scholarly journals VARIATION ON A THEME BY KISELEV AND NAZAROV: HÖLDER ESTIMATES FOR NONLOCAL TRANSPORT-DIFFUSION, ALONG A NON-DIVERGENCE-FREE BMO FIELD

2021 ◽  
pp. 1-25
Author(s):  
Ioann Vasilyev ◽  
François Vigneron
Keyword(s):  
Fractional Diffusion ◽  
Bounded Mean Oscillation ◽  
Negative Part ◽  
Hölder Continuous ◽  
Diffusion Operator ◽  
Mean Oscillation ◽  
Regularity Estimates ◽  
Transport Diffusion ◽  
Dual Evolution ◽  
Hölder Estimates

Abstract We prove uniform Hölder regularity estimates for a transport-diffusion equation with a fractional diffusion operator and a general advection field in of bounded mean oscillation, as long as the order of the diffusion dominates the transport term at small scales; our only requirement is the smallness of the negative part of the divergence in some critical Lebesgue space. In comparison to a celebrated result by Silvestre, our advection field does not need to be bounded. A similar result can be obtained in the supercritical case if the advection field is Hölder continuous. Our proof is inspired by Kiselev and Nazarov and is based on the dual evolution technique. The idea is to propagate an atom property (i.e., localisation and integrability in Lebesgue spaces) under the dual conservation law, when it is coupled with the fractional diffusion operator.

scholarly journals Variable Hardy Spaces Associated with Operators Satisfying Davies–Gaffney Estimates

2018 ◽  
Vol 61 (3) ◽  
pp. 759-810 ◽  
Author(s):  
Dachun Yang ◽  
Junqiang Zhang ◽  
Ciqiang Zhuo
Keyword(s):  
Hardy Space ◽  
Adjoint Operator ◽  
Riesz Transform ◽  
Bounded Mean Oscillation ◽  
Hölder Continuous ◽  
Measurable Coefficients ◽  
Mean Oscillation ◽  
Function Characterization ◽  
Bounded Holomorphic

AbstractLetLbe a one-to-one operator of type ω inL2(ℝn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. Letp(·): ℝn→(0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space$H_L^{p(\cdot )} ({\open R}^n)$associated withL. By means of variable tent spaces, the authors establish the molecular characterization of$H_L^{p(\cdot )} ({\open R}^n)$. Then the authors show that the dual space of$H_L^{p(\cdot )} ({\open R}^n)$is the bounded mean oscillation (BMO)-type space${\rm BM}{\rm O}_{p(\cdot ),{\kern 1pt} L^ * }({\open R}^n)$, whereL* denotes the adjoint operator ofL. In particular, whenLis the second-order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of$H_L^{p(\cdot )} ({\open R}^n)$and show that the fractional integralL−αfor α∈(0, (1/2)] is bounded from$H_L^{p(\cdot )} ({\open R}^n)$to$H_L^{q(\cdot )} ({\open R}^n)$with (1/p(·))−(1/q(·))=2α/n, and the Riesz transform ∇L−1/2is bounded from$H_L^{p(\cdot )} ({\open R}^n)$to the variable Hardy spaceHp(·)(ℝn).

scholarly journals Some estimates for the commutators of multilinear maximal function on Morrey-type space

2021 ◽  
Vol 19 (1) ◽  
pp. 515-530
Author(s):  
Xiao Yu ◽  
Pu Zhang ◽  
Hongliang Li
Keyword(s):  
Maximal Function ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Bounded Mean Oscillation ◽  
Generalized Morrey Spaces ◽  
Type Space ◽  
Bmo Function ◽  
Mean Oscillation ◽  
Multilinear Maximal Function ◽  
Equivalent Conditions

Abstract In this paper, we study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.

scholarly journals EXPLICIT INTERPOLATION BOUNDS BETWEEN HARDY SPACE AND

2013 ◽  
Vol 95 (2) ◽  
pp. 158-168
Author(s):  
H.-Q. BUI ◽  
R. S. LAUGESEN
Keyword(s):  
Growth Rate ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Mean Oscillation ◽  
Complex Interpolation ◽  
Bounded Linear ◽  
Exponential Bound ◽  
Mean Oscillation ◽  
Good Lambda Inequality ◽  
Interpolation Result

AbstractEvery bounded linear operator that maps ${H}^{1} $ to ${L}^{1} $ and ${L}^{2} $ to ${L}^{2} $ is bounded from ${L}^{p} $ to ${L}^{p} $ for each $p\in (1, 2)$, by a famous interpolation result of Fefferman and Stein. We prove ${L}^{p} $-norm bounds that grow like $O(1/ (p- 1))$ as $p\downarrow 1$. This growth rate is optimal, and improves significantly on the previously known exponential bound $O({2}^{1/ (p- 1)} )$. For $p\in (2, \infty )$, we prove explicit ${L}^{p} $ estimates on each bounded linear operator mapping ${L}^{\infty } $ to bounded mean oscillation ($\mathit{BMO}$) and ${L}^{2} $ to ${L}^{2} $. This $\mathit{BMO}$ interpolation result implies the ${H}^{1} $ result above, by duality. In addition, we obtain stronger results by working with dyadic ${H}^{1} $ and dyadic $\mathit{BMO}$. The proofs proceed by complex interpolation, after we develop an optimal dyadic ‘good lambda’ inequality for the dyadic $\sharp $-maximal operator.

On Analytic Functions of Bergman BMO in the Ball

1999 ◽  
Vol 42 (1) ◽  
pp. 97-103 ◽  
Author(s):  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Composition Operator ◽  
Analytic Functions ◽  
Bloch Space ◽  
Bounded Mean Oscillation ◽  
Open Unit ◽  
Bergman Metric ◽  
Open Unit Ball ◽  
Mean Oscillation ◽  
Volume Measure

AbstractLet B = Bn be the open unit ball of Cn with volume measure v, U = B1 and B be the Bloch space on , 1 ≤ α < 1, is defined as the set of holomorphic f : B → C for whichif 0 < α < 1 and , the Hardy space. Our objective of this note is to characterize, in terms of the Bergman distance, those holomorphic f : B → U for which the composition operator defined by , is bounded. Our result has a corollary that characterize the set of analytic functions of bounded mean oscillation with respect to the Bergman metric.

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