Two-weight Inequalities Research Articles

Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent

Necessary and sufficient conditions governing two-weight inequalities with general-type weights for fractional maximal functions and Riesz potentials with variable parameters are established in the Lebesgue spaces with variable exponent. In two-weight inequalities the right-hand side weight to the certain power satisfies the reverse doubling condition. In particular, from the general results we have: generalization of the Sobolev inequality for potentials; criteria governing the trace inequality for fractional maximal functions and potential operators; theorem of Muckenhoupt–Wheeden type (one-weight inequality) for fractional maximal functions defined on a bounded interval when the parameter satisfies the Dini–Lipschitz condition. Sawyer-type two-weight criteria for fractional maximal functions are also derived.

Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture

We prove two-weight norm inequalities for Calderón–Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.

The Bellman functions for a certain two-weight inequality: A case study

A formula is presented for the exact Bellman function of a certain toy two-weight problem. This adds one more function to a short list of other Bellman functions for which the precise expressions have recently been found. The case study reveals essential features of finding Bellman functions in general and gives the ex- tremal sequences for the problem. Some open questions are posed.

Two-weight inequalities for convolution operators in Lebesgue space

In this paper, we prove a theorem on the boundedness of a convolution operator in a weighted Lebesgue space with kernel satisfying a certain version of Hormander’s condition.

Two-weight norm inequalities for commutators of potential type integral operators

We give a condition which is sufficient for the two-weight ( p , q ) inequalities for commutators of potential type integral operators.

THE MINIMAL OPERATOR AND WEIGHTED INEQUALITIES FOR MARTINGALES

In this article the authors introduce the minimal operator on martingale spaces, discuss some one-weight and two-weight inequalities for the minimal operator and characterize the conditions which make the inequalities hold.

The Bochner–Riesz means for Fourier–Bessel expansions

The Bochner–Riesz means for Fourier–Bessel expansions are analyzed. We prove a uniform two-weight inequality, with potential weights, for these means. The result provides necessary and sufficient conditions for boundedness. Moreover, we obtain some corollaries regarding the convergence of these means and the boundedness of other operators related to the Fourier–Bessel series.

Conductor inequalities and criteria for Sobolev type two-weight imbeddings

A typical inequality handled in this article connects the L p -norm of the gradient of a function to a one-dimensional integral of the p -capacitance of the conductor between two level surfaces of the same function. Such conductor inequalities lead to necessary and sufficient conditions for multi-dimensional and one-dimensional Sobolev type inequalities involving two arbitrary measures. Compactness criteria and two-sided estimates for the essential norm of the related imbedding operator are obtained. Some counterexamples are presented to illustrate the peculiarities arising in the case of higher derivatives. Criteria for two-weight inequalities with fractional Sobolev norms of order l < 2 are found.

A trace inequality for generalized potentials in Lebesgue spaces with variable exponent

A trace inequality for the generalized Riesz potentials Iα(x) is established in spaces Lp(x) defined on spaces of homogeneous type. The results are new even in the case of Euclidean spaces. As a corollary a criterion for a two‐weighted inequality in classical Lebesgue spaces for potentials Iα(x) defined on fractal sets is derived.

On the existence of principal values for the cauchy integral on weighted lebesgue spaces for non-doubling measures

Let T be a Calderon-Zygmund operator in a "non-homogeneous" space (X,d,µ), where, in particular, the measure µ may be non- doubling. Much of the classical theory of singular integrals has been recently extended to this context by F. Nazarov, S. Treil and A. Vol- berg and, independently by X. Tolsa. In the present work we study some weighted inequalities for T?, which is the supremum of the trun- cated operators associated with T. Specifically, for 1 < p < 1, we obtain sucient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresponding L p weighted inequality holds for T?. The main tool to deal with this problem is the theory of vector-valued inequalities for T? and some re- lated operators. We discuss it first by showing how these operators are connected to the general theory of vector-valued Calderon-Zygmund operators in non-homogeneous spaces, developed in our previous pa- per (GM). For the Cauchy integral operator C, which is the main example, we apply the two-weight inequalities for C? to characterize the existence of principal values for functions in weighted L p .

