Hardy-Littlewood Maximal Operator Research Articles

Criteria of a Two-Weight, Weak-Type Inequality in Orlicz Classes for Maximal Functions Defined on Homogeneous Spaces

In this study, some new necessary and sufficient conditions for a two-weight, weak-type maximal inequality of the form φ1(λ)∫{x∈X:Mf(x)>λ}ϱ(x)dμ(x)≤c∫Xφ2c|f(x)|σ(x)dμ(x) are obtained in Orlicz classes, where Mf is a Hardy–Littlewood maximal function defined on homogeneous spaces and ϱ is a weight function.

A remark on the boundedness of the Hardy–Littlewood maximal operator on Orlicz–Lorentz spaces

A remark on the boundedness of the Hardy–Littlewood maximal operator on Orlicz–Lorentz spaces

Triangular maximal operators on locally finite trees

AbstractWe introduce the centred and the uncentred triangular maximal operators and , respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both and are bounded on for every in , that is also bounded on , and that is not of weak type (1, 1) on homogeneous trees. Our proof of the boundedness of hinges on the geometric approach of Córdoba and Fefferman. We also establish bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy–Littlewood maximal operators (on balls) may be unbounded on for every even on some trees where the number of neighbours is uniformly bounded.

Extrapolation of Compactness on Banach Function Spaces

We prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator T in the weighted Lebesgue scale and the compactness of T in the unweighted Lebesgue scale yields compactness of T on a very general class of Banach function spaces. As our main new tool, we prove various characterizations of the boundedness of the Hardy-Littlewood maximal operator on such spaces and their associate spaces, using a novel sparse self-improvement technique. We apply our main results to prove compactness of the commutators of singular integral operators and pointwise multiplication by functions of vanishing mean oscillation on, for example, weighted variable Lebesgue spaces.

Limiting Weak-Type Behavior of the Centered Hardy–Littlewood Maximal Function of General Measures on the Positive Real Line

Limiting Weak-Type Behavior of the Centered Hardy–Littlewood Maximal Function of General Measures on the Positive Real Line

The Hardy–Littlewood maximal operator on discrete weighted Morrey spaces

The Hardy–Littlewood maximal operator on discrete weighted Morrey spaces

Higher order elliptic equations in weighted Banach spaces

AbstractWe consider m-th order linear, uniformly elliptic equations $$\mathcal {L}u=f$$ L u = f with non-smooth coefficients in Banach–Sobolev spaces $$W_{X_w}^m (\Omega )$$ W X w m ( Ω ) generated by weighted Banach Function Spaces (BFS) $$X_w (\Omega )$$ X w ( Ω ) on a bounded domain $$\Omega \subset {\mathbb R}^{n}$$ Ω ⊂ R n . Supposing boundedness of the Hardy–Littlewood Maximal operator and the Calderón–Zygmund singular integrals in $$X_w (\Omega )$$ X w ( Ω ) we obtain solvability in the small in $$W_{X_w}^m (\Omega )$$ W X w m ( Ω ) and establish interior Schauder type a priori estimates. These results will be used in order to obtain Fredholmness of the operator $$\mathcal {L}$$ L in $$X_w (\Omega )$$ X w ( Ω ) .

Capacitary maximal inequalities and applications

In this paper we introduce capacitary analogues of the Hardy-Littlewood maximal function,MCf(x):=supr>0⁡1C(B(x,r))∫B(x,r)|f|dC, for C= the Hausdorff content or a Riesz capacity. For these maximal functions, we prove a strong-type (p,p) bound for 1<p≤+∞ on the capacitary integration spaces Lp(C) and a weak-type (1,1) bound on the capacitary integration space L1(C). We show how these estimates clarify and improve the existing literature concerning maximal function estimates on capacitary integration spaces. As a consequence, we deduce correspondingly stronger differentiation theorems of Lebesgue-type, which in turn, by classical capacitary inequalities, yield more precise estimates concerning Lebesgue points for functions in Sobolev spaces.

Extrapolation to mixed Herz spaces and its applications

AbstractIn this paper, we extend the extrapolation theory to mixed Herz spaces and . To prove the main result, we first study the dual spaces of mixed Herz spaces, and then give the boundedness of the Hardy–Littlewood maximal operator on mixed Herz spaces. By using the extrapolation theorems, we obtain the boundedness of many integral operators on mixed Herz spaces. We also give a new characterization of via the boundedness of commutators of some operators on mixed Herz spaces.

Density of smooth functions in Musielak–Orlicz–Sobolev spaces Wk,Φ(Ω)$W^{k,\Phi }(\Omega)$

AbstractWe consider here Musielak–Orlicz–Sobolev (MOS) spaces , where is an open subset of , and is a Musielak–Orlicz function. The main outcomes consist of the results on density of the space of compactly supported smooth functions in . One section is devoted to compare the various conditions on appearing in the literature in the context of maximal operator and density theorems in MOS spaces. The assumptions on we apply here are substantially weaker than in the earlier papers on the topics of approximation by smooth functions. One of the reasons is that in the process of proving density theorems, we do not use the fact that the Hardy–Littlewood maximal operator on Musielak–Orlicz space is bounded, a standard tool employed in density results for different types of Sobolev spaces. We show in particular that under some regularity assumptions on , (A1) and that are not sufficient for the maximal operator to be bounded, the space of is dense in . In the case of variable exponent Sobolev space , we obtain the similar density result under the assumption that , , , , satisfies the log‐Hölder condition and the exponent function is essentially bounded.

