On the exponential integrability of fractional integrals on spaces of homogeneous type

Abstract. Let X 1 , X 2 , . . . , Xq be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain Ω ⊂ R n (n > q). We consider the differential operatorwhere the coefficients a ij (x) are real valued, bounded measurable functions, satisfying the uniform ellipticity condition:for a.e. x ∈ Ω, every ξ ∈ R q , some constant µ. Moreover, we assume that the coefficients a ij belong to the space VMO ("Vanishing Mean Oscillation"), defined with respect to the subelliptic metric induced by the vector fields X 1 , X 2 , . . . , Xq. We prove the following local L p -estimate:for every Ω ⊂⊂ Ω, 1 < p < ∞. We also prove the local Hölder continuity for solutions to Lf = g for any g ∈ L p with p large enough. Finally, we prove L p -estimates for higher order derivatives of f , whenever g and the coefficients a ij are more regular. Introduction and main resultsLet X 0 , X 1 ,. . . , X q be a system of C ∞ real vector fields defined in R n (n ≥ q+1), that is, . . . , q; j = 1, 2, . . . , n), and letRecall that a linear differential operator P with C ∞ coefficients is said to be hypoelliptic in an open set Ω if, whenever the equation P u = f is satisfied, in the distributional sense, in the neighborhood of a point in Ω and f is C ∞ in that neighborhood, then u is also C ∞ in that neighborhood. A famous theorem proved by

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“…Estimates (2.6), (2.7) have been proved, respectively, in [23] and [7]. Theorem 2.14 (Commutators of operators with positive kernels, see [3]).…”

Section: The Constant C In (26)-(28) Has the Form C = C(p S α)mentioning

confidence: 99%

𝐿^{𝑝} estimates for nonvariational hypoelliptic operators with 𝑉𝑀𝑂 coefficients

Bramanti

Brandolini

1999

Trans. Amer. Math. Soc.

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“…It is known that I α is bounded from L p (X) to Lp(X) if 1 < p <p < ∞ and −1/p + α = −1/p (see [9]). This boundedness is well known as the Hardy-Littlewood-Sobolev theorem in R n case.…”

Section: Fractional Integral Operatorsmentioning

confidence: 98%

Singular and fractional integral operators on Campanato spaces with variable growth conditions

Nakai

2009

Rev Mat Complut

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Let X be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we prove boundedness of singular and fractional integral operators on Campanato spaces over X with variable growth conditions. The function spaces contain generalized Lipschitz spaces with variable exponent as special cases. Moreover, by using the function spaces, we can deal with functions which are L p -functions locally on one subset in X, BMO-functions locally on one another subset and Lip α -functions locally on the other one. Our results are new even for R n case.

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“…Other generalizations of these classical results can be found in [6], [7], [9], [14], [17]- [19]. Our paper is organized as follows: In Section 2 we define the spaces and the classes of weights used throughout and we state the main result concerning the characterization of weighted Besov spaces by means of weighted Lipschitz spaces.…”

Section: Introductionmentioning

confidence: 99%