One-sided Maximal Operators Research Articles

A Weighted Weak-Type Inequality for One-Sided Maximal Operators

A Weighted Weak-Type Inequality for One-Sided Maximal Operators

A weighted weak-type inequality for the one-sided maximal operators

UDC 517.5 We obtain some necessary and sufficient conditions for a weighted weak-type inequality of the form ∫ { M g + ( f ) > λ } φ ˜ ( λ ω 3 ( x ) ω 4 ( x ) ) ω 4 ( x ) ⅆ x ≤ C 1 ∫ - ∞ + ∞ φ ˜ ( C 1 | f ( x ) | ω 1 ( x ) ω 2 ( x ) ) ω 2 ( x ) ⅆ x to be true, which generalize some known results.

A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

We investigate the regularity properties of the two-dimensional one-sided Hardy-Littlewood maximal operator. We point out that the above operator is bounded and continuous on the Sobolev spaces Ws,p(R2) for 0≤s≤1 and 1<p<∞. More importantly, we establish the sharp boundedness and continuity for the discrete two-dimensional one-sided Hardy-Littlewood maximal operator from l1(Z2) to BV(Z2). Here BV(Z2) denotes the set of all functions of bounded variation on Z2.

Sharp weighted bounds for one-sided operators

Abstract In this paper, we establish sharp weighted bounds (Buckley-type theorems) for one-sided maximal and fractional integral operators in terms of one-sided A p {A_{p}} characteristics.

Two-Weight Norm Inequality for the One-Sided Hardy-Littlewood Maximal Operators in Variable Lebesgue Spaces

The authors establish the two-weight norm inequalities for the one-sided Hardy-Littlewood maximal operators in variable Lebesgue spaces. As application, they obtain the two-weight norm inequalities of variable Riemann-Liouville operator and variable Weyl operator in variable Lebesgue spaces on bounded intervals.

Sharp weighted bounds for one-sided maximal operators

In this note we prove some results about the best constants for the boundedness of the one-sided Hardy–Littlewood maximal operator in \(L^p(\mu )\), where \(\mu \) is a locally finite Borel measure, that in the two-sided weights have been obtained by Buckley (Trans Am Math Soc 340(1):253–272, 1993) and more recently by Hytonen and Perez (Anal PDE 6(4):777–818, 2013) and Hytonen et al. (J Funct Anal 263(12):3883–3899, 2012). To prove Bucley’s theorem for one-sided maximal operators, we follow the ideas of Lerner (Proc Am Math Soc 136(8):2829–2833, 2008). To obtain a better estimate in terms of mixed constants we follow the steps in Hytonen and Perez (Anal PDE 6(4):777–818, 2013) and Hytonen et al. (J Funct Anal 263(12):3883–3899, 2012) i.e., (a) getting a sharp estimate for the constant for the weak type type, in terms of the one-sided \(A_p\) constant, (b) obtaining a sharp reverse Holder inequality and (c) using Marcinkiewicz interpolation theorem. Our proofs of these facts are different from those in Hytonen and Perez (Anal PDE 6(4):777–818, 2013) and Hytonen et al. (J Funct Anal 263(12):3883–3899, 2012) and apply to more general measures.

The one-sided dyadic Hardy—Littlewood maximal operator

The main aim of this paper is to introduce an appropriate dyadic one-sided maximal operator , smaller than the one-sided Hardy–Littlewood maximal operator M+ but such that it controls M+ in a similar way to how the usual dyadic maximal operator controls the Hardy-Littlewood maximal operator. We characterize the weighted inequalities for this dyadic one-sided maximal operator.

Weighted Inequalities for One-Sided Vector-Valued Maximal Operators with Respect to a Function

The purpose of this paper is to find necessary and sufficient conditions on the weight \(w\) for the weak type \(\left( p,p\right) \) with \(1\leq p<+\infty \) and for the strong type \(\left( p,p\right)\) with \(1<p<+\infty \) respect to a measure \(w\,dx\) of the vector-valued one-sided maximal operator \(\left(M_{g}^{+}\right) _{r}\).

Norm inequalities for some one-sided operators

We show that the one-sided maximal operators associated with Borel measures are of strong type (p, p) , 1 < p < ∞ , with constant p∗ , and the related one-sided geometric maximal operators are of strong type (p, p) , 0 < p < ∞ , with constant e1/p . We also investigate norm inequalities for integral operators with three measures on the cone of nonnegative nonincreasing functions. Our results show that if we restrict the measures in the inequalities to some particular classes, then a simple characterization for these inequalities to hold can be obtained. Mathematics subject classification (2010): Primary 26D15; Secondary 26D10, 42B25.

The one-sided Ap conditions and local maximal operator

AbstractWe introduce the one-sided local maximal operator and study its connection to the one-sided Ap conditions. We get a new characterization of the boundedness of the one-sided maximal operator on a quasi-Banach function space. We obtain applications to weighted Lebesgue spaces and variable-exponent Lebesgue spaces.