Corrigendum to “Improvements on Sawyer‐type estimates for generalized maximal functions”

AbstractThe purpose of this note is to correct a mixed estimate involving the generalized maximal function , when Φ is a Young function satisfying certain properties. The family considered include type functions. Although the obtained estimates turn out to be slightly different, they are still good extensions of mixed inequalities for the classical Hardy–Littlewood maximal operators corresponding to the power functions , with . They also allow us to derive the boundedness properties of between Lebesgue spaces by using interpolation techniques related to endpoint estimates of modular type.

The sharp weighted maximal inequalities for noncommutative martingales

The purpose of the paper is to establish weighted maximal Lp-inequalities in the context of operator-valued martingales on semifinite von Neumann algebras. The main emphasis is put on the optimal dependence of the Lp constants on the characteristic of the weight involved. As applications, we establish weighted estimates for the noncommutative version of Hardy-Littlewood maximal operator and weighted bounds for noncommutative maximal truncations of a wide class of singular integrals.

Approximation by exponential-type polynomials

In this paper, a family of exponential-type polynomials is introduced and studied. Both the uniform and the Lp convergence are established in suitable function spaces. In the Lp-case, some estimates are also achieved using an exponentially weighted version of the p-norm. Further, a Voronovskaja type formula is proved, finding the exact order of pointwise approximation in case of continuous functions having second derivative at some points. Finally, quantitative estimates for the order of approximation are established in both the continuous and the Lp-cases in terms of the modulus of continuity and K-functionals of the involved functions. In the latter estimate, a crucial role is played by the so-called Hardy-Littlewood maximal function.

Gagliardo representation of norms of ball quasi-Banach function spaces

Let X be a ball quasi-Banach function space on Rn. In this article, under some mild assumptions about both X and the boundedness of the Hardy–Littlewood maximal operator on the associate space of the convexification of X, the authors prove that, for any f∈(X/R)∩⋃s∈(0,1)W˙Xs,q(Rn),‖f‖X/R≲lim infs→0+s1q‖[∫Rn|f(⋅)−f(y)|q|⋅−y|n+sqdy]1q‖X≤lim sups→0+s1q‖[∫Rn|f(⋅)−f(y)|q|⋅−y|n+sqdy]1q‖X≲‖f‖X/R and, for any γ∈R∖{0} and f∈X/R,supλ∈(0,∞)⁡λ‖[∫Rn1Ef(λ,q,γ)(⋅,y)|⋅−y|γ−ndy]1q‖X∼‖f‖X/R with the positive equivalence constants independent of f, where f∈X/R if and only if there exists a∈R such that f+a∈X and where ‖f‖X/R:=infa∈R⁡‖f+a‖X<∞, the index q∈(0,∞) is related to X, W˙Xs,q(Rn) is the homogeneous fractional Sobolev space associated with the ball quasi-Banach function space X, andEf(λ,q,γ):={(x,y)∈Rn×Rn:|f(x)−f(y)|>λ|x−y|γq}. In the case when X:=Lp(Rn) with 1≤q=p<∞ and f∈X, the first formula is closely related to the celebrated classical formula of V. Maz'ya and T. Shaposhnikova and the second formula is exactly the recent formula of H. Brezis et al. These results are new even when X=Lp(Rn) with 1≤q<p<∞ and 0<q≤p<1. All these results are of quite wide generality and, even when they are applied to various specific function spaces, most of the obtained results are new. To obtain these results, the authors overcome those obstacles caused by the deficiency of both the translation invariance and an explicit expression of the quasi-norm of X under consideration via establishing some weighted estimates and new decompositions, which depend on the extrapolation and the exact operator norm of the Hardy–Littlewood maximal operator.

Some characterizations of Lipschitz spaces via commutators of the Hardy-Littlewood maximal operator on slice spaces

Let M be the Hardy-Littlewood maximal operator and b be a locally integrable function. Denote by M_b and [b,M] the maximal commutator and the nonlinear commutator of M with b. In this paper, we give necessary and sufficient conditions for the boundedness of M_b and [b,M] on slice spaces when the function b belongs to Lipschitz spaces, by which a new characterization of non-negative Lipschitz functions is obtained.

Optimality problems in Orlicz spaces

In mathematical modelling, the data and solutions are represented as measurable functions and their quality is oftentimes captured by the membership to a certain function space. One of the core questions for an analysis of a model is the mutual relationship between the data and solution quality. The optimality of the obtained results deserves a special focus. It requires a careful choice of families of function spaces balancing between their expressivity, i.e. the ability to capture fine properties of the model, and their accessibility, i.e. its technical difficulty for practical use. This paper presents a unified and general approach to optimality problems in Orlicz spaces. Orlicz spaces are parametrized by a single convex function and neatly balance the expressivity and accessibility. We prove a general principle that yields an easily verifiable necessary and sufficient condition for the existence or the non-existence of an optimal Orlicz space in various tasks. We demonstrate its use in specific problems, including the continuity of Sobolev embeddings and boundedness of integral operators such as the Hardy–Littlewood maximal operator and the Laplace transform.