Weighted Inequalities on John Domains

We study two-weighted inequalities on John domains. We first introduce domains that generalize the C(λ,M) domains defined by HajLasz and Koskela (J. London Math. Soc., to appear) and then show that our domains are actually just John domains. We then extend an interesting and nice result of HajLasz and Koskela by modifying their proof and then obtain some interesting consequences that include weighted Sobolev interpolation inequalities and an almost necessary and sufficient condition for two-weighted Poincaré inequality on cubes.

Two-weighted inequalities for the derivatives of holomorphic functions and Carleson measures on the unit ball

AbstractWe characterize those positive measure µ’s on the higher dimensional unit ball such that “two-weighted inequalities” hold for holomorphic functions and their derivatives. Characterizations are given in terms of the Carleson measure conditions. The results of this paper also distinguish between the fractional and the tangential derivatives.

A Characterization of a Two-Weight Inequality for Discrete Two-Dimensional Hardy Operators

We obtain a characterization of non-negative double sequences V = V(n_1, n_2))_{n_1, n_2} and U = (U(n_1 ,n_2))_{n_1,n_2} for which the two-dimensional discrete Hardy operator \mathbb H is bounded from l^p(V) into l^q(U) whenever 1 &lt; p ≤ q &lt; \infty .

The Bellman functions and two-weight inequalities for Haar multipliers

The Bellman functions and two-weight inequalities for Haar multipliers

Two-Weighted Inequalities for Integral Operators in Lorentz Spaces Defined on Homogeneous Groups

Abstract The optimal sufficient conditions are found for weights, which guarantee the validity of two-weighted inequalities for singular integrals in the Lorentz spaces defined on homogeneous groups. In some particular case the found conditions are necessary for the corresponding inequalities to be valid. Also, the necessary and sufficient conditions are found for pairs of weights, which provide the validity of two-weighted inequalities for the generalized Hardy operator in the Lorentz spaces defined on homogeneous groups.

Two-Weighted Inequalities for Integral Operators in Lorentz Spaces Defined on Homogeneous Groups

The optimal sucient conditions are found for weights, which guarantee the validity of two-weighted inequalities for singular integrals in the Lorentz spaces deÞned on homogeneous groups. In some particular case the found conditions are necessary for the cor- responding inequalities to be valid. Also, the necessary and sucient conditions are found for pairs of weights, which provide the validity of two-weighted inequalities for the generalized Hardy operator in the Lorentz spaces deÞned on homogeneous groups.

Characterization of a Two-Weighted Vector-Valued Inequality for Fractional Maximal Operators

Abstract We give a characterization of the weights 𝑢(·) and 𝑣(·) for which the fractional maximal operator 𝑀𝑠 is bounded from the weighted Lebesgue spaces 𝐿𝑝(𝑙𝑟, 𝑣𝑑𝑥) into 𝐿𝑞(𝑙𝑟, 𝑢𝑑𝑥) whenever 0 ≤ 𝑠 &lt; 𝑛, 1 &lt; 𝑝, 𝑟 &lt; ∞, and 1 ≤ 𝑞 &lt; ∞.

A Note on Two-Weight Inequalities for Maximal Functions and Singular Integrals

A Note on Two-Weight Inequalities for Maximal Functions and Singular Integrals

A Note on Two-Weight Inequalities for Maximal Functions and Singular Integrals

A Note on Two-Weight Inequalities for Maximal Functions and Singular Integrals

Criteria of Strong Type Two-Weighted Inequalities for Fractional Maximal Functions

Abstract A strong type two-weight problem is solved for fractional maximal functions defined in homogeneous type general spaces. A similar problem is also solved for one-sided fractional maximal functions.