Centered Hardy-Littlewood Maximal Operator Research Articles

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.

Weak-Type Lower Bounds for High-Dimensional Hardy–Littlewood Maximal Operators on Certain Measures via Averaging Operators

Consider mathbb {R}^d with the euclidean distance, and let 0<alpha <1. We study the behavior of the averaging operators given by the radial density dmu (x)=|x|^{-alpha d}dx. When 1le p<infty is such that (1-2alpha )p<1-alpha , we show that the weak (p, p) bounds grow exponentially with the dimension d. As a consequence, the corresponding results follow for the centered Hardy–Littlewood maximal operator. The lower bounds obtained here are new for averaging operators, and when 0<alpha le 1/2 and p>1, they are also new for the maximal operator.

Sharp inequalities for maximal operators on finite graphs, II

Let MG be the centered Hardy-Littlewood maximal operator on a finite graph G. We find limp→∞‖MG‖pp when G is the start graph (Sn) and the complete graph (Kn), and we fully describe ‖MSn‖p and the corresponding extremizers for p∈(1,2). We prove that limp→∞‖MSn‖pp=1+n2 when n≥25. Also, we compute the best constant CSn,2 such that for every f:V→R we have Var2MSnf≤CSn,2Var2f. We prove that CSn,2=(n2−n−1)1/2n for all n≥3 and characterize the extremizers. Moreover, when M is the Hardy-Littlewood maximal operator on Z, we compute the best constant Cp such that VarpMf≤Cp‖f‖p for p∈(12,1) and we describe the extremizers.

Maximal operators on Lorentz spaces in non-doubling setting

We study mapping properties of the centered Hardy–Littlewood maximal operator \(\mathcal {M}\) acting on Lorentz spaces \(L^{p,q}({\mathfrak {X}})\) in the context of certain non-doubling metric measure spaces \({\mathfrak {X}}\). The special class of spaces for which these properties are very peculiar is introduced and many examples are given. In particular, for each \(p_0, q_0, r_0 \in (1, \infty )\) with \(r_0 \ge q_0\) we construct a space \({\mathfrak {X}}\) for which the associated operator \(\mathcal {M}\) is bounded from \(L^{p_0,q_0}({\mathfrak {X}})\) to \(L^{p_0,r}({\mathfrak {X}})\) if and only if \(r \ge r_0\).

Lower bounds for the centered Hardy-Littlewood maximal operator on the real line

Let 1<p<∞. We prove that there exists an εp>0 such that for each f∈Lp(R), the centered Hardy-Littlewood maximal operator M on R satisfies the lower bound ‖Mf‖Lp(R)≥(1+εp)‖f‖Lp(R).

Equivalence of operator norm for Hardy-Littlewood maximal operators and their truncated operators on Morrey spaces

We will prove that for 1 < p < ∞ and 0 < λ < n, the central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mγc equals that of the centered Hardy-Littlewood maximal operator for all 0 < γ < +∞. When p = 1 and 0 < λ < n, it turns out that the weak central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mγc equals that of the centered Hardy-Littlewood maximal operator for all 0 < γ < +∞. Moreover, the same results are true for the truncated uncentered Hardy-Littlewood maximal operator. Our work extends the previous results of Lebesgue spaces to Morrey spaces.

Lower Bounds and Fixed Points for the Centered Hardy–Littlewood Maximal Operator

For all \(p>1\) and all centrally symmetric convex bodies, \(K\subset {\mathbb {R}}^d\) defined Mf as the centered maximal function associated to K. We show that when \(d=1\) or \(d=2\), we have \(||Mf||_p\ge (1+\epsilon (p,K))||f||_p\). For \(d\ge 3\), let \(q_0(K)\) be the infimum value of p for which M has a fixed point. We show that for generic shapes K, we have \(q_0(K)>q_0(B(0,1))\).

Centered Hardy–Littlewood maximal operator on the real line: Lower bounds

For 1<p<∞ and M the centered Hardy–Littlewood maximal operator on R, we consider whether there is some ε=ε(p)>0 such that ||Mf||p≥(1+ε)||f||p. We prove this for 1<p<2. For 2≤p<∞, we prove the inequality for indicator functions and for unimodal functions.

Geometric properties of infinite graphs and the Hardy–Littlewood maximal operator

We study different geometric properties on infinite graphs, related to the weak-type boundedness of the Hardy–Littlewood maximal averaging operator. In particular, we analyze the connections between the doubling condition, having finite dilation and overlapping indices, uniformly bounded degree, the equidistant comparison property and the weak-type boundedness of the centered Hardy–Littlewood maximal operator. Several non-trivial examples of infinite graphs are given to illustrate the differences among these properties.

