Carleson type estimates for p-harmonic functions and the conformal Martin boundary of John domains in metric measure spaces

Aikawa, Hiroaki; Shanmugalingam, Nageswari

doi:10.1307/mmj/1114021091

Cited by 32 publications

(65 citation statements)

References 21 publications

(13 reference statements)

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“…In Theorem 3.7, we show the main result of this section, namely the variable exponent Carleson estimate. Such estimates play a crucial role in studies of positive p-harmonic functions, see, e.g., Aikawa and Shanmugalingam [8], also Garofalo [29] for an application of Carleson estimates for a class of parabolic equations. According to our best knowledge, Carleson estimates in the setting of equations with nonstandard growth have not been known so far in the literature.…”

Section: Oscillation and Carleson Estimates For P(·)-harmonic Functionsmentioning

confidence: 99%

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

Adamowicz

Lundström

2015

Annali di Matematica

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We investigate various boundary decay estimates for p(·)-harmonic functions. For domains in R n , n ≥ 2 satisfying the ball condition (C 1,1 -domains), we show the boundary Harnack inequality for p(·)-harmonic functions under the assumption that the variable exponent p is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson-type estimate for p(·)-harmonic functions in NTA domains in R n and provide lower and upper growth estimates and a doubling property for a p(·)-harmonic measure.

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Section: Oscillation and Carleson Estimates For P(·)-harmonic Functionsmentioning

confidence: 99%

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

Adamowicz

Lundström

2015

Annali di Matematica

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“…Our argument is based on a recurrence type scheme often attributed to Carleson -Domar, see [7], [9], in the complex world, and to Caffarelli et. al., see [6], in the PDE world (see also [1] for references). Given the shifted boxQ(a) we let b j,1 = x j + iδ * Im a, j = 1, 2, and note that b j,1 , j = 1, 2, are points on the vertical sides ofQ(a).…”

Section: Proof Of (410)mentioning

confidence: 99%

“…There is only one caveat. Namely, the path λ 1,1 is required to be contained in the half-plane { Re z < Re b 1,1 }, i.e., to stay entirely to the left of b 1,1 . This extra caveat is easily achieved in view of Lemma 4.7. λ 2,1 with endpoints, b 2,1 , x 2,1 is constructed similarly, to lie in { Re z > Re b 2,1 } (see Picture 5.1).…”

Section: Proof Of (410)mentioning

confidence: 99%

p Harmonic Measure in Simply Connected Domains

Lewis¹,

Nyström²,

Poggi‐Corradini³

2011

Ann. inst. Fourier

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Let Ω be a bounded simply connected domain in the complex plane, C. Let N be a neighborhood of ∂Ω, let p be fixed, 1 < p < ∞, and let û be a positive weak solution to the p Laplace equation in Ω ∩ N. Assume that û has zero boundary values on ∂Ω in the Sobolev sense and extend û to N \ Ω by putting û ≡ 0 on N \ Ω. Then there exists a positive finite Borel measure μ on C with support contained in ∂Ω and such that |∇û| p−2 ∇û, ∇φ dA = − φ dμ whenever φ ∈ C ∞ 0 (N ). If p = 2 and if û is the Green function for Ω with pole at x ∈ Ω \ N then the measure μ coincides with harmonic measure at x, ω = ω x , associated to the Laplace equation. In this paper we continue the studies in [BL05], [L06] by establishing new results, in simply connected domains, concerning the Hausdorff dimension of the support of the measure μ. In particular, we prove results, for 1 < p < ∞, p = 2, reminiscent of the famous result of Makarov [Mak85] concerning the Hausdorff dimension of the support of harmonic measure in simply connected domains.

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“…1 We say that a bounded domain ⊂ H n satisfies the uniform outer (interior) ball condition if there exists R 0 > 0 such that for every g 0 ∈ ∂ and every…”

Section: Nta and Adp Domainsmentioning

confidence: 99%

“…In the framework of metrics associated with a system of smooth vector fields satisfying the finite rank condition, a detailed study of such domains was developed in [8], and we refer the reader to that source for all relevant results. One should also consult the paper [1] for further generalizations. Here, we will focus on the special yet basic setting of H n with its gauge metric.…”

Section: Definition 52 We Say That a Bounded Domainmentioning

confidence: 99%

Boundary behavior of p-harmonic functions in the Heisenberg group

Garofalo

Phuc

2010

Math. Ann.

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In this paper we study the boundary behavior of nonnegative p-harmonic functions in a bounded domain in the Heisenberg group H n . Under suitable geometric assumptions on the ground domain ⊂ H n our main contributions can be summarized as follows: (1) In Theorem 1.1 we obtain an estimate from above stating that any such function should vanish at most linearly like the sub-Riemannian distance from the boundary:where A r (g 0 ) ∈ is a non-tangential point relative to g 0 ∈ ∂ . (2) In Theorem 1.2 we establish an estimate from below which states that, away from the characteristic set of , the order of vanishing is exactly linear, i.e.:u(g) u(A r (g 0 )) ≥ C d(g, ∂ ) r .

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Carleson type estimates for p-harmonic functions and the conformal Martin boundary of John domains in metric measure spaces

Cited by 32 publications

References 21 publications

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

p Harmonic Measure in Simply Connected Domains

Boundary behavior of p-harmonic functions in the Heisenberg group

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