysis of stochastic equations and backward stochastic differential equations

The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations in $${\mathcal{R}^p}$$ ( $${p\in [1,\infty)}$$ ) and backward stochastic differential equations (BSDEs) in $${\mathcal{R}^p\times \mathcal{H}^p}$$ ( $${p\in (1, \infty)}$$ ) and in $${\mathcal{R}^\infty\times\overline{L^\infty}^{\rm BMO}}$$ , with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman’s inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear cases for stochastic differential equations (SDEs) and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse Holder inequality for some suitable exponent p ≥ 1. Finally, some relations are established between Kazamaki’s quadratic critical exponent b(M) of a BMO martingale M and the spectral radius of the stochastic integral operator with respect to M, which lead to a characterization of Kazamaki’s quadratic critical exponent of BMO martingales being infinite.

On Extending the Inequalities of Payne, Pólya, and Weinberger Using Spherical Harmonics

Using spherical harmonics, rearrangement techniques, the Sobolev in- equality, and Chiti's reverse Holder inequality, we obtain extensions of a classical result of Payne, Polya, and Weinberger bounding the gap between consecutive eigen- values of the Dirichlet Laplacian in terms of moments of the preceding ones. The extensions yield domain-dependent inequalities.

A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable

We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally Holder continuous with Holder exponent depending only on the growth of the Hamiltonian. The proof relies on a reverse Holder inequality.

The Reverse Hölder Inequality for the Solution to p-Harmonic Type System

Some inequalities to -harmonic equation have been proved. The -harmonic equation is a particular form of -harmonic type system only when and . In this paper, we will prove the Poincare inequality and the reverse Holder inequality for the solution to the -harmonic type system.

Abstract duality Sawyer formula and its applications

The main subject of this paper is to present some general results on duality and Banach envelopes of a large class of symmetric quasi-Banach spaces. An abstract duality formula, a general form of the Sawyer’s characterization of the reverse Holder inequality for Lp-spaces, 1 < p < ∞, in function ideals is proved. This formula is then applied for characterization of Kothe duals of rearrangement invariant (r.i.) function lattices. A duality formula for 1-concave quasi-Banach function lattices is shown and it is employed to characterize Banach envelopes of r.i. quasi-Banach lattices. As applications of the duality formulas to quasi-normed Orlicz-Lorentz spaces, a description of their Kothe duals and of their Banach envelopes is presented. The conditions for normability as well as for boundedness of Hardy-Littlewood maximal operator are also given in some class of Orlicz-Lorentz spaces.

Mappings of finite distortion: Reverse inequalities for the Jacobian

Let f be a nonconstant mapping of finite distortion. We establish integrability results on 1/Jf by studying weights that satisfy a weak reverse Holder inequality where the associated constant can depend on the ball in question. Here Jf is the Jacobian determinant of f.

Maximal functions and Calderon-Zygmund theory for vector-valued functions with operator weights

We generalize the Hardy-Littlewood maximal functions to vector-valued functions taking values in a Banach space with a varying weight, t →ρ t , which satisfies a reverse Holder condition, and prove estimates on the bounds of the maximal function. We also prove the existence of a Calderon-Zygmund decomposition for functions in L P (t → ρ t ) when the weight satisfies a reverse Holder inequality. This decomposition is used to prove an extrapolation theorem for the martingale transform on weighted spaces.

Regularity for very weak solutions to A-harmonic equation

In this paper, the following result is given by using Hodge decomposition and weak reverse Holder inequality: For every r 1 with p−(2 n+1 100 n 2 p(23+n/(p−1)+1)b/a)−1 p, such that for every very weak solution u∈ W r 1 ,loc/1(Ω) to A-harmonic equation, u also belongs to W r/ 2 /,loc/1/(/gW). In particular, u is the weak solution to A-harmonic equation in the usual sense.