A note on one-sided maximal operator in L^p(.) (ℝ)

Consider one-sided Hardy-Littlewood maximal operator on the general Lebesgue space with variable exponent. It is known a local sufficient condition to the function p(.) for the boundedness of the one-sided maximal operator on Lp(.)(R) provided p(.) is a constant function in a neighborhood of infinity. Our main aim is to find a weaker condition to p(.) at infinity to guarantee the boundedness of the one-sided maximal operator on Lp(.)(R) . We will show two different sufficient conditions to the behavior of p(.) at infinity under which the one-sided maximal operator is bounded on Lp(.)(R) . Mathematics subject classification (2010): 46E30, 26D15.

Sharp Two Weight Inequalities for Commutators of Riemann-Liouville and Weyl Fractional Integral Operators

Let b be a BMO function, \(0 1) and weighted endpoint estimates (p = 1) for the operator \(I^{+,k}_{\alpha,b}\) and for the pairs of weights of the type (w, \({\mathcal{M}_w}\)), where w is any weight and \(\mathcal{M}\) is a suitable one-sided maximal operator. We also prove that, for \(A^{+}_{\infty}\) weights, the operator \(I^{+,k}_{\alpha,b}\) is controlled in the Lp(w) norm by a composition of the one-sided fractional maximal operator and the one-sided Hardy-Littlewood maximal operator iterated k times. These results improve those obtained by an immediate application of the corresponding two-sided results and provide a different way to obtain known results about the operators \(I^{+,k}_{\alpha,b}\). The same results can be obtained for the commutator of order k for the Riemann-Liouville fractional integral \(I^{-,k}_{\alpha,b}\)

Two extrapolation theorems for related weights and applications

In this paper we prove two extrapolation theorems for related weights. The theorems proved by C. Segovia and J.L. Torrea in [C. Segovia and J.L. Torrea, Weighted inequalities for commutators of fractional and singular integrals. Publ. Mat. 35 (1991) 209-235] are adapted for one-sided weights. We apply these extrapolation theorems to improve weighted inequalities for commutators (with symbol b depending on the related weights) of several one-sided operators such as the Weyl and the Riemann-Liouville fractional integrals, or one-sided maximal operators given by the convolution with a smooth function. We also characterize the symbols b for which the commutators of these one-sided operators are bounded.

Weights for commutators of the one-sided discrete square function, the Weyl fractional integral and other one-sided operators

The purpose of this paper is to prove strong-type inequalities with one-sided weights for commutators (with symbol b ∈ BMO) of several one-sided operators, such as the one-sided discrete square function, the one-sided fractional operators, or one-sided maximal operators given by the convolution with a smooth function. We also prove that b ∈ BMO is a necessary condition for the boundedness of commutators of these one-sided operators.

Weights for commutators of the one-sided discrete square function, the Weyl fractional integral and other one-sided operators

The purpose of this paper is to prove strong-type inequalities with one-sided weights for commutators (with symbol b ∈ BMO) of several one-sided operators, such as the one-sided discrete square function, the one-sided fractional operators, or one-sided maximal operators given by the convolution with a smooth function. We also prove that b ∈ BMO is a necessary condition for the boundedness of commutators of these one-sided operators.

Two weight inequalities for fractional one-sided maximal operators on Orlicz and Lorentz spaces

In the paper, we characterize two weight weak and strong type inequalities for the fractional one-sided maximal operators on Lorentz and Orlicz space.

Weighted Lorentz norm inequalities for general maximal operators associated with certain families of Borel measures

Given a certain family ℱ of positive Borel measures and γ ∈ [0, 1), we define a general onesided maximal operatorand we study weighted inequalities inLp,qspaces for these operators. Our results contain, as particular cases, the characterisation of weighted Lorentz norm inequalities for some well-known one-sided maximal operators such as the one-sided Hardy–Littlewood maximal operator associated with a general measure, the one-sided fractional maximal operatorand the maximal operatorassociated with the Cesèro-α averages.

On the Uniqueness of Maximal Functions

Abstract The uniqueness theorem for the one-sided maximal operator has been proved.

On the uniqueness of maximal functions

The uniqueness theorem for the one-sided maximal operator has been proved. Let L be the class of real 2π-periodic integrable functions and let M be the one-sided maximal operator M(f)(x) = sup b>x 1 b− x b ∫ x fdm, f ∈ L, x ∈ R (m denotes the Lebesgue measure on the line R). In this paper we shall prove the following uniqueness Theorem 1. Let f, g ∈ L and M(f) = M(g). Then f = g a.e. on R. Sets of the type {x ∈ R : M(f)(x) > t} = {x ∈ R : M(g)(x) > t} will be briefly denoted by (M > t). Obviously (M > t)t∈R is a class of bundled open sets continuous from the right, i.e., ∪ t>τ (M > t) = (M > τ). Let t0 = inf{M(f) : x ∈ R} = inf{M(g)(x) : x ∈ R}. For an arbitrary integrable function f if t = 1 2π ∫ 2π 0 fdm, then M(f) ≥ t on the whole line and M(f)(x0) = t for x0 being the point of maximum of the function x 7→ ∫ x 0 fdm− tx. Thus we can conclude that 2π ∫

A note on the weighted norm inequality for the one-sided maximal operator

Let M g + M_g^+ be the one-sided maximal function. In this note we obtain some necessary and sufficient conditions in order that the weighted weak type inequality holds for M g + M_g^+ . Meanwhile, some necessary or sufficient conditions for the weighted inequality for M g + M_g^+ are given.