A new characterization of ${\rm RBMO}(\mu )$ by John-Strömberg sharp maximal functions

Guoen Hu; Dachun Yang; Dongyong Yang

Displaying similar documents to “A new characterization of $RBMO (μ)$ by John-Strömberg sharp maximal functions”

The trace inequality and eigenvalue estimates for Schrödinger operators

R. Kerman, Eric T. Sawyer (1986)

Annales de l'institut Fourier

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Suppose $Φ$ is a nonnegative, locally integrable, radial function on $R^{n}$ , which is nonincreasing in $| x |$ . Set $(T f) (x) = \int_{R^{n}} Φ (x - y) f (y) d y$ when $f \geq 0$ and $x \in R^{n}$ . Given $1 < p < \infty$ and $v \geq 0$ , we show there exists $C > 0$ so that $\int_{R^{n}} (T f) (x)^{p} v (x) d x \leq C \int_{R^{n}} f (x)^{p} d x$ for all $f \geq 0$ , if and only if $C^{'} > 0$ exists with $\int_{Q} T (x_{Q} v) (x)^{p^{'}} d x \leq C^{'} \int_{Q} v (x) d x < \infty$ for all dyadic cubes Q, where $p^{'} = p / (p - 1)$ . This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

Some remarks on the dyadic Rademacher maximal function

Mikko Kemppainen (2013)

Colloquium Mathematicae

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Properties of a maximal function for vector-valued martingales were studied by the author in an earlier paper. Restricting here to the dyadic setting, we prove the equivalence between (weighted) $L^{p}$ inequalities and weak type estimates, and discuss an extension to the case of locally finite Borel measures on ℝⁿ. In addition, to compensate for the lack of an $L^{\infty}$ inequality, we derive a suitable BMO estimate. Different dyadic systems in different dimensions are also considered.

The John-Nirenberg type inequality for non-doubling measures

Yoshihiro Sawano, Hitoshi Tanaka (2007)

Studia Mathematica

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X. Tolsa defined a space of BMO type for positive Radon measures satisfying some growth condition on $ℝ^{d}$ . This new BMO space is very suitable for the Calderón-Zygmund theory with non-doubling measures. Especially, the John-Nirenberg type inequality can be recovered. In the present paper we introduce a localized and weighted version of this inequality and, as applications, we obtain some vector-valued inequalities and weighted inequalities for Morrey spaces.

Triebel-Lizorkin spaces with non-doubling measures

Yongsheng Han, Dachun Yang (2004)

Studia Mathematica

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Suppose that μ is a Radon measure on $ℝ^{d}$ , which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces $Ḟ_{p q}^{s} (μ)$ for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function...

On the maximal function for rotation invariant measures in $ℝ^{n}$

Ana Vargas (1994)

Studia Mathematica

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Given a positive measure μ in $ℝ^{n}$ , there is a natural variant of the noncentered Hardy-Littlewood maximal operator $M_{μ} f (x) = s u p_{x \in B} 1 / μ (B) ʃ_{B} | f | d μ$ , where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in $ℝ^{n}$ . We give some necessary and sufficient conditions for $M_{μ}$ to be bounded from $L^{1} (d μ)$ to $L^{1, \infty} (d μ)$ .

Calderón-Zygmund operators acting on generalized Carleson measure spaces

Chin-Cheng Lin, Kunchuan Wang (2012)

Studia Mathematica

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We study Calderón-Zygmund operators acting on generalized Carleson measure spaces $C M O_{r}^{α, q}$ and show a necessary and sufficient condition for their boundedness. The spaces $C M O_{r}^{α, q}$ are a generalization of BMO, and can be regarded as the duals of homogeneous Triebel-Lizorkin spaces as well.

Mean values and associated measures of $δ$ -subharmonic functions

Neil A. Watson (2002)

Mathematica Bohemica

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Let $u$ be a $δ$ -subharmonic function with associated measure $μ$ , and let $v$ be a superharmonic function with associated measure $ν$ , on an open set $E$ . For any closed ball $B (x, r)$ , of centre $x$ and radius $r$ , contained in $E$ , let $ℳ (u, x, r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s, t \to 0$ with $0 < s < t$ of the quotient $(ℳ (u, x, s) - ℳ (u, x, t)) / (ℳ (v, x, s) - ℳ (v, x, t))$ , lie between the upper and lower limits as $r \to 0 +$ of the quotient $μ (B (x, r)) / ν (B (x, r))$ . This enables us to use some well-known measure-theoretic results to prove new variants...

Weighted norm inequalities for maximal singular integrals with nondoubling measures

Guoen Hu, Dachun Yang (2008)

Studia Mathematica

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Let μ be a nonnegative Radon measure on $ℝ^{d}$ which satisfies μ(B(x,r)) ≤ Crⁿ for any $x \in ℝ^{d}$ and r > 0 and some positive constants C and n ∈ (0,d]. In this paper, some weighted norm inequalities with $A_{p}^{ϱ} (μ)$ weights of Muckenhoupt type are obtained for maximal singular integral operators with such a measure μ, via certain weighted estimates with $A_{\infty}^{ϱ} (μ)$ weights of Muckenhoupt type involving the John-Strömberg maximal operator and the John-Strömberg sharp maximal operator, where ϱ,p ∈ [1,∞).

Symmetric and Zygmund measures in several variables

Evgueni Doubtsov, Artur Nicolau (2002)

Annales de l’institut Fourier

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Let $ω : (0, \infty) \to (0, \infty)$ be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure $μ \in ℝ^{n}$ is called $ω$ -Zygmund if there exists a positive constant $C$ such that $| μ (Q_{+}) - μ (Q_{-}) | \leq C ω (ℓ (Q_{+})) | Q_{+} |$ for any pair $Q_{+}, Q_{-} \subset ℝ^{n}$ of adjacent cubes of the same size. Similarly, $μ$ is called an $ω$ - symmetric measure if there exists a positive constant $C$ such that $| μ (Q_{+}) / μ (Q_{-}) - 1 | \leq C ω (ℓ (Q_{+}))$ for any pair $Q_{+}, Q_{-} \subset ℝ^{n}$ of adjacent cubes of the same size, $ℓ (Q_{+}) = ℓ (Q_{-}) < 1$ . We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic...

Non-compact Littlewood-Paley theory for non-doubling measures

Michael Wilson (2007)

Studia Mathematica

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We prove weighted Littlewood-Paley inequalities for linear sums of functions satisfying mild decay, smoothness, and cancelation conditions. We prove these for general “regular” measure spaces, in which the underlying measure is not assumed to satisfy any doubling condition. Our result generalizes an earlier result of the author, proved on $ℝ^{d}$ with Lebesgue measure. Our proof makes essential use of the technique of random dyadic grids, due to Nazarov, Treil, and Volberg.

Displaying similar documents to “A new characterization of RBMO ( μ ) by John-Strömberg sharp maximal functions”

Displaying similar documents to “A new characterization of $RBMO (μ)$ by John-Strömberg sharp maximal functions”