Extrapolation in general quasi-Banach function spaces

In this work we prove off-diagonal, limited range, multilinear, vector-valued, and two-weight versions of the Rubio de Francia extrapolation theorem in general quasi-Banach function spaces. We prove mapping properties of the generalization of the Hardy-Littlewood maximal operator to very general bases that includes a method to obtain self-improvement results that are sharp with respect to its operator norm. Furthermore, we prove bounds for the Hardy-Littlewood maximal operator in weighted Lorentz, variable Lebesgue, and Morrey spaces, and recover and extend several extrapolation theorems in the literature. Finally, we provide an application of our results to the Riesz potential and the Bilinear Hilbert transform.

Euclidean Structures and Operator Theory in Banach Spaces

We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators Γ \Gamma on a Hilbert space. Our assumption on Γ \Gamma is expressed in terms of α \alpha -boundedness for a Euclidean structure α \alpha on the underlying Banach space X X . This notion is originally motivated by R \mathcal {R} - or γ \gamma -boundedness of sets of operators, but for example any operator ideal from the Euclidean space ℓ n 2 \ell ^2_n to X X defines such a structure. Therefore, our method is quite flexible. Conversely we show that Γ \Gamma has to be α \alpha -bounded for some Euclidean structure α \alpha to be representable on a Hilbert space. By choosing the Euclidean structure α \alpha accordingly, we get a unified and more general approach to the Kwapień–Maurey factorization theorem and the factorization theory of Maurey, Nikišin and Rubio de Francia. This leads to an improved version of the Banach function space-valued extension theorem of Rubio de Francia and a quantitative proof of the boundedness of the lattice Hardy–Littlewood maximal operator. Furthermore, we use these Euclidean structures to build vector-valued function spaces. These enjoy the nice property that any bounded operator on L 2 L^2 extends to a bounded operator on these vector-valued function spaces, which is in stark contrast to the extension problem for Bochner spaces. With these spaces we define an interpolation method, which has formulations modelled after both the real and the complex interpolation method. Using our representation theorem, we prove a transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We for example extend and refine the known theory based on R \mathcal {R} -boundedness for the joint and operator-valued H ∞ H^\infty -calculus. Moreover, we extend the classical characterization of the boundedness of the H ∞ H^\infty -calculus on Hilbert spaces in terms of B I P BIP , square functions and dilations to the Banach space setting. Furthermore we establish, via the H ∞ H^\infty -calculus, a version of Littlewood–Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our abstract setup allows us to reduce assumptions on the geometry of X X , such as (co)type and UMD. We conclude with some sophisticated counterexamples for sectorial operators, with as a highlight the construction of a sectorial operator of angle 0 0 on a closed subspace of L p L^p for 1 &gt; p &gt; ∞ 1&gt;p&gt;\infty with a bounded H ∞ H^\infty -calculus with optimal angle ω H ∞ ( A ) &gt; 0 \omega _{H^\infty }(A) &gt;0 .

Boundedness properties of maximal operators on Lorentz spaces

We study mapping properties of the centered Hardy-Littlewood maximal operator \(M\) acting on Lorentz spaces. Given \(p \in (1,\infty)\) and a metric measure space \(X = (X, \rho, \mu)\) we let \(\Omega^p_{\rm HL}(X) \subset [0,1]^2\) be the set of all pairs \((\frac{1}{q},\frac{1}{r})\) such that \(M\) is bounded from \(L^{p,q}(X)\) to \(L^{p,r}(X)\). Under mild assumptions on \(\mu\), for each fixed \(p\) all possible shapes of \(\Omega^p_{\rm HL}(X)\) are characterized. Namely, we show that the boundary of \(\Omega^p_{\rm HL}(X)\) either is empty or takes the form \(\{ \delta \} \times [0, \lim_{u \rightarrow \delta} F(u)] \cup \{(u, F(u)) \colon u \in (\delta, 1] \}\), where \(\delta \in [0,1]\) and \(F \colon [\delta, 1] \rightarrow [0,1]\) is concave, nondecreasing, and satisfies \(F(u) \leq u\). Conversely, for each such \(F\) we find \(X\) such that \(M\) is bounded from \(L^{p,q}(X)\) to \(L^{p,r}(X)\) if and only if the point \((\frac{1}{q}, \frac{1}{r})\) lies on or under the graph of \(F\), that is, \(\frac{1}{q} \geq \delta\) and \(\frac{1}{r} \leq F\big(\frac{1}{q}\big)\).

Continuity for the one-dimensional centered Hardy-Littlewood maximal operator at the derivative level

We prove the continuity of the map f↦(Mf)′ from W1,1(R) to L1(R), where M is the centered Hardy-Littlewood maximal operator. This solves a question posed by Carneiro, Madrid and Pierce.