The $L^p$ Dirichlet Problem for Elliptic Systems on Lipschitz Domains

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in $R^n$. For $n\ge 4$ and $2-\epsilon<p<2(n-1)/(n-3)+\epsilon$, we establish the solvability of the Dirichlet problem with boundary value data in $L^p(\partial\Omega)$. In the case of the polyharmonic equation $\Delta^{\ell}u=0$ with $\ell\ge 2$, the range of $p$ is sharp if $4\le n \le 2\ell +1$.

BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality

We identify the dual space of the Hardy-type space related to the time independent Schrodinger operator =−Δ+V, with V a potential satisfying a reverse Holder inequality, as a BMO-type space . We prove the boundedness in this space of the versions of some classical operators associated to (Hardy-Littlewood, semigroup and Poisson maximal functions, square function, fractional integral operator). We also get a characterization of in terms of Carlesson measures.

The reserse Hölder classes in martingale setting

In martingale setting, it has been shown thatAp weights can be factorized in terms ofA1 weights. This factorization benefits many problems very much. In this paper, the new class of RH∞ plays the same role for RHs, which makes the reverse Holder inequalities hold with exponents>1, that the classA1 does forAp class. Therefore, the Jones' factorization theorem forAp weights was extended to include some information about the reverse Holder classes. And it is the most convenient object in weight theory indeed.

Weighted integral inequalities for differential forms

In this paper, we obtain some weighted integral inequalities for differential forms, which can be considered as generalizations of the Poincare inequality, the Caccioppoli-type estimate, and the weak reverse Holder inequality, respectively. These results can be used to study the integrability of differential forms and to estimate the integrals of differential forms. We also give some applications of the above results.

The sharp constant in the reverse Hölder inequality for Muckenhoupt weights

Coifman and Fefferman proved that the “reverse Holder inequality” is fulfilled for any weight satisfying the Muckenhoupt condition. In order to illustrate the power of the Bellman function technique, Nazarov, Volberg, and Treil showed (among other things) how this technique leads to the reverse Holder inequality for the weights satisfying the dyadic Muckenhoupt condition on the real line. In this paper the proof of the reverse Holder inequality with sharp constants is presented for the weights satisfying the usual (rather than dyadic) Muckenhoupt condition on the line. The results are a consequence of the calculation of the true Bellman function for the corresponding extremal problem. In [1] Coifman and Fefferman proved that for any weight (i.e., for any nonnegative function) w on R satisfying the Muckenhoupt condition (Ap) sup Q⊂Rn { 〈w〉Q〈w 1 1−p 〉p−1 Q } 1, with a constant C independent of the cube Q. Here and later, 〈φ〉Q stands for the average of the function φ over the set Q:

On biharmonic maps and their generalizations

We give a new proof of regularity of biharmonic maps from four-dimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Holder inequalities is used next to prove continuity of solutions and higher integrability of their second order derivatives. As a byproduct, we also prove that a weak limit of biharmonic maps into a sphere is again biharmonic. The proof of regularity can be adapted to biharmonic maps on the Heisenberg group, and to other functionals leading to fourth order elliptic equations with critical nonlinearities in lower order derivatives.

A multidirectional Dirichlet problem

B.E.J. Dahlberg’s theorems on the mutual absolute continuity of harmonic and surface measures, and on the unique solvability of the Dirichlet problem for Laplace’s equation with data taken in Lp spaces p > 2 − δ are extended to compact polyhedral domains of ℝn. Consequently, for q < 2 + δ, Dahlberg’s reverse Holder inequality for the density of harmonic measure is established for compact polyhedra that additionally satisfy the Harnack chain condition. It is proved that a compact polyhedral domain satisfies the Harnack chain condition if its boundary is a topological manifold. The double suspension of the Mazur manifold is invoked to indicate that perhaps such a polyhedron need not itself be a manifold with boundary; see the footnote in Section 9. A theorem on approximating compact polyhedra by Lipschitz domains in a certain weak sense is proved, along with other geometric lemmas.