A note on Hardy-Littlewood maximal operators

In this paper, we will prove that, for $1< p<\infty$ , the $L^{p}$ norm of the truncated centered Hardy-Littlewood maximal operator $M^{c}_{\gamma}$ equals the norm of the centered Hardy-Littlewood maximal operator for all $0<\gamma<\infty$ . When $p=1$ , we also find that the weak $(1,1)$ norm of the truncated centered Hardy-Littlewood maximal operator $M^{c}_{\gamma}$ equals the weak $(1,1)$ norm of the centered Hardy-Littlewood maximal operator for $0<\gamma<\infty$ . Moreover, the same is true for the truncated uncentered Hardy-Littlewood maximal operator. Finally, we investigate the properties of the iterated Hardy-Littlewood maximal function.

On high dimensional maximal operators

In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy-Littlewood maximal operator associated to certain families of doubling, radial decreasing measures, and acting on radial functions. In fact, we precisely determine when the weak type $(1,1)$ bounds are uniform in the dimension.

The best constant for the centered maximal operator on radial decreasing functions

We show that the lowest constant appearing in the weak type (1,1) inequality satisfied by the centered Hardy-Littlewood maximal operator on radial integrable functions is 1.

Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures

As shown in Aldaz (Bull. Lond. Math. Soc. 39:203–208, 2007), the lowest constants appearing in the weak type (1, 1) inequalities satisfied by the centered Hardy-Littlewood maximal operator associated to certain finite radial measures, grow exponentially fast with the dimension. Here we extend this result to a wider class of radial measures and to some values of p > 1. Furthermore, we improve the previously known bounds for p = 1. Roughly speaking, whenever \({p\in (1, 1.03]}\), if μ is defined by a radial, radially decreasing density satisfying some mild growth conditions, then the best constants c p,d,μ in the weak type (p, p) inequalities satisfy c p,d,μ ≥ 1.005d for all d sufficiently large. We also show that exponential increase of the best constants occurs for certain families of doubling measures, and for arbitrarily high values of p.

Singular Measures and Convolution Operators

We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type (1, 1) inequalities. As an application, we prove that the best constants for the centered Hardy–Littlewood maximal operator associated with parallelotopes do not decrease with the dimension.

Outer measures and weak type (1,1) estimates of Hardy-Littlewood maximal operators

We will introduce the times modified centered and uncentered Hardy-Littlewood maximal operators on nonhomogeneous spaces for. We will prove that the times modified centered Hardy-Littlewood maximal operator is weak type bounded with constant when if the Radon measure of the space has "continuity" in some sense. In the proof, we will use the outer measure associated with the Radon measure. We will also prove other results of Hardy-Littlewood maximal operators on homogeneous spaces and on the real line by using outer measures.

The best constant for the centered Hardy–Littlewood maximal inequality

We find the exact value of the best possible constant C for the weak-type (1, 1) inequality for the one-dimensional centered Hardy-Littlewood maximal operator. We prove that C is the largest root of the quadratic equation 12C 2 − 22C +5 =0 thus obtaining C =1 .5675208 ... . This is the first time the best constant for one of the fundamental inequalities satisfied by a centered maximal operator is precisely evaluated.

On the centered Hardy-Littlewood maximal operator

We will study the centered Hardy-Littlewood maximal operator acting on positive linear combinations of Dirac deltas. We will use this to obtain improvements in both the lower and upper bounds or the best constant C in the L 1 → weak L1 inequality for this operator. In fact we will show that 11+√61/12 = 1.5675208... ≤ C ≤ 5/3 = 1.66....

A remark on the centered n-dimensional Hardy-Littlewood maximal function

We study the behaviour of the n-dimensional centered Hardy-Littlewood maximal operator associated to the family of cubes with sides parallel to the axes, improving the previously known lower bounds for the best constants c n that appear in the weak type (1,1) inequalities.

Weighted norm inequalities for homogeneous singular integrals

We prove weighted norm inequalities for homogeneous singular integrals when only a size condition is assumed on the restriction of the kernel to the unit sphere. The same results hold for the operator obtained by modifying the centered Hardy-Littlewood maximal operator over balls with a degree zero homogeneous function and also for the maximal singular